text: 18 JOHN A. BALDWIN AND STEVEN SIVEK − − and (2) a contact closure of a sutured manifold obtained from ( M, Γ) via contact 1-handle − − attachments is also a contact closure of ( M, Γ), as explained in [3]. We now describe a Heegaard triple diagram which encodes closures of the sutured manifolds − speciﬁed by the diagrams in (6) and (7) as well as the cobordism W . Let T be a compact, ≥ oriented, connected surface with boundary and g(T ) 2, such that ∼ ∼ π (∂T ) = π (∂Σ) = π (∂S). 0 0 0 Let R be the closed, oriented surface formed by gluing T to S by a diﬀeomorphism of their S boundaries. Let R be the surface formed by gluing T to Σ in a similar manner. Let D and Σ a D be two disjoint disks in T . Let R and R denote the complements of these disks in R b S Σ S and R , Σ (cid:114) (cid:114) R = R D D , S S a b (cid:114) (cid:114) R = R D D , Σ Σ a b and let ∪ Σ = R R S Σ be the closed surface formed by gluing these complements together by the identity maps on ∂D and ∂D . In other words, Σ is obtained by connecting R and R via two tubes. We will a b ⊂ S Σ think of the α , β , γ curves above as lying in R Σ. See the middle diagram in Figure 4 i i i ∪ Σ for an illustration of Σ = R R in the case that S is an annulus, h is a right-handed Dehn { S Σ} twist around the core, c , . . . , c consists of just the cocore, and T is a genus 2 surface with 1 n 2 boundary components. Suppose R has genus g. Then R and Σ have genera n + g and n + 2g + 1, respectively. S Σ Let { } { } (8) a , . . . , a and b , . . . , b n+1 n+2g n+1 n+2g be the sets of pairwise disjoint, properly embedded arcs in R shown in Figure 5. In particular, S ◦ the two sets of arcs are related by rotation of the surface by 180 around the axis shown in the ﬁgure. We index the arcs so that each a intersects b in exactly one place and is disjoint (cid:54) i i from b for j = i. Note that the a have endpoints on ∂D and cut R into an annulus with j i b S ∂D as a boundary component. Likewise, the b have endpoints on ∂D and cut R into an a j a S annulus with ∂D as a boundary component. b Observe that the regular neighborhoods ∪ ⊂ U = N(α β ) Σ, i i i ∪ ⊂ V = N(β γ ) Σ i i i ⊂ are once-punctured tori. In particular, there exist disks D , . . . , D S such that 1 n ∼ ∼ (cid:114) ∪ (cid:114) ∪ (cid:114) ∪ (9) R ( n D ) = R ( n U ) = R ( n V ). S i=1 i Σ i=1 i Σ i=1 i We may assume that these D , as disks in R , are disjoint from the arcs in (8). Let i S (cid:114) ∪ → (cid:114) ∪ ϕ : R ( n D ) R ( n U ), α S i=1 i Σ i=1 i (cid:114) ∪ → (cid:114) ∪ ϕ : R ( n D ) R ( n V ) γ S i=1 i Σ i=1 i ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JOHN

text: A.

text: BALDWIN

text: AND

text: STEVEN

text: SIVEK

text: and

text: a

text: contact

text: closure

text: of

text: a

text: sutured

text: manifold

text: obtained

text: from

text: M,

text: Γ)

text: via

text: contact

text: attachments

text: is

text: also

text: a

text: contact

text: closure

text: of

text: M,

text: Γ),

text: as

text: explained

text: in

text: We

text: now

text: describe

text: a

text: Heegaard

text: triple

text: diagram

text: which

text: encodes

text: closures

text: of

text: the

text: sutured

text: manifolds

text: speciﬁed

text: by

text: the

text: diagrams

text: in

text: and

text: as

text: well

text: as

text: the

text: cobordism

text: W

text: Let

text: T

text: be

text: a

text: compact,

text: oriented,

text: connected

text: surface

text: with

text: boundary

text: and

text: g(T

text: such

text: that

text: π

text: (∂T

text: π

text: (∂Σ)

text: π

text: (∂S).

text: Let

text: R

text: be

text: the

text: closed,

text: oriented

text: surface

text: formed

text: by

text: gluing

text: T

text: to

text: S

text: by

text: a

text: diﬀeomorphism

text: of

text: their

text: S

text: boundaries.

text: Let

text: R

text: be

text: the

text: surface

text: formed

text: by

text: gluing

text: T

text: to

text: Σ

text: in

text: a

text: similar

text: manner.

text: Let

text: D

text: and

text: Σ

text: a

text: D

text: be

text: two

text: disjoint

text: disks

text: in

text: T

text: Let

text: R

text: and

text: R

text: denote

text: the

text: complements

text: of

text: these

text: disks

text: in

text: R

text: b

text: S

text: Σ

text: S

text: and

text: R

text: Σ

text: R

text: R

text: D

text: D

text: S

text: S

text: a

text: b

text: R

text: R

text: D

text: D

text: Σ

text: Σ

text: a

text: b

text: and

text: let

text: Σ

text: R

text: R

text: S

text: Σ

text: be

text: the

text: closed

text: surface

text: formed

text: by

text: gluing

text: these

text: complements

text: together

text: by

text: the

text: identity

text: maps

text: on

text: ∂D

text: and

text: ∂D

text: In

text: other

text: words,

text: Σ

text: is

text: obtained

text: by

text: connecting

text: R

text: and

text: R

text: via

text: two

text: tubes.

text: We

text: will

text: a

text: b

text: S

text: Σ

text: think

text: of

text: the

text: α

text: β

text: γ

text: curves

text: above

text: as

text: lying

text: in

text: R

text: Σ.

text: See

text: the

text: middle

text: diagram

text: in

text: Figure

text: i

text: i

text: i

text: Σ

text: for

text: an

text: illustration

text: of

text: Σ

text: R

text: R

text: in

text: the

text: case

text: that

text: S

text: is

text: an

text: annulus,

text: h

text: is

text: a

text: right-handed

text: Dehn

text: S

text: Σ}

text: twist

text: around

text: the

text: core,

text: c

text: c

text: consists

text: of

text: just

text: the

text: cocore,

text: and

text: T

text: is

text: a

text: genus

text: surface

text: with

text: n

text: boundary

text: components.

text: Suppose

text: R

text: has

text: genus

text: g.

text: Then

text: R

text: and

text: Σ

text: have

text: genera

text: n

text: g

text: and

text: n

text: respectively.

text: S

text: Σ

text: Let

text: a

text: a

text: and

text: b

text: b

text: be

text: the

text: sets

text: of

text: pairwise

text: disjoint,

text: properly

text: embedded

text: arcs

text: in

text: R

text: shown

text: in

text: Figure

text: In

text: particular,

text: S

text: the

text: two

text: sets

text: of

text: arcs

text: are

text: related

text: by

text: rotation

text: of

text: the

text: surface

text: by

text: around

text: the

text: axis

text: shown

text: in

text: the

text: ﬁgure.

text: We

text: index

text: the

text: arcs

text: so

text: that

text: each

text: a

text: intersects

text: b

text: in

text: exactly

text: one

text: place

text: and

text: is

text: disjoint

text: i

text: i

text: from

text: b

text: for

text: j

text: i.

text: Note

text: that

text: the

text: a

text: have

text: endpoints

text: on

text: ∂D

text: and

text: cut

text: R

text: into

text: an

text: annulus

text: with

text: j

text: i

text: b

text: S

text: ∂D

text: as

text: a

text: boundary

text: component.

text: Likewise,

text: the

text: b

text: have

text: endpoints

text: on

text: ∂D

text: and

text: cut

text: R

text: into

text: an

text: a

text: j

text: a

text: S

text: annulus

text: with

text: ∂D

text: as

text: a

text: boundary

text: component.

text: b

text: Observe

text: that

text: the

text: regular

text: neighborhoods

text: U

text: N(α

text: β

text: Σ,

text: i

text: i

text: i

text: V

text: N(β

text: γ

text: Σ

text: i

text: i

text: i

text: are

text: once-punctured

text: tori.

text: In

text: particular,

text: there

text: exist

text: disks

text: D

text: D

text: S

text: such

text: that

text: n

text: R

text: n

text: D

text: R

text: n

text: U

text: R

text: n

text: V

text: S

text: i

text: Σ

text: i

text: Σ

text: i

text: We

text: may

text: assume

text: that

text: these

text: D

text: as

text: disks

text: in

text: R

text: are

text: disjoint

text: from

text: the

text: arcs

text: in

text: Let

text: i

text: S

text: ϕ

text: R

text: n

text: D

text: R

text: n

text: U

text: α

text: S

text: i

text: Σ

text: i

text: ϕ

text: R

text: n

text: D

text: R

text: n

text: V

text: γ

text: S

text: i

text: Σ

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON THE EQUIVALENCE OF CONTACT INVARIANTS IN SUTURED FLOER HOMOLOGY THEORIES 3 Theorem 1.2 (Main Theorem). There is an isomorphism of Λ-modules − − ⊗ → − − SF H( M, Γ) Λ SHM( M, Γ), ⊗ sending c (ξ) 1 to c (ξ). HF HM In order to deﬁne c (ξ)—in both the closed and with boundary settings—one ﬁrst chooses HF an open book compatible with ξ. The existing proofs that c (ξ) does not depend on this HF choice rely on Giroux’s correspondence—in particular, on its assertion that any two open books compatible with ξ are related by a sequence of positive stabilizations. By contrast, our construction of c (ξ) is logically independent of this assertion. One immediate application HM of Theorem 1.2 is therefore a proof of the following well-deﬁnedness result from [19] that is logically independent of Giroux’s correspondence. Theorem 1.3 (Honda–Kazez–Mati´c). For any sutured contact manifold (M, Γ, ξ), the vanish- (cid:3) ing or non-vanishing of c (ξ) does not depend on the choices made in its construction. HF The main topological application of Theorem 1.2 in this paper is to the study of Lagrangian ⊂ concordance initiated by Chantraine in [8]. Recall that for Legendrian knots K , K (Y, ξ), ≺ 1 2 K is Lagrangian concordant to K —written K K —if there is an embedded Lagrangian 1 2 1 2 cylinder C in the symplectization of (Y, ξ) such that ∩ −∞ − × −∞ − × C (( , T ) Y ) = ( , T ) K , 1 ∩ ∞ × ∞ × C ((T, ) Y ) = (T, ) K 2 for some T > 0. Two Legendrian knots related by Lagrangian concordance must have the same classical invariants (Thurston-Bennequin and rotation numbers) [8]. A challenging problem, therefore, which has attracted a lot of recent attention, is to ﬁnd tools for deciding whether two knots with the same classical invariants are Lagrangian concordant. Note that this is more diﬃcult than the already formidable task of deciding whether two smoothly isotopic knots with the same classical invariants are Legendrian isotopic. ⊂ In [5], we deﬁned a Legendrian invariant which assigns to a Legendrian K (Y, ξ) a class L ∈ − (K) KHM( Y, K) HM in monopole knot Floer homology with local coeﬃcients. It is deﬁned in terms of a certain contact structure ξ on the sutured knot complement Y (K) with two meridional sutures, K L ∈ − − (1) (K) := c (ξ ) SHM( Y (K)) =: KHM( Y, K). HM HM K Furthermore, we used the functoriality of c under exact symplectic cobordism (a fea- HM ture whose analogue in Heegaard Floer homology has not been established independently ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

label: paragraph

text: THE

text: EQUIVALENCE

text: OF

text: CONTACT

text: INVARIANTS

text: IN

text: SUTURED

text: FLOER

text: HOMOLOGY

text: THEORIES

text: Theorem

text: (Main

text: Theorem).

text: There

text: is

text: an

text: isomorphism

text: of

text: Λ-modules

text: SF

text: H(

text: M,

text: Γ)

text: Λ

text: SHM(

text: M,

text: Γ),

text: sending

text: c

text: (ξ)

text: to

text: c

text: (ξ).

text: HF

text: HM

text: In

text: order

text: to

text: deﬁne

text: c

text: (ξ)—in

text: both

text: the

text: closed

text: and

text: with

text: boundary

text: settings—one

text: ﬁrst

text: chooses

text: HF

text: an

text: open

text: book

text: compatible

text: with

text: ξ.

text: The

text: existing

text: proofs

text: that

text: c

text: (ξ)

text: does

text: not

text: depend

text: on

text: this

text: HF

text: choice

text: rely

text: on

text: Giroux’s

text: correspondence—in

text: particular,

text: on

text: its

text: assertion

text: that

text: any

text: two

text: open

text: books

text: compatible

text: with

text: ξ

text: are

text: related

text: by

text: a

text: sequence

text: of

text: positive

text: stabilizations.

text: By

text: contrast,

text: our

text: construction

text: of

text: c

text: (ξ)

text: is

text: logically

text: independent

text: of

text: this

text: assertion.

text: One

text: immediate

text: application

text: HM

text: of

text: Theorem

text: is

text: therefore

text: a

text: proof

text: of

text: the

text: following

text: well-deﬁnedness

text: result

text: from

text: that

text: is

text: logically

text: independent

text: of

text: Giroux’s

text: correspondence.

text: Theorem

text: (Honda–Kazez–Mati´c).

text: For

text: any

text: sutured

text: contact

text: manifold

text: (M,

text: Γ,

text: ξ),

text: the

text: vanish-

text: ing

text: or

text: non-vanishing

text: of

text: c

text: (ξ)

text: does

text: not

text: depend

text: on

text: the

text: choices

text: made

text: in

text: its

text: construction.

text: HF

text: The

text: main

text: topological

text: application

text: of

text: Theorem

text: in

text: this

text: paper

text: is

text: to

text: the

text: study

text: of

text: Lagrangian

text: concordance

text: initiated

text: by

text: Chantraine

text: in

text: Recall

text: that

text: for

text: Legendrian

text: knots

text: K

text: K

text: (Y,

text: ξ),

text: K

text: is

text: Lagrangian

text: concordant

text: to

text: K

text: —written

text: K

text: K

text: —if

text: there

text: is

text: an

text: embedded

text: Lagrangian

text: cylinder

text: C

text: in

text: the

text: symplectization

text: of

text: (Y,

text: ξ)

text: such

text: that

text: C

text: T

text: Y

text: T

text: K

text: C

text: ((T,

text: Y

text: (T,

text: K

text: for

text: some

text: T

text: Two

text: Legendrian

text: knots

text: related

text: by

text: Lagrangian

text: concordance

text: must

text: have

text: the

text: same

text: classical

text: invariants

text: (Thurston-Bennequin

text: and

text: rotation

text: numbers)

text: A

text: challenging

text: problem,

text: therefore,

text: which

text: has

text: attracted

text: a

text: lot

text: of

text: recent

text: attention,

text: is

text: to

text: ﬁnd

text: tools

text: for

text: deciding

text: whether

text: two

text: knots

text: with

text: the

text: same

text: classical

text: invariants

text: are

text: Lagrangian

text: concordant.

text: Note

text: that

text: this

text: is

text: more

text: diﬃcult

text: than

text: the

text: already

text: formidable

text: task

text: of

text: deciding

text: whether

text: two

text: smoothly

text: isotopic

text: knots

text: with

text: the

text: same

text: classical

text: invariants

text: are

text: Legendrian

text: isotopic.

text: In

text: we

text: deﬁned

text: a

text: Legendrian

text: invariant

text: which

text: assigns

text: to

text: a

text: Legendrian

text: K

text: (Y,

text: ξ)

text: a

text: class

text: L

text: (K)

text: KHM(

text: Y,

text: K)

text: HM

text: in

text: monopole

text: knot

text: Floer

text: homology

text: with

text: local

text: coeﬃcients.

text: It

text: is

text: deﬁned

text: in

text: terms

text: of

text: a

text: certain

text: contact

text: structure

text: ξ

text: on

text: the

text: sutured

text: knot

text: complement

text: Y

text: (K)

text: with

text: two

text: meridional

text: sutures,

text: K

text: L

text: (K)

text: c

text: (ξ

text: SHM(

text: Y

text: (K))

text: KHM(

text: Y,

text: K).

text: HM

text: HM

text: K

text: Furthermore,

text: we

text: used

text: the

text: functoriality

text: of

text: c

text: under

text: exact

text: symplectic

text: cobordism

text: (a

text: fea-

text: HM

text: ture

text: whose

text: analogue

text: in

text: Heegaard

text: Floer

text: homology

text: has

text: not

text: been

text: established

text: independently

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON THE EQUIVALENCE OF CONTACT INVARIANTS IN SUTURED FLOER HOMOLOGY THEORIES 43 Let A and B be the points in R and R shown in Figure 9. Given generators S Σ ∪ { } ∪ ∪ { (cid:48)} ∪ | [x s x , i] and [y s x , j] of CF +(γ, α R ; Γ ) γ γ γ η (cid:48) ∈ { } representing the same Spinc structure, for x, y as in (32) and s, s u , v , choose a Whitney γ γ disk ∈ ∪ { } ∪ ∪ { (cid:48)} ∪ φ π (x s x , y s x ) 2 γ γ − with n (φ) = i j, and deﬁne the relative grading z F ∪ { } ∪ ∪ { (cid:48)} ∪ − ([x s x , i], [y s x , j]) = 2n (φ) n (φ). γ γ A B − This relative grading is well-deﬁned since 2n n is zero on the periodic domains in (33). A B For each Spinc structure, we choose some lift of this relative grading to an absolute grading, F which we also denote by . Lemma 3.11. Suﬃciently large g and suﬃcient winding guarantees that the absolute grading F deﬁnes a ﬁltration. Proof. Fix x, y as in (32). We must show that for suﬃciently large g and suﬃcient winding, the following is true: if the coeﬃcient (cid:104) ∪ { } ∪ ∪ { (cid:48)} ∪ (cid:105) (37) [ ∂([x s x , i]), y s x , j] γ γ is nonzero, then F ∪ { } ∪ ≥ F ∪ { (cid:48)} ∪ ([x s x , i]) ([y s x , j]). γ γ (cid:48) (cid:48) We will break the proof into three cases, according to whether s = s or (s, s ) = (u , v ) or (cid:48) γ γ (s, s ) = (v , u ). γ γ (cid:48) (cid:48) Case 1: s = s . Suppose s = s . Suppose the coeﬃcient in (37) is nonzero. Then there is a Whitney disk ∈ ∪ { } ∪ ∪ { } ∪ φ π (x s x , y s x ) 2 γ γ − ≥ with µ(φ) = 1, with no negative multiplicities, and with n (φ) = i j 0. Borrowing notation (cid:88) z (cid:88) from the proof of Lemma 3.10, we can then write the domain D(φ) as a linear combination q r (cid:48) − D(φ) = D + (i j)Σ + a P + b S + c S γ γ i i i i i=1 i=q+1 (cid:48) with integer coeﬃcients. As the multiplicities of D at A and B are zero, we have that F ∪ { } ∪ − F ∪ { } ∪ − − ≥ ([x s x , i]) ([y s x , j]) = 2n (φ) n (φ) = i j 0, γ γ A B as desired. (cid:48) (cid:48) Case 2: (s, s ) = (u , v ). Suppose s = u and s = v . The proof of Lemma 3.10 shows that γ γ γ γ for g large enough and suﬃcient winding, no Whitney disk ∈ ∪ { } ∪ ∪ { } ∪ φ π (x u x , y v x ) 2 γ γ γ γ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: THE

text: EQUIVALENCE

text: OF

text: CONTACT

text: INVARIANTS

text: IN

text: SUTURED

text: FLOER

text: HOMOLOGY

text: THEORIES

text: Let

text: A

text: and

text: B

text: be

text: the

text: points

text: in

text: R

text: and

text: R

text: shown

text: in

text: Figure

text: Given

text: generators

text: S

text: Σ

text: [x

text: s

text: x

text: i]

text: and

text: [y

text: s

text: x

text: j]

text: of

text: CF

text: +(γ,

text: α

text: R

text: Γ

text: γ

text: γ

text: γ

text: η

text: representing

text: the

text: same

text: Spinc

text: structure,

text: for

text: x,

text: y

text: as

text: in

text: and

text: s,

text: s

text: u

text: v

text: choose

text: a

text: Whitney

text: γ

text: γ

text: disk

text: φ

text: π

text: (x

text: s

text: x

text: y

text: s

text: x

text: γ

text: γ

text: with

text: n

text: (φ)

text: i

text: j,

text: and

text: deﬁne

text: the

text: relative

text: grading

text: z

text: F

text: ([x

text: s

text: x

text: i],

text: [y

text: s

text: x

text: j])

text: (φ)

text: n

text: (φ).

text: γ

text: γ

text: A

text: B

text: This

text: relative

text: grading

text: is

text: well-deﬁned

text: since

text: n

text: is

text: zero

text: on

text: the

text: periodic

text: domains

text: in

text: A

text: B

text: For

text: each

text: Spinc

text: structure,

text: we

text: choose

text: some

text: lift

text: of

text: this

text: relative

text: grading

text: to

text: an

text: absolute

text: grading,

text: F

text: which

text: we

text: also

text: denote

text: by

text: Lemma

text: Suﬃciently

text: large

text: g

text: and

text: suﬃcient

text: winding

text: guarantees

text: that

text: the

text: absolute

text: grading

text: F

text: deﬁnes

text: a

text: ﬁltration.

text: Proof.

text: Fix

text: x,

text: y

text: as

text: in

text: We

text: must

text: show

text: that

text: for

text: suﬃciently

text: large

text: g

text: and

text: suﬃcient

text: winding,

text: the

text: following

text: is

text: true:

text: if

text: the

text: coeﬃcient

text: ∂([x

text: s

text: x

text: i]),

text: y

text: s

text: x

text: j]

text: γ

text: γ

text: is

text: nonzero,

text: then

text: F

text: F

text: ([x

text: s

text: x

text: i])

text: ([y

text: s

text: x

text: j]).

text: γ

text: γ

text: We

text: will

text: break

text: the

text: proof

text: into

text: three

text: cases,

text: according

text: to

text: whether

text: s

text: s

text: or

text: (s,

text: s

text: (u

text: v

text: or

text: γ

text: γ

text: (s,

text: s

text: (v

text: u

text: γ

text: γ

text: Case

text: s

text: s

text: Suppose

text: s

text: s

text: Suppose

text: the

text: coeﬃcient

text: in

text: is

text: nonzero.

text: Then

text: there

text: is

text: a

text: Whitney

text: disk

text: φ

text: π

text: (x

text: s

text: x

text: y

text: s

text: x

text: γ

text: γ

text: with

text: µ(φ)

text: with

text: no

text: negative

text: multiplicities,

text: and

text: with

text: n

text: (φ)

text: i

text: j

text: Borrowing

text: notation

text: z

text: from

text: the

text: proof

text: of

text: Lemma

text: we

text: can

text: then

text: write

text: the

text: domain

text: D(φ)

text: as

text: a

text: linear

text: combination

text: q

text: r

text: D(φ)

text: D

text: (i

text: j)Σ

text: a

text: P

text: b

text: S

text: c

text: S

text: γ

text: γ

text: i

text: i

text: i

text: i

text: with

text: integer

text: coeﬃcients.

text: As

text: the

text: multiplicities

text: of

text: D

text: at

text: A

text: and

text: B

text: are

text: zero,

text: we

text: have

text: that

text: F

text: F

text: ([x

text: s

text: x

text: i])

text: ([y

text: s

text: x

text: j])

text: (φ)

text: n

text: (φ)

text: i

text: j

text: γ

text: γ

text: A

text: B

text: as

text: desired.

text: Case

text: (s,

text: s

text: (u

text: v

text: Suppose

text: s

text: u

text: and

text: s

text: v

text: The

text: proof

text: of

text: Lemma

text: shows

text: that

text: γ

text: γ

text: γ

text: γ

text: for

text: g

text: large

text: enough

text: and

text: suﬃcient

text: winding,

text: no

text: Whitney

text: disk

text: φ

text: π

text: (x

text: u

text: x

text: y

text: v

text: x

text: γ

text: γ

text: γ

text: γ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON THE EQUIVALENCE OF CONTACT INVARIANTS IN SUTURED FLOER HOMOLOGY THEORIES 49 In particular, the last condition implies by Theorem 1.11 that there is no Lagrangian concor- dance from K to K . 1 2 The fact that K in Theorem 4.1 is a stabilization implies that its Legendrian contact ho- 1 mology DGA is trivial. Legendrian contact homology therefore fails to obstruct a Lagrangian concordance from K to K for these examples. 1 2 In our proof of Theorem 4.1 below, we will adopt the convention that the (p, q)-cable of a knot K, denoted by C (K), has longitudinal winding p and meridional winding q. We will p,q also, for notational convenience, we will denote the (r, s)-cable of C (K) simply by p,q C (K) := C (C (K)). p,q;r,s r,s p,q Given a Legendrian knot K, we will use K+ to denote its transverse pushoﬀ which satisﬁes − sl(K+) = tb(K) r(K). Finally, given a knot K we will denote by tb(K) and sl(K) the maximal Thurston-Bennequin and self-linking numbers among Legendrian and transverse knots smoothly isotopic to K. Proof of Theorem 4.1. In our examples, K will be a Legendrian representative of the iterated 1 torus knot C (U) and K will be a Legendrian representative of 3,2;3,2 2 − − C (P ( m, 3, 3))#6 , 3,2;3,2 1 − − ≥ where P ( m, 3, 3) is the usual pretzel knot, for any m 3. Such K and K are smoothly 1 2 concordant as these pretzel knots and the twist knot 6 are all smoothly slice. 1 To deﬁne the Legendrian representative K , we rely on the following result of Ng, Ozsv´ath, 1 and Thurston from [39, Section 3.3]. The Legendrian K in the proposition below was ﬁrst discovered and studied by Etnyre and Honda in [16]. Proposition 4.2 (Ng–Ozsv´ath–Thurston). There is a Legendrian representative K of the (cid:54) iterated torus knot C (U) with (tb(K), r(K)) = (5, 2) and Θ (K) = 0. 3,2;3,2 HF We then deﬁne K to be the Legendrian knot obtained by negatively stabilizing this knot 1 K three times. It follows that − (cid:54) (tb(K ), r(K )) = (2, 1) and Θ (K ) = 0 1 1 HF 1 since Θ is preserved by negative stabilization. HF We now record some facts that will be relevant in deﬁning K . First, we record that 2 sl(C (U)) = 7, where this maximal self-linking number is realized by the transverse pushoﬀ 3,2;3,2 − of the unique Legendrian representative with (tb, r) = (6, 1); see [16] for the full Legendrian classiﬁcation of C (U). Since g(C (U)) = 4, the three inequalities 3,2;3,2 3,2;3,2 ≤ − ≤ − ≤ − sl(C (U)) 2τ(C (U)) 1 2g (C (U)) 1 2g(C (U)) 1 3,2;3,2 3,2;3,2 s 3,2;3,2 3,2;3,2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: THE

text: EQUIVALENCE

text: OF

text: CONTACT

text: INVARIANTS

text: IN

text: SUTURED

text: FLOER

text: HOMOLOGY

text: THEORIES

text: In

text: particular,

text: the

text: last

text: condition

text: implies

text: by

text: Theorem

text: that

text: there

text: is

text: no

text: Lagrangian

text: concor-

text: dance

text: from

text: K

text: to

text: K

text: The

text: fact

text: that

text: K

text: in

text: Theorem

text: is

text: a

text: stabilization

text: implies

text: that

text: its

text: Legendrian

text: contact

text: ho-

text: mology

text: DGA

text: is

text: trivial.

text: Legendrian

text: contact

text: homology

text: therefore

text: fails

text: to

text: obstruct

text: a

text: Lagrangian

text: concordance

text: from

text: K

text: to

text: K

text: for

text: these

text: examples.

text: In

text: our

text: proof

text: of

text: Theorem

text: below,

text: we

text: will

text: adopt

text: the

text: convention

text: that

text: the

text: (p,

text: q)-cable

text: of

text: a

text: knot

text: K,

text: denoted

text: by

text: C

text: (K),

text: has

text: longitudinal

text: winding

text: p

text: and

text: meridional

text: winding

text: q.

text: We

text: will

text: p,q

text: also,

text: for

text: notational

text: convenience,

text: we

text: will

text: denote

text: the

text: (r,

text: s)-cable

text: of

text: C

text: (K)

text: simply

text: by

text: p,q

text: C

text: (K)

text: C

text: (C

text: (K)).

text: p,q;r,s

text: r,s

text: p,q

text: Given

text: a

text: Legendrian

text: knot

text: K,

text: we

text: will

text: use

text: K+

text: to

text: denote

text: its

text: transverse

text: pushoﬀ

text: which

text: satisﬁes

text: sl(K+)

text: tb(K)

text: r(K).

text: Finally,

text: given

text: a

text: knot

text: K

text: we

text: will

text: denote

text: by

text: tb(K)

text: and

text: sl(K)

text: the

text: maximal

text: Thurston-Bennequin

text: and

text: self-linking

text: numbers

text: among

text: Legendrian

text: and

text: transverse

text: knots

text: smoothly

text: isotopic

text: to

text: K.

text: Proof

text: of

text: Theorem

text: In

text: our

text: examples,

text: K

text: will

text: be

text: a

text: Legendrian

text: representative

text: of

text: the

text: iterated

text: torus

text: knot

text: C

text: (U)

text: and

text: K

text: will

text: be

text: a

text: Legendrian

text: representative

text: of

text: C

text: (P

text: m,

text: where

text: P

text: m,

text: is

text: the

text: usual

text: pretzel

text: knot,

text: for

text: any

text: m

text: Such

text: K

text: and

text: K

text: are

text: smoothly

text: concordant

text: as

text: these

text: pretzel

text: knots

text: and

text: the

text: twist

text: knot

text: are

text: all

text: smoothly

text: slice.

text: To

text: deﬁne

text: the

text: Legendrian

text: representative

text: K

text: we

text: rely

text: on

text: the

text: following

text: result

text: of

text: Ng,

text: Ozsv´ath,

text: and

text: Thurston

text: from

text: Section

text: The

text: Legendrian

text: K

text: in

text: the

text: proposition

text: below

text: was

text: ﬁrst

text: discovered

text: and

text: studied

text: by

text: Etnyre

text: and

text: Honda

text: in

text: Proposition

text: (Ng–Ozsv´ath–Thurston).

text: There

text: is

text: a

text: Legendrian

text: representative

text: K

text: of

text: the

text: iterated

text: torus

text: knot

text: C

text: (U)

text: with

text: (tb(K),

text: r(K))

text: and

text: Θ

text: (K)

text: HF

text: We

text: then

text: deﬁne

text: K

text: to

text: be

text: the

text: Legendrian

text: knot

text: obtained

text: by

text: negatively

text: stabilizing

text: this

text: knot

text: K

text: three

text: times.

text: It

text: follows

text: that

text: (tb(K

text: r(K

text: and

text: Θ

text: (K

text: HF

text: since

text: Θ

text: is

text: preserved

text: by

text: negative

text: stabilization.

text: HF

text: We

text: now

text: record

text: some

text: facts

text: that

text: will

text: be

text: relevant

text: in

text: deﬁning

text: K

text: First,

text: we

text: record

text: that

text: sl(C

text: (U))

text: where

text: this

text: maximal

text: self-linking

text: number

text: is

text: realized

text: by

text: the

text: transverse

text: pushoﬀ

text: of

text: the

text: unique

text: Legendrian

text: representative

text: with

text: (tb,

text: r)

text: see

text: for

text: the

text: full

text: Legendrian

text: classiﬁcation

text: of

text: C

text: (U).

text: Since

text: g(C

text: (U))

text: the

text: three

text: inequalities

text: sl(C

text: (U))

text: (U))

text: (C

text: (U))

text: (U))

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 JOHN A. BALDWIN AND STEVEN SIVEK knots with the same classical invariants which satisfy the hypotheses of Theorem 1.11 [48, 39, 23, 1]. The results of [48] imply that such K and K are not Legendrian isotopic, whereas 1 2 Theorem 1.11 implies the much stronger fact that K is not Lagrangian concordant to K . 1 2 It bears mentioning that the Legendrian contact homology DGA of Chekanov and Eliash- berg [14] enjoys a similar sort of functoriality under Lagrangian concordance [13]. However, it can be diﬃcult to apply this DGA obstruction in practice. Consider, for example, the two Leg- endrian representatives K and K of m(10 ) with (tb, r) = (1, 0) described by Ng, Ozsv´ath, 1 2 132 and Thurston in [39]. One can show that K is not Lagrangian concordant to K by showing 1 2 that the DGA is trivial for K while nontrivial for K . But proving this nontriviality is tricky 2 1 as the DGA for K does not even admit any nontrivial ﬁnite-dimensional representations [51]. 1 By contrast, it is quite easy to check that Θ vanishes K but not for K , and in so doing, HF 2 1 apply the Heegaard Floer obstruction in Theorem 1.11. Another advantage of Θ is that it is preserved under negative Legendrian stabilization, HF whereas the Legendrian contact homology DGA is trivial for stabilized knots. In particular, for K and K satisfying the hypotheses of Theorem 1.11, we may also conclude that no negative 1 2 stabilization of K is Lagrangian concordant to any negative stabilization of K . The DGA, 1 2 by contrast, cannot tell us anything about Lagrangian concordances between stabilized knots. In Section , we give another demonstration of our obstruction, providing several additional examples of Legendrian knots with the same classical invariants which are not smoothly iso- topic or Lagrangian concordant, but which are smoothly concordant. In these examples, La- grangian concordance is obstructed by Theorem 1.11 while the Legendrian contact homology DGA provides no such obstruction. We next outline our proof of Theorem 1.2, starting with a very brief review of sutured monopole Floer homology and of our contact invariant. A closure of a balanced sutured manifold (M, Γ), as deﬁned by Kronheimer and Mrowka ⊂ in [27], is a closed manifold Y together with a distinguished surface R Y , formed from (M, Γ) in a certain manner, and containing M as a submanifold. The sutured monopole Floer homology of (M, Γ) is deﬁned as the monopole Floer (cid:77)homology of Y in the bottom Spinc structures with respect to R, (cid:100) (cid:100) | SHM(M, Γ) := HM •(Y R; Γ ) := HM •(Y ; Γ ), η η (cid:104) (cid:105) − c1(s),[R] =2 2g(R) ⊂ for some curve η R.6 Given a sutured contact manifold (M, Γ, ξ), we gave a procedure in [3] for extending ξ to a contact structure ξ¯ on a closure (Y, R) with respect to which R is convex and such that (cid:104) (cid:105) − c (s ), [R] = 2 2g(R). 1 ξ¯ 6In [27] and [6], these sutured Floer homology modules are deﬁned in terms of the top Spinc structures with respect to R; this is not an important diﬀerence as the two diﬀerent choices result in isomorphic modules. ##LTLine## ##LTLine##

text: JOHN

text: A.

text: BALDWIN

text: AND

text: STEVEN

text: SIVEK

text: knots

text: with

text: the

text: same

text: classical

text: invariants

text: which

text: satisfy

text: the

text: hypotheses

text: of

text: Theorem

text: The

text: results

text: of

text: imply

text: that

text: such

text: K

text: and

text: K

text: are

text: not

text: Legendrian

text: isotopic,

text: whereas

text: Theorem

text: implies

text: the

text: much

text: stronger

text: fact

text: that

text: K

text: is

text: not

text: Lagrangian

text: concordant

text: to

text: K

text: It

text: bears

text: mentioning

text: that

text: the

text: Legendrian

text: contact

text: homology

text: DGA

text: of

text: Chekanov

text: and

text: Eliash-

text: berg

text: enjoys

text: a

text: similar

text: sort

text: of

text: functoriality

text: under

text: Lagrangian

text: concordance

text: However,

text: it

text: can

text: be

text: diﬃcult

text: to

text: apply

text: this

text: DGA

text: obstruction

text: in

text: practice.

text: Consider,

text: for

text: example,

text: the

text: two

text: Leg-

text: endrian

text: representatives

text: K

text: and

text: K

text: of

text: with

text: (tb,

text: r)

text: described

text: by

text: Ng,

text: Ozsv´ath,

text: and

text: Thurston

text: in

text: One

text: can

text: show

text: that

text: K

text: is

text: not

text: Lagrangian

text: concordant

text: to

text: K

text: by

text: showing

text: that

text: the

text: DGA

text: is

text: trivial

text: for

text: K

text: while

text: nontrivial

text: for

text: K

text: But

text: proving

text: this

text: nontriviality

text: is

text: tricky

text: as

text: the

text: DGA

text: for

text: K

text: does

text: not

text: even

text: admit

text: any

text: nontrivial

text: ﬁnite-dimensional

text: representations

text: By

text: contrast,

text: it

text: is

text: quite

text: easy

text: to

text: check

text: that

text: Θ

text: vanishes

text: K

text: but

text: not

text: for

text: K

text: and

text: in

text: so

text: doing,

text: HF

text: apply

text: the

text: Heegaard

text: Floer

text: obstruction

text: in

text: Theorem

text: Another

text: advantage

text: of

text: Θ

text: is

text: that

text: it

text: is

text: preserved

text: under

text: negative

text: Legendrian

text: stabilization,

text: HF

text: whereas

text: the

text: Legendrian

text: contact

text: homology

text: DGA

text: is

text: trivial

text: for

text: stabilized

text: knots.

text: In

text: particular,

text: for

text: K

text: and

text: K

text: satisfying

text: the

text: hypotheses

text: of

text: Theorem

text: we

text: may

text: also

text: conclude

text: that

text: no

text: negative

text: stabilization

text: of

text: K

text: is

text: Lagrangian

text: concordant

text: to

text: any

text: negative

text: stabilization

text: of

text: K

text: The

text: DGA,

text: by

text: contrast,

text: cannot

text: tell

text: us

text: anything

text: about

text: Lagrangian

text: concordances

text: between

text: stabilized

text: knots.

text: In

text: Section

text: we

text: give

text: another

text: demonstration

text: of

text: our

text: obstruction,

text: providing

text: several

text: additional

text: examples

text: of

text: Legendrian

text: knots

text: with

text: the

text: same

text: classical

text: invariants

text: which

text: are

text: not

text: smoothly

text: iso-

text: topic

text: or

text: Lagrangian

text: concordant,

text: but

text: which

text: are

text: smoothly

text: concordant.

text: In

text: these

text: examples,

text: La-

text: grangian

text: concordance

text: is

text: obstructed

text: by

text: Theorem

text: while

text: the

text: Legendrian

text: contact

text: homology

text: DGA

text: provides

text: no

text: such

text: obstruction.

text: We

text: next

text: outline

text: our

text: proof

text: of

text: Theorem

text: starting

text: with

text: a

text: very

text: brief

text: review

text: of

text: sutured

text: monopole

text: Floer

text: homology

text: and

text: of

text: our

text: contact

text: invariant.

text: A

text: closure

text: of

text: a

text: balanced

text: sutured

text: manifold

text: (M,

text: Γ),

text: as

text: deﬁned

text: by

text: Kronheimer

text: and

text: Mrowka

text: in

text: is

text: a

text: closed

text: manifold

text: Y

text: together

text: with

text: a

text: distinguished

text: surface

text: R

text: Y

text: formed

text: from

text: (M,

text: Γ)

text: in

text: a

text: certain

text: manner,

text: and

text: containing

text: M

text: as

text: a

text: submanifold.

text: The

text: sutured

text: monopole

text: Floer

text: homology

text: of

text: (M,

text: Γ)

text: is

text: deﬁned

text: as

text: the

text: monopole

text: Floer

text: of

text: Y

text: in

text: the

text: bottom

text: Spinc

text: structures

text: with

text: respect

text: to

text: R,

text: SHM(M,

text: Γ)

text: HM

text: •(Y

text: R;

text: Γ

text: HM

text: •(Y

text: Γ

text: η

text: η

text: for

text: some

text: curve

text: η

text: Given

text: a

text: sutured

text: contact

text: manifold

text: (M,

text: Γ,

text: ξ),

text: we

text: gave

text: a

text: procedure

text: in

text: for

text: extending

text: ξ

text: to

text: a

text: contact

text: structure

text: ξ¯

text: on

text: a

text: closure

text: (Y,

text: R)

text: with

text: respect

text: to

text: which

text: R

text: is

text: convex

text: and

text: such

text: that

text: c

text: (s

text: [R]

text: ξ¯

text: and

text: these

text: sutured

text: Floer

text: homology

text: modules

text: are

text: deﬁned

text: in

text: terms

text: of

text: the

text: top

text: Spinc

text: structures

text: with

text: respect

text: to

text: R;

text: this

text: is

text: not

text: an

text: important

text: diﬀerence

text: as

text: the

text: two

text: diﬀerent

text: choices

text: result

text: in

text: isomorphic

text: modules.

text: ##LTLine##

text: ##LTLine##


text: 7 N in (b). Omnians are indicated as square nodes and reticulations are marked with a pink shading around the nodes. From the bipartite graph in Figure 9 it follows with Hall’s Theorem, with S = U, that there exists no matching in B that covers U. Therefore, with Theorem 2.4 it follows that N in Figure 9(a) is not tree- { } based. Indeed, we can directly see in N that the omians in S = a, i, h, f, g have only four different { } children b, c, d, e (note that, in general, the set S and the set of children of vertices in S do not have to be disjoint). Hence this network is not tree-based. a i b U R p h f a x b a c q c f g i y d h e e d w d g f Fig. 9. Example of a non-tree-based binary phylogenetic network N and the bipartite graph B associated to N. In a previous article, the following necessary condition for a network to be tree-based was found. Proposition 2.8. [7] If a binary phylogenetic network over leaf set X is tree-based, then it satisﬁes the antichain-to- leaf property. On the other hand, if a network is not tree-based, it can still satisfy the antichain-to-leaf property, as shown by the example from [7] in Figure 10. However, it is difﬁcult to see in this network what is going on. Therefore, we show in Figure 11 two examples of local structures that cause a network to be non-tree-based. The example in Figure 11(a) is similar to the local structure that causes the network of Figure 10 not to be tree-based, which can be checked using Corollary 2.7. At ﬁrst sight, it might look like networks containing the local structures in Figure 11 cannot satisfy the antichain-to-leaf property. However, in Figure 12 we show that it is indeed possible that they do satisfy this property. Although it was already known that networks that satisfy the antichain-to-leaf property are not necessarily tree-based, these ﬁgures illuminate why this is possible. Looking at the examples in Figure 11, we see that a pattern has emerged. In (a) the pattern is marked dashed in green. Starting at vertex o and ending at vertex o , we see a zigzag starting with an omnian, 1 5 alternating between reticulations and omnians, eventually ending with an omnian. The last omnian in the pattern can be a reticulation that is already part of the path, as can be seen in Figure 11(b). The next theorem shows that every binary phylogenetic network that is not tree-based contains a local structure as in the examples in Figure 11. ∪ Theorem 2.9. Let N be a binary phylogenetic network and B = (U R, E) the bipartite graph associated to N. Network N is tree-based if and only if B contains no maximal path which starts and ends in U. Proof. Notice that every vertex in B is of degree at most 2. Therefore, B is a disjoint union of paths and (cid:48) (cid:48) ∪ (cid:48) (cid:48) cycles. Hence, for each connected component B = (U R , E ) of B, there are four possible topologies: i) A maximal path that begins and ends in R. ii) A maximal path that begins in U and ends in R. iii) A maximal path that begins and ends in U. ##LTLine## ##LTLine##

font: UWJTTJ+NimbusSanL-Regu

text: N

text: in

font: UTFAXN+URWPalladioL-Roma

text: (b).

text: Omnians

text: are

text: indicated

text: as

text: square

text: nodes

text: and

text: reticulations

text: are

text: marked

text: with

text: a

text: pink

text: shading

text: around

text: the

text: nodes.

text: From

text: the

text: bipartite

text: graph

text: in

text: Figure

text: it

text: follows

text: with

text: Hall’s

text: Theorem,

text: with

text: S

text: U,

text: that

text: there

text: exists

text: no

text: matching

text: in

text: B

text: that

text: covers

text: U.

text: Therefore,

text: with

text: Theorem

text: it

text: follows

text: that

text: N

text: in

text: Figure

text: is

text: not

text: tree-

text: based.

text: Indeed,

text: we

text: can

text: directly

text: see

text: in

text: N

text: that

text: the

text: omians

text: in

text: S

text: a,

text: i,

text: h,

text: f,

text: g

text: have

text: only

text: four

text: different

text: children

text: b,

text: c,

text: d,

text: e

text: (note

text: that,

text: in

text: general,

text: the

text: set

text: S

text: and

text: the

text: set

text: of

text: children

text: of

text: vertices

text: in

text: S

text: do

text: not

text: have

text: to

text: be

text: disjoint).

text: Hence

text: this

text: network

text: is

text: not

text: tree-based.

text: a

text: i

text: b

text: U

text: R

text: p

text: h

text: f

text: a

text: x

text: b

text: a

text: c

text: q

text: c

text: f

text: g

text: i

text: y

text: d

text: h

text: e

text: e

text: d

text: w

text: d

text: g

text: f

text: Fig.

text: Example

text: of

text: a

text: non-tree-based

text: binary

text: phylogenetic

text: network

text: N

text: and

text: the

text: bipartite

text: graph

text: B

text: associated

text: to

text: N.

text: In

text: a

text: previous

text: article,

text: the

text: following

text: necessary

text: condition

text: for

text: a

text: network

text: to

text: be

text: tree-based

text: was

text: found.

text: Proposition

text: If

text: a

text: binary

text: phylogenetic

text: network

text: over

text: leaf

text: set

text: X

text: is

text: tree-based,

text: then

text: it

text: satisﬁes

text: the

text: antichain-to-

text: leaf

text: property.

text: On

text: the

text: other

text: hand,

text: if

text: a

text: network

text: is

text: not

text: tree-based,

text: it

text: can

text: still

text: satisfy

text: the

text: antichain-to-leaf

text: property,

text: as

text: shown

text: by

text: the

text: example

text: from

text: in

text: Figure

text: However,

text: it

text: is

text: difﬁcult

text: to

text: see

text: in

text: this

text: network

text: what

text: is

text: going

text: on.

text: Therefore,

text: we

text: show

text: in

text: Figure

text: two

text: examples

text: of

text: local

text: structures

text: that

text: cause

text: a

text: network

text: to

text: be

text: non-tree-based.

text: The

text: example

text: in

text: Figure

text: is

text: similar

text: to

text: the

text: local

text: structure

text: that

text: causes

text: the

text: network

text: of

text: Figure

text: not

text: to

text: be

text: tree-based,

text: which

text: can

text: be

text: checked

text: using

text: Corollary

text: At

text: ﬁrst

text: sight,

text: it

text: might

text: look

text: like

text: networks

text: containing

text: the

text: local

text: structures

text: in

text: Figure

text: cannot

text: satisfy

text: the

text: antichain-to-leaf

text: property.

text: However,

text: in

text: Figure

text: we

text: show

text: that

text: it

text: is

text: indeed

text: possible

text: that

text: they

text: do

text: satisfy

text: this

text: property.

text: Although

text: it

text: was

text: already

text: known

text: that

text: networks

text: that

text: satisfy

text: the

text: antichain-to-leaf

text: property

text: are

text: not

text: necessarily

text: tree-based,

text: these

text: ﬁgures

text: illuminate

text: why

text: this

text: is

text: possible.

text: Looking

text: at

text: the

text: examples

text: in

text: Figure

text: we

text: see

text: that

text: a

text: pattern

text: has

text: emerged.

text: In

text: (a)

text: the

text: pattern

text: is

text: marked

text: dashed

text: in

text: green.

text: Starting

text: at

text: vertex

text: o

text: and

text: ending

text: at

text: vertex

text: o

text: we

text: see

text: a

text: zigzag

text: starting

text: with

text: an

text: omnian,

text: alternating

text: between

text: reticulations

text: and

text: omnians,

text: eventually

text: ending

text: with

text: an

text: omnian.

text: The

text: last

text: omnian

text: in

text: the

text: pattern

text: can

text: be

text: a

text: reticulation

text: that

text: is

text: already

text: part

text: of

text: the

text: path,

text: as

text: can

text: be

text: seen

text: in

text: Figure

text: The

text: next

text: theorem

text: shows

text: that

text: every

text: binary

text: phylogenetic

text: network

text: that

text: is

text: not

text: tree-based

text: contains

text: a

text: local

text: structure

text: as

text: in

text: the

text: examples

text: in

text: Figure

text: Theorem

text: Let

text: N

text: be

text: a

text: binary

text: phylogenetic

text: network

text: and

text: B

text: (U

text: R,

text: E)

text: the

text: bipartite

text: graph

text: associated

text: to

text: N.

text: Network

text: N

text: is

text: tree-based

text: if

text: and

text: only

text: if

text: B

text: contains

text: no

text: maximal

text: path

text: which

text: starts

text: and

text: ends

text: in

text: U.

text: Proof.

text: Notice

text: that

text: every

text: vertex

text: in

text: B

text: is

text: of

text: degree

text: at

text: most

text: Therefore,

text: B

text: is

text: a

text: disjoint

text: union

text: of

text: paths

text: and

text: cycles.

text: Hence,

text: for

text: each

text: connected

text: component

text: B

text: (U

text: R

text: E

text: of

text: B,

text: there

text: are

text: four

text: possible

text: topologies:

text: i)

text: A

text: maximal

text: path

text: that

text: begins

text: and

text: ends

text: in

text: R.

text: ii)

text: A

text: maximal

text: path

text: that

text: begins

text: in

text: U

text: and

text: ends

text: in

text: R.

text: iii)

text: A

text: maximal

text: path

text: that

text: begins

text: and

text: ends

text: in

text: U.

text: ##LTLine##

text: ##LTLine##


text: 10 • vertices with indegree 1 and outdegree 0, called leaves (coloured blue in Figure 13), which are labelled; • vertices with outdegree 1 and indegree 2 or more, called reticulations (marked in pink in Figure 13); • vertices with indegree 1 and outdegree 2 or more, called tree-vertices. Root a b d e f g c Fig. 13. Example of a nonbinary phylogenetic network. A nonbinary phylogenetic tree is a nonbinary phylogenetic network without reticulations. We will consider two different variants of tree-basedness of nonbinary networks, which we name “tree- based” and “strictly-tree-based”. Deﬁnition 4. A nonbinary phylogenetic network N is called tree-based with base-tree T, when N can be obtained from T via the following steps: (a) Add some vertices to arcs of T. These vertices, called attachment points, have in- and outdegree 1. (b) Add arcs, called linking arcs, between pairs of attachments points and from tree-vertices to attachment points, so that N remains acyclic and so that attachment points have indegree or outdegree 1. (c) Suppress every attachment point that is not incident to a linking arc. Deﬁnition 5. A nonbinary phylogenetic network N is called strictly tree-based with base-tree T, when N can be obtained from T via the following steps: (1) Add some vertices to arcs in T. These vertices, called attachment points, have in- and outdegree 1. (2) Add arcs, called linking arcs, between pairs of attachments points, so that N remains acyclic and so that exactly one linking arc is attached to each attachment point. A nonbinary phylogenetic network is tree-based if it is tree-based with base-tree T for some nonbinary phylogenetic tree T. Similarly, a nonbinary phylogenetic network is strictly tree-based if it is strictly tree-based with base-tree T for some nonbinary rooted phylogenetic tree T. The distinction between tree-based and strictly tree-based is illustrated by two examples in Figure 14. An example of a strictly-tree-based nonbinary network can be found in Figure 18(a). a b c d e a b c d (a) This network is not strictly (b) This network is not strictly tree-based because the linking tree-based because two link- arc is attached to a vertex of ing arcs are attached to the the base-tree, rather than to same attachment point, and an attachment point, and this this cannot be avoided. cannot be avoided. Fig. 14. Two nonbinary networks that are both tree-based but not strictly tree-based. The black solid lines indicate possible base-trees, while the dashed, grey lines are the linking arcs. The following observations are easily veriﬁed. Observation 3.1. Let N be a nonbinary phylogenetic network. If N is strictly tree-based, then N is tree-based. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: vertices

font: ISMPFW+URWPalladioL-Ital

text: with

text: indegree

text: and

text: outdegree

text: called

text: leaves

text: (coloured

text: blue

text: in

text: Figure

text: which

text: are

text: labelled;

text: vertices

text: with

text: outdegree

text: and

text: indegree

text: or

text: more,

text: called

text: reticulations

text: (marked

text: in

text: pink

text: in

text: Figure

text: vertices

text: with

text: indegree

text: and

text: outdegree

text: or

text: more,

text: called

text: tree-vertices.

text: Root

text: a

text: b

text: d

text: e

text: f

text: g

text: c

text: Fig.

text: Example

text: of

text: a

text: nonbinary

text: phylogenetic

text: network.

text: A

text: nonbinary

text: phylogenetic

text: tree

text: is

text: a

text: nonbinary

text: phylogenetic

text: network

text: without

text: reticulations.

text: We

text: will

text: consider

text: two

text: different

text: variants

text: of

text: tree-basedness

text: of

text: nonbinary

text: networks,

text: which

text: we

text: name

text: “tree-

text: based”

text: and

text: “strictly-tree-based”.

text: Deﬁnition

text: A

text: nonbinary

text: phylogenetic

text: network

text: N

text: is

text: called

text: tree-based

text: with

text: base-tree

text: T,

text: when

text: N

text: can

text: be

text: obtained

text: from

text: T

text: via

text: the

text: following

text: steps:

text: (a)

text: Add

text: some

text: vertices

text: to

text: arcs

text: of

text: T.

text: These

text: vertices,

text: called

text: attachment

text: points,

text: have

text: in-

text: and

text: outdegree

text: (b)

text: Add

text: arcs,

text: called

text: linking

text: arcs,

text: between

text: pairs

text: of

text: attachments

text: points

text: and

text: from

text: tree-vertices

text: to

text: attachment

text: points,

text: so

text: that

text: N

text: remains

text: acyclic

text: and

text: so

text: that

text: attachment

text: points

text: have

text: indegree

text: or

text: outdegree

text: (c)

text: Suppress

text: every

text: attachment

text: point

text: that

text: is

text: not

text: incident

text: to

text: a

text: linking

text: arc.

text: Deﬁnition

text: A

text: nonbinary

text: phylogenetic

text: network

text: N

text: is

text: called

text: strictly

text: tree-based

text: with

text: base-tree

text: T,

text: when

text: N

text: can

text: be

text: obtained

text: from

text: T

text: via

text: the

text: following

text: steps:

text: Add

text: some

text: vertices

text: to

text: arcs

text: in

text: T.

text: These

text: vertices,

text: called

text: attachment

text: points,

text: have

text: in-

text: and

text: outdegree

text: Add

text: arcs,

text: called

text: linking

text: arcs,

text: between

text: pairs

text: of

text: attachments

text: points,

text: so

text: that

text: N

text: remains

text: acyclic

text: and

text: so

text: that

text: exactly

text: one

text: linking

text: arc

text: is

text: attached

text: to

text: each

text: attachment

text: point.

text: A

text: nonbinary

text: phylogenetic

text: network

text: is

text: tree-based

text: if

text: it

text: is

text: tree-based

text: with

text: base-tree

text: T

text: for

text: some

text: nonbinary

text: phylogenetic

text: tree

text: T.

text: Similarly,

text: a

text: nonbinary

text: phylogenetic

text: network

text: is

text: strictly

text: tree-based

text: if

text: it

text: is

text: strictly

text: tree-based

text: with

text: base-tree

text: T

text: for

text: some

text: nonbinary

text: rooted

text: phylogenetic

text: tree

text: T.

text: The

text: distinction

text: between

text: tree-based

text: and

text: strictly

text: tree-based

text: is

text: illustrated

text: by

text: two

text: examples

text: in

text: Figure

text: An

text: example

text: of

text: a

text: strictly-tree-based

text: nonbinary

text: network

text: can

text: be

text: found

text: in

text: Figure

text: a

text: b

text: c

text: d

text: e

text: a

text: b

text: c

text: d

text: (a)

text: This

text: network

text: is

text: not

text: strictly

text: (b)

text: This

text: network

text: is

text: not

text: strictly

text: tree-based

text: because

text: the

text: linking

text: tree-based

text: because

text: two

text: link-

text: arc

text: is

text: attached

text: to

text: a

text: vertex

text: of

text: ing

text: arcs

text: are

text: attached

text: to

text: the

text: the

text: base-tree,

text: rather

text: than

text: to

text: same

text: attachment

text: point,

text: and

text: an

text: attachment

text: point,

text: and

text: this

text: this

text: cannot

text: be

text: avoided.

text: cannot

text: be

text: avoided.

text: Fig.

text: Two

text: nonbinary

text: networks

text: that

text: are

text: both

text: tree-based

text: but

text: not

text: strictly

text: tree-based.

text: The

text: black

text: solid

text: lines

text: indicate

text: possible

text: base-trees,

text: while

text: the

text: dashed,

text: grey

text: lines

text: are

text: the

text: linking

text: arcs.

text: The

text: following

text: observations

text: are

text: easily

text: veriﬁed.

text: Observation

text: Let

text: N

text: be

text: a

text: nonbinary

text: phylogenetic

text: network.

text: If

text: N

text: is

text: strictly

text: tree-based,

text: then

text: N

text: is

text: tree-based.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 Bae et al. − Fig. 7.— Distributions of (left) gas, (middle) 300 µm dust in φ R coordinates, and (right) azimuthally averaged radial proﬁles ×of the gas and×dust distributions at t = 50.6 Tp. The same model parameters are used×as in Figure 1, but with twice higher resolution (nR nφ) = (57×6 1376). In the right panel, the solid curves show high-resolution (576 1376) results and the dashed curves show low-resolution (288 688) results. Mu¨ller, A., van den Ancker, M. E., Launhardt, R., et al. 2011, van der Marel, N., van Dishoeck, E. F., Bruderer, S., et al. 2015, A&A, 530, A85 A&A, 579, A106 Muto, T., Grady, C. A., Hashimoto, J. 2012, ApJL, 748, L22 van der Marel, N., van Dishoeck, E. F., Bruderer, S., et al. 2016, Ogilvie, G. I., & Lubow, S. H. 1999, ApJ, 515, 767 A&A, 585, A58 Paardekooper, S.-J., & Papaloizou, J. C. N. 2008, A&A, 485, 877 Vicente, S., Mer’in, B., Hartung, M., et al. 2011, A&A, 533, A135 P´erez, L. M., Isella, A., Carpenter, J. M., & Chandler, C. J. Youdin, A. N., & Lithwick, Y. 2007, Icar, 192, 588 2014, ApJL, 783, L13 Zhu, Z., Nelson, R. P., Dong, R., Espaillat, C. & Hartmann, L. Pinilla, P., van der Marel, N., van Dishoeck, E. F., et al. 2015, 2012, ApJ, 755, 6 A&A, 584, A16 Zhu, Z., & Stone, J. M. 2014, ApJ, 795, 53 Reg´aly, Zs., Juha´sz, A., Sa´ndor, Zs., & Dullemond, C. P. 2012, Zhu, Z., Stone, J. M., Raﬁkov, R. R., & Bai, X. 2014, ApJ, 785, MNRAS, 419, 1701 122 Zhu, Z., Dong, R., Stone, J. M., & Raﬁkov, R. R. 2015, ApJ, 813, 88 ##LTFigure## ##LTFigure##

text: Bae

text: et

text: al.

label: caption

text: Fig.

text: Distributions

text: of

text: (left)

text: gas,

text: (middle)

text: µm

text: dust

text: in

text: φ

text: R

text: coordinates,

text: and

text: (right)

text: azimuthally

text: averaged

text: radial

text: proﬁles

text: ×of

text: the

text: gas

text: and×dust

text: distributions

text: at

text: t

text: Tp.

text: The

text: same

text: model

text: parameters

text: are

text: used×as

text: in

text: Figure

text: but

text: with

text: twice

text: higher

text: resolution

text: (nR

text: nφ)

text: In

text: the

text: right

text: panel,

text: the

text: solid

text: curves

text: show

text: high-resolution

text: results

text: and

text: the

text: dashed

text: curves

text: show

text: low-resolution

text: results.

text: Mu¨ller,

label: reference

text: A.,

text: van

text: den

text: Ancker,

text: M.

text: E.,

text: Launhardt,

text: R.,

text: et

text: al.

text: van

text: der

text: Marel,

text: N.,

text: van

text: Dishoeck,

text: E.

text: F.,

text: Bruderer,

text: S.,

text: et

text: al.

text: A&A,

text: A&A,

text: Muto,

text: T.,

text: Grady,

text: C.

text: A.,

text: Hashimoto,

text: J.

text: ApJL,

text: van

text: der

text: Marel,

text: N.,

text: van

text: Dishoeck,

text: E.

text: F.,

text: Bruderer,

text: S.,

text: et

text: al.

text: Ogilvie,

text: G.

text: I.,

text: Lubow,

text: S.

text: H.

text: ApJ,

text: A&A,

text: Paardekooper,

text: S.-J.,

text: Papaloizou,

text: J.

text: C.

text: N.

text: A&A,

text: Vicente,

text: S.,

text: Mer’in,

text: B.,

text: Hartung,

text: M.,

text: et

text: al.

text: A&A,

text: P´erez,

text: L.

text: M.,

text: Isella,

text: A.,

text: Carpenter,

text: J.

text: M.,

text: Chandler,

text: C.

text: J.

text: Youdin,

text: A.

text: N.,

text: Lithwick,

text: Y.

text: Icar,

text: ApJL,

text: Zhu,

text: Z.,

text: Nelson,

text: R.

text: P.,

text: Dong,

text: R.,

text: Espaillat,

text: C.

text: Hartmann,

text: L.

text: Pinilla,

text: P.,

text: van

text: der

text: Marel,

text: N.,

text: van

text: Dishoeck,

text: E.

text: F.,

text: et

text: al.

text: ApJ,

text: A&A,

text: Zhu,

text: Z.,

text: Stone,

text: J.

text: M.

text: ApJ,

text: Reg´aly,

text: Zs.,

text: Juha´sz,

text: A.,

text: Sa´ndor,

text: Zs.,

text: Dullemond,

text: C.

text: P.

text: Zhu,

text: Z.,

text: Stone,

text: J.

text: M.,

text: Raﬁkov,

text: R.

text: R.,

text: Bai,

text: X.

text: ApJ,

text: MNRAS,

text: Zhu,

text: Z.,

text: Dong,

text: R.,

text: Stone,

text: J.

text: M.,

text: Raﬁkov,

text: R.

text: R.

text: ApJ,

text: ##LTFigure##

text: ##LTFigure##


text: (cid:80) 12 MASAHARU ISHIKAWA AND HIROKAZU YANAGI Lemma 4.3 (Byberi-Chernov [?]). Set P (K) (cid:88)= m∈N bmtm. Suppose that K is virtually null-homotopic. Then the following inequality holds: ≥ 1 | | vu(K) b . m 2 ∈N m In this section, we complete the proof of Theorem ?? by showing the lower bound of the equality in the theorem. The deﬁnition of i(c) is interpreted in terms of a virtual knot diagram D as follows: A chord c corresponds to a classical crossing p of D. Start from p and follow the strand into one of the possible directions, then we come back to p again at some moment. Denote this loop by γ and the rest part of the virtual knot by γ . For each classical crossing 1 2 consisting of γ and γ , if γ is an understrand then we apply a crossing change so that 1 2 1 γ is an overstrand. We denote the obtained virtual knot diagram by D . Then, i(c) 1 c is the sum of signs of all classical crossings consisting of γ and γ . Let i (γ , γ ) and 1 2 o 1 2 i (γ , γ ) be the number of classical crossings on D consisting of γ and γ such that γ u 1 2 1 2 1 is an overstand and an understrand, respectively. Then we have | | | − | i(c) = i (γ , γ ) i (γ , γ ) . o 1 2 u 1 2 A closed positive braid is a link, or a diagram, obtained as the closure of a braid consisting of only positive words. Remark that all classical crossings on (i, j, k) are positive and they are given by positive words in its braid word presentation. Lemma 4.4. Let B be the diagram of a closed positive braid and suppose that B is a knot. Let GD(B) be the Gauss diagram of B. Then i(c) = 0 for any chord c in GD(B). Proof. Choose a crossing point p of B. We choose γ by following the strand from p in the 1 right-bottom direction. There are two important remarks: One is that, since all crossings on B are positive, γ is always an overstrand when γ directs to the right-bottom, and it 1 is always an understrand when γ directs to the right-top. The other remark is that, since B is a knot, γ comes back to p from the left-bottom. If γ has no self-crossing, then 1 1 i (γ , γ ) = i (γ , γ ). Furthermore, this equality holds even if γ has self-crossings since o 1 2 u 1 2 1 each self-crossing consists of a strand going to the right-top and a strand going to the (cid:3) right-bottom. Thus we have i(c) = 0. Lemma 4.5. Suppose that (i, j, k) is a virtual knot diagram and let GD((i, j, k)) be its (cid:54) Gauss diagram. Then i(c) = 0 for any chord c in GD((i, j, k)). Proof. Assume that there exists a chord c such that i(c) = 0. Choose γ such that it 1 does not contain the vertical strand a which passes though all virtual crossings. This 1 is possible by exchanging the roles of γ and γ if necessary. Suppose that γ meets a 1 2 1 1 at virtual crossings N times, where N is at least 1. If we regard the virtual crossings ##LTLine##

text: MASAHARU

text: ISHIKAWA

text: AND

text: HIROKAZU

text: YANAGI

text: Lemma

text: (Byberi-Chernov

text: Set

text: P

text: (K)

text: m∈N

text: bmtm.

text: Suppose

text: that

text: K

text: is

text: virtually

text: null-homotopic.

text: Then

text: the

text: following

text: inequality

text: holds:

text: vu(K)

text: b

text: m

text: ∈N

text: m

text: In

text: this

text: section,

text: we

text: complete

text: the

text: proof

text: of

text: Theorem

text: by

text: showing

text: the

text: lower

text: bound

text: of

text: the

text: equality

text: in

text: the

text: theorem.

text: The

text: deﬁnition

text: of

text: i(c)

text: is

text: interpreted

text: in

text: terms

text: of

text: a

text: virtual

text: knot

text: diagram

text: D

text: as

text: follows:

text: A

text: chord

text: c

text: corresponds

text: to

text: a

text: classical

text: crossing

text: p

text: of

text: D.

text: Start

text: from

text: p

text: and

text: follow

text: the

text: strand

text: into

text: one

text: of

text: the

text: possible

text: directions,

text: then

text: we

text: come

text: back

text: to

text: p

text: again

text: at

text: some

text: moment.

text: Denote

text: this

text: loop

text: by

text: γ

text: and

text: the

text: rest

text: part

text: of

text: the

text: virtual

text: knot

text: by

text: γ

text: For

text: each

text: classical

text: crossing

text: consisting

text: of

text: γ

text: and

text: γ

text: if

text: γ

text: is

text: an

text: understrand

text: then

text: we

text: apply

text: a

text: crossing

text: change

text: so

text: that

text: γ

text: is

text: an

text: overstrand.

text: We

text: denote

text: the

text: obtained

text: virtual

text: knot

text: diagram

text: by

text: D

text: Then,

text: i(c)

text: c

text: is

text: the

text: sum

text: of

text: signs

text: of

text: all

text: classical

text: crossings

text: consisting

text: of

text: γ

text: and

text: γ

text: Let

text: i

text: (γ

text: γ

text: and

text: o

text: i

text: (γ

text: γ

text: be

text: the

text: number

text: of

text: classical

text: crossings

text: on

text: D

text: consisting

text: of

text: γ

text: and

text: γ

text: such

text: that

text: γ

text: u

text: is

text: an

text: overstand

text: and

text: an

text: understrand,

text: respectively.

text: Then

text: we

text: have

text: i(c)

text: i

text: (γ

text: γ

text: i

text: (γ

text: γ

text: o

text: u

text: A

text: closed

text: positive

text: braid

text: is

text: a

text: link,

text: or

text: a

text: diagram,

text: obtained

text: as

text: the

text: closure

text: of

text: a

text: braid

text: consisting

text: of

text: only

text: positive

text: words.

text: Remark

text: that

text: all

text: classical

text: crossings

text: on

text: (i,

text: j,

text: k)

text: are

text: positive

text: and

text: they

text: are

text: given

text: by

text: positive

text: words

text: in

text: its

text: braid

text: word

text: presentation.

text: Lemma

text: Let

text: B

text: be

text: the

text: diagram

text: of

text: a

text: closed

text: positive

text: braid

text: and

text: suppose

text: that

text: B

text: is

text: a

text: knot.

text: Let

text: GD(B)

text: be

text: the

text: Gauss

text: diagram

text: of

text: B.

text: Then

text: i(c)

text: for

text: any

text: chord

text: c

text: in

text: GD(B).

text: Proof.

text: Choose

text: a

text: crossing

text: point

text: p

text: of

text: B.

text: We

text: choose

text: γ

text: by

text: following

text: the

text: strand

text: from

text: p

text: in

text: the

text: right-bottom

text: direction.

text: There

text: are

text: two

text: important

text: remarks:

text: One

text: is

text: that,

text: since

text: all

text: crossings

text: on

text: B

text: are

text: positive,

text: γ

text: is

text: always

text: an

text: overstrand

text: when

text: γ

text: directs

text: to

text: the

text: right-bottom,

text: and

text: it

text: is

text: always

text: an

text: understrand

text: when

text: γ

text: directs

text: to

text: the

text: right-top.

text: The

text: other

text: remark

text: is

text: that,

text: since

text: B

text: is

text: a

text: knot,

text: γ

text: comes

text: back

text: to

text: p

text: from

text: the

text: left-bottom.

text: If

text: γ

text: has

text: no

text: self-crossing,

text: then

text: i

text: (γ

text: γ

text: i

text: (γ

text: γ

text: Furthermore,

text: this

text: equality

text: holds

text: even

text: if

text: γ

text: has

text: self-crossings

text: since

text: o

text: u

text: each

text: self-crossing

text: consists

text: of

text: a

text: strand

text: going

text: to

text: the

text: right-top

text: and

text: a

text: strand

text: going

text: to

text: the

text: right-bottom.

text: Thus

text: we

text: have

text: i(c)

text: Lemma

text: Suppose

text: that

text: (i,

text: j,

text: k)

text: is

text: a

text: virtual

text: knot

text: diagram

text: and

text: let

text: GD((i,

text: j,

text: k))

text: be

text: its

text: Gauss

text: diagram.

text: Then

text: i(c)

text: for

text: any

text: chord

text: c

text: in

text: GD((i,

text: j,

text: k)).

text: Proof.

text: Assume

text: that

text: there

text: exists

text: a

text: chord

text: c

text: such

text: that

text: i(c)

text: Choose

text: γ

text: such

text: that

text: it

text: does

text: not

text: contain

text: the

text: vertical

text: strand

text: a

text: which

text: passes

text: though

text: all

text: virtual

text: crossings.

text: This

text: is

text: possible

text: by

text: exchanging

text: the

text: roles

text: of

text: γ

text: and

text: γ

text: if

text: necessary.

text: Suppose

text: that

text: γ

text: meets

text: a

text: at

text: virtual

text: crossings

text: N

text: times,

text: where

text: N

text: is

text: at

text: least

text: If

text: we

text: regard

text: the

text: virtual

text: crossings

text: ##LTLine##


text: VIRTUAL UNKNOTTING NUMBERS OF CERTAIN VIRTUAL TORUS KNOTS 13 as classical crossings, then the diagram is a closed positive braid and we have i(c) = 0 by Lemma ??. This means that i(c) for (i, j, k) is calculated from that for the closed positive braid by subtracting the signed number of virtual crossings on γ . Thus we have | | ≥ 1 (cid:3) i(c) = N 1. Proof of Theorem ??. As mentioned in Remark ??, (i, j, k) is virtually null-homotopic. This also follows from Lemma ??. The upper bound of the equality in Theorem ?? (cid:80) follows from Proposition ??. For the lower bound, we use the inequality in Lemma ??. − − The diagram (i, j, k) has (i 1)(j 1) + k classical crossings and the signs of all classical (cid:54) | | cro−ssings−are positive. Since i(c) = 0 for any chord c by Lemma ??, we have m∈N bm (cid:3)= (i 1)(j 1) + k. Thus the lower bound follows from the inequality in Lemma ??. Remark 4.6. The assertion in Theorem ?? holds in case pq < 0 also since the virtual knot diagram V 1 with pq < 0 is the mirror image of V|1| | |. p,q p , q 5. Further discussions ≤ ≤ Lemma 5.1. Suppose that p and q are coprime and n is an integer such that 1 n q. ≥ − − If p and n are coprime then vu(V T n ) (p 1)(q n)/2. p,q ≥ Proof. Let V be the braid part of V T n having only virtual crossings. If n p then we p,n p,q can apply generalized Reidemeister moves to V such that it becomes V − . Therefore, p,n p,n p we can assume that n < p. Assume that there exists a chord c such that i(c) = 0. Choose γ such that it does not 1 contain the vertical strand a . This is possible by exchanging the roles of γ and γ if 1 1 2 necessary. Observing how γ passes though V , we have two cases: 1 p,n ∈ { } (1) When γ passes a vertical strand a , for some i 1, . . . , n , the contribution on 1 i the virtual crossings on a to i (γ , γ ) by virtualizations from the closed positive − i o 1 2 braid D is p 1. The strand containing a in the braid part V also has the p,q − i p,n contribution n 1 to i (γ , γ ) since it has one crossing with each of the other (cid:54) u 1 2 vertical strands a , j = i, in V . j p,n (2) When γ passes though V horizontally, the contribution to i (γ , γ ) is n and 1 p,n u 1 2 there is no contribution to i (γ , γ ). o 1 2 Let N and N be the number of case (1) and (2) appearing along γ , respectively. Since 1 2 ≤ ≤ − ≤ ≤ −1 γ does not contain a , 0 N n 1. We have 1 N p n. As in the proof 1 1 1 2 of Lemma ??, we check the diﬀerence of i(c) for V T n and D(T ), where i(c) = 0 for p,q p,q D(T ) by Lemma ??. From the above observation, we have p,q | | | − | 0 = i(c) = i (γ , γ ) i (γ , γ ) o 1 2 u 1 2 | − − − | | − | = (p 1)N ((n 1)N + nN ) = pN n(N + N ) . 1 1 2 1 1 2

text: VIRTUAL

text: UNKNOTTING

text: NUMBERS

text: OF

text: CERTAIN

text: VIRTUAL

text: TORUS

text: KNOTS

text: as

text: classical

text: crossings,

text: then

text: the

text: diagram

text: is

text: a

text: closed

text: positive

text: braid

text: and

text: we

text: have

text: i(c)

text: by

text: Lemma

text: This

text: means

text: that

text: i(c)

text: for

text: (i,

text: j,

text: k)

text: is

text: calculated

text: from

text: that

text: for

text: the

text: closed

text: positive

text: braid

text: by

text: subtracting

text: the

text: signed

text: number

text: of

text: virtual

text: crossings

text: on

text: γ

text: Thus

text: we

text: have

text: i(c)

text: N

text: Proof

text: of

text: Theorem

text: As

text: mentioned

text: in

text: Remark

text: (i,

text: j,

text: k)

text: is

text: virtually

text: null-homotopic.

text: This

text: also

text: follows

text: from

text: Lemma

text: The

text: upper

text: bound

text: of

text: the

text: equality

text: in

text: Theorem

text: follows

text: from

text: Proposition

text: For

text: the

text: lower

text: bound,

text: we

text: use

text: the

text: inequality

text: in

text: Lemma

text: The

text: diagram

text: (i,

text: j,

text: k)

text: has

text: (i

text: k

text: classical

text: crossings

text: and

text: the

text: signs

text: of

text: all

text: classical

text: cro−ssings−are

text: positive.

text: Since

text: i(c)

text: for

text: any

text: chord

text: c

text: by

text: Lemma

text: we

text: have

text: m∈N

text: bm

text: (i

text: k.

text: Thus

text: the

text: lower

text: bound

text: follows

text: from

text: the

text: inequality

text: in

text: Lemma

text: Remark

text: The

text: assertion

text: in

text: Theorem

text: holds

text: in

text: case

text: pq

text: also

text: since

text: the

text: virtual

text: knot

text: diagram

text: V

text: with

text: pq

text: is

text: the

text: mirror

text: image

text: of

text: p,q

text: p

text: q

text: Further

text: discussions

text: Lemma

text: Suppose

text: that

text: p

text: and

text: q

text: are

text: coprime

text: and

text: n

text: is

text: an

text: integer

text: such

text: that

text: n

text: q.

text: If

text: p

text: and

text: n

text: are

text: coprime

text: then

text: vu(V

text: T

text: n

text: (p

text: p,q

text: Proof.

text: Let

text: V

text: be

text: the

text: braid

text: part

text: of

text: V

text: T

text: n

text: having

text: only

text: virtual

text: crossings.

text: If

text: n

text: p

text: then

text: we

text: p,n

text: p,q

text: can

text: apply

text: generalized

text: Reidemeister

text: moves

text: to

text: V

text: such

text: that

text: it

text: becomes

text: V

text: Therefore,

text: p,n

text: p,n

text: p

text: we

text: can

text: assume

text: that

text: n

text: p.

text: Assume

text: that

text: there

text: exists

text: a

text: chord

text: c

text: such

text: that

text: i(c)

text: Choose

text: γ

text: such

text: that

text: it

text: does

text: not

text: contain

text: the

text: vertical

text: strand

text: a

text: This

text: is

text: possible

text: by

text: exchanging

text: the

text: roles

text: of

text: γ

text: and

text: γ

text: if

text: necessary.

text: Observing

text: how

text: γ

text: passes

text: though

text: V

text: we

text: have

text: two

text: cases:

text: p,n

text: When

text: γ

text: passes

text: a

text: vertical

text: strand

text: a

text: for

text: some

text: i

text: n

text: the

text: contribution

text: on

text: i

text: the

text: virtual

text: crossings

text: on

text: a

text: to

text: i

text: (γ

text: γ

text: by

text: virtualizations

text: from

text: the

text: closed

text: positive

text: i

text: o

text: braid

text: D

text: is

text: p

text: The

text: strand

text: containing

text: a

text: in

text: the

text: braid

text: part

text: V

text: also

text: has

text: the

text: p,q

text: i

text: p,n

text: contribution

text: n

text: to

text: i

text: (γ

text: γ

text: since

text: it

text: has

text: one

text: crossing

text: with

text: each

text: of

text: the

text: other

text: u

text: vertical

text: strands

text: a

text: j

text: i,

text: in

text: V

text: j

text: p,n

text: When

text: γ

text: passes

text: though

text: V

text: horizontally,

text: the

text: contribution

text: to

text: i

text: (γ

text: γ

text: is

text: n

text: and

text: p,n

text: u

text: there

text: is

text: no

text: contribution

text: to

text: i

text: (γ

text: γ

text: o

text: Let

text: N

text: and

text: N

text: be

text: the

text: number

text: of

text: case

text: and

text: appearing

text: along

text: γ

text: respectively.

text: Since

text: −1

text: γ

text: does

text: not

text: contain

text: a

text: N

text: n

text: We

text: have

text: N

text: p

text: n.

text: As

text: in

text: the

text: proof

text: of

text: Lemma

text: we

text: check

text: the

text: diﬀerence

text: of

text: i(c)

text: for

text: V

text: T

text: n

text: and

text: D(T

text: where

text: i(c)

text: for

text: p,q

text: p,q

text: D(T

text: by

text: Lemma

text: From

text: the

text: above

text: observation,

text: we

text: have

text: p,q

text: i(c)

text: i

text: (γ

text: γ

text: i

text: (γ

text: γ

text: o

text: u

text: (p

text: ((n

text: nN

text: pN

text: n(N

text: N


text: VIRTUAL UNKNOTTING NUMBERS OF CERTAIN VIRTUAL TORUS KNOTS 15 (cid:88) originally from [?], that the polynomial | | u(K) = sign(n(c))t n(c) c does not depend on the choice of a Gauss diagram of K and also does not change under (cid:54) ¯ crossing changes, where the sum runs all chords c on D with n(c) = 0. This polynomial is called the u-invariant of K. In, particular, if a virtual knot K is virtually null-homotopic then u(K) = 0. We can check that the u-invariant of the virtual torus knot in Figure ?? − is t2 2t and hence it is not virtually null-homotopic. Figure 8. The closure of this braid is a virtual torus knot which is not virtually homotopic to the unknot. For the set of virtually null-homotopic virtual knots, we can deﬁne a virtual version of the Gordian complex as follows: set a vertex for each virtually null-homotopic virtual knot and connect two vertices by an edge if one can be obtained from the other by one crossing change. We then attach a 2-simplex along a triangle consisting of three edges, and ≥ continue this process m-simplices with m 3 inductively. By construction, if there is an m-simplex then the m+1 virtual knots corresponding to the vertices of the m-simplex are related by a single crossing change. The Gordian complex of classical knots is embedded in this complex canonically. However, we do not know if the embedding preserves the distance of two vertices (i.e., the minimal number of edges between two vertices). For example, we may ask the following: Is the distance from the vertex corresponding to the − − (p, q)-torus knot to the vertex of the unknot is (p 1)(q 1)/2 in the virtual Gordian complex? If this is true then our main theorem means that the virtualization from D(T ) p,q to V T 1 does not change the distance from the vertices of these knots to the vertex of the p,q unknot in the virtual Gordian complex. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VIRTUAL

text: UNKNOTTING

text: NUMBERS

text: OF

text: CERTAIN

text: VIRTUAL

text: TORUS

text: KNOTS

text: originally

text: from

text: that

text: the

text: polynomial

text: u(K)

text: sign(n(c))t

text: n(c)

text: c

text: does

text: not

text: depend

text: on

text: the

text: choice

text: of

text: a

text: Gauss

text: diagram

text: of

text: K

text: and

text: also

text: does

text: not

text: change

text: under

font: default

text: crossing

text: changes,

text: where

text: the

text: sum

text: runs

text: all

text: chords

text: c

text: on

text: D

text: with

text: n(c)

text: This

text: polynomial

text: is

text: called

text: the

text: u-invariant

text: of

text: K.

text: In,

text: particular,

text: if

text: a

text: virtual

text: knot

text: K

text: is

text: virtually

text: null-homotopic

text: then

text: u(K)

text: We

text: can

text: check

text: that

text: the

text: u-invariant

text: of

text: the

text: virtual

text: torus

text: knot

text: in

text: Figure

text: is

text: and

text: hence

text: it

text: is

text: not

text: virtually

text: null-homotopic.

text: Figure

text: The

text: closure

text: of

text: this

text: braid

text: is

text: a

text: virtual

text: torus

text: knot

text: which

text: is

text: not

text: virtually

text: homotopic

text: to

text: the

text: unknot.

text: For

text: the

text: set

text: of

text: virtually

text: null-homotopic

text: virtual

text: knots,

text: we

text: can

text: deﬁne

text: a

text: virtual

text: version

text: of

text: the

text: Gordian

text: complex

text: as

text: follows:

text: set

text: a

text: vertex

text: for

text: each

text: virtually

text: null-homotopic

text: virtual

text: knot

text: and

text: connect

text: two

text: vertices

text: by

text: an

text: edge

text: if

text: one

text: can

text: be

text: obtained

text: from

text: the

text: other

text: by

text: one

text: crossing

text: change.

text: We

text: then

text: attach

text: a

text: along

text: a

text: triangle

text: consisting

text: of

text: three

text: edges,

text: and

text: continue

text: this

text: process

text: m-simplices

text: with

text: m

text: inductively.

text: By

text: construction,

text: if

text: there

text: is

text: an

text: m-simplex

text: then

text: the

text: virtual

text: knots

text: corresponding

text: to

text: the

text: vertices

text: of

text: the

text: m-simplex

text: are

text: related

text: by

text: a

text: single

text: crossing

text: change.

text: The

text: Gordian

text: complex

text: of

text: classical

text: knots

text: is

text: embedded

text: in

text: this

text: complex

text: canonically.

text: However,

text: we

text: do

text: not

text: know

text: if

text: the

text: embedding

text: preserves

text: the

text: distance

text: of

text: two

text: vertices

text: (i.e.,

text: the

text: minimal

text: number

text: of

text: edges

text: between

text: two

text: vertices).

text: For

text: example,

text: we

text: may

text: ask

text: the

text: following:

text: Is

text: the

text: distance

text: from

text: the

text: vertex

text: corresponding

text: to

text: the

text: (p,

text: q)-torus

text: knot

text: to

text: the

text: vertex

text: of

text: the

text: unknot

text: is

text: (p

text: in

text: the

text: virtual

text: Gordian

text: complex?

text: If

text: this

text: is

text: true

text: then

text: our

text: main

text: theorem

text: means

text: that

text: the

text: virtualization

text: from

text: D(T

text: p,q

text: to

text: V

text: T

text: does

text: not

text: change

text: the

text: distance

text: from

text: the

text: vertices

text: of

text: these

text: knots

text: to

text: the

text: vertex

text: of

text: the

text: p,q

text: unknot

text: in

text: the

text: virtual

text: Gordian

text: complex.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: VIRTUAL UNKNOTTING NUMBERS OF CERTAIN VIRTUAL TORUS KNOTS 5 3. Upper bound ≤ Let i, j and k be integers such that 0 < j and 0 k < i. We denote by V B1 the virtual i,j braid diagram obtained from the braid part of the standard diagram of the (i, j)-torus link by changing all classical crossings on the left-most overstrand into virtual crossings. For · · · example, the braid in Figure ?? is V B1 . Let B be the braid given by σ σ − σ σ as 7,3 k k k 1 2 1 shown in Figure ??, where we read the braid words from the left to the right. We denote by (i, j, k) the virtual link diagram obtained as the closure of the product of V B1 and i,j B . Hereafter, we always assume that (i, j, k) is a virtual knot, that is, it has only one k link component. Figure 3. The braid B . k Remark 3.1. It is known by Masuda in [?, Lemma 5.8] that if the supporting genus of a virtual knot is at most one then it is virtually null-homotopic. The proof is done by realizing the virtual knot as a knot on a torus and straightening it in the torus. If it is homotopic to a straight line in the torus then the corresponding virtual knot is virtually homotopic to the unknot. If it is homotopic to a straight line with multiplicity m > 1 then the corresponding virtual knot is virtually homotopic to the (m, 1)-torus knot, which i : b ra id in d e x (cid:27) (cid:27) (cid:27) 1 2 k ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VIRTUAL

text: UNKNOTTING

text: NUMBERS

text: OF

text: CERTAIN

text: VIRTUAL

text: TORUS

text: KNOTS

label: section

text: Upper

text: bound

text: Let

text: i,

text: j

text: and

text: k

text: be

text: integers

text: such

text: that

text: j

text: and

text: k

text: i.

text: We

text: denote

text: by

text: V

text: the

text: virtual

text: i,j

text: braid

text: diagram

text: obtained

text: from

text: the

text: braid

text: part

text: of

text: the

text: standard

text: diagram

text: of

text: the

text: (i,

text: j)-torus

text: link

text: by

text: changing

text: all

text: classical

text: crossings

text: on

text: the

text: left-most

text: overstrand

text: into

text: virtual

text: crossings.

text: For

text: example,

text: the

text: braid

text: in

text: Figure

text: is

text: V

text: Let

text: B

text: be

text: the

text: braid

text: given

text: by

text: σ

text: σ

text: σ

text: σ

text: as

text: k

text: k

text: k

text: shown

text: in

text: Figure

text: where

text: we

text: read

text: the

text: braid

text: words

text: from

text: the

text: left

text: to

text: the

text: right.

text: We

text: denote

text: by

text: (i,

text: j,

text: k)

text: the

text: virtual

text: link

text: diagram

text: obtained

text: as

text: the

text: closure

text: of

text: the

text: product

text: of

text: V

text: and

text: i,j

text: B

text: Hereafter,

text: we

text: always

text: assume

text: that

text: (i,

text: j,

text: k)

text: is

text: a

text: virtual

text: knot,

text: that

text: is,

text: it

text: has

text: only

text: one

text: k

text: link

text: component.

text: Figure

text: The

text: braid

text: B

text: k

text: Remark

text: It

text: is

text: known

text: by

text: Masuda

text: in

text: Lemma

text: that

text: if

text: the

text: supporting

text: genus

text: of

text: a

text: virtual

text: knot

text: is

text: at

text: most

text: one

text: then

text: it

text: is

text: virtually

text: null-homotopic.

text: The

text: proof

text: is

text: done

text: by

text: realizing

text: the

text: virtual

text: knot

text: as

text: a

text: knot

text: on

text: a

text: torus

text: and

text: straightening

text: it

text: in

text: the

text: torus.

text: If

text: it

text: is

text: homotopic

text: to

text: a

text: straight

text: line

text: in

text: the

text: torus

text: then

text: the

text: corresponding

text: virtual

text: knot

text: is

text: virtually

text: homotopic

text: to

text: the

text: unknot.

text: If

text: it

text: is

text: homotopic

text: to

text: a

text: straight

text: line

text: with

text: multiplicity

text: m

text: then

text: the

text: corresponding

text: virtual

text: knot

text: is

text: virtually

text: homotopic

text: to

text: the

text: (m,

text: knot,

text: which

text: i

text: b

text: ra

text: id

text: in

text: d

text: e

text: x

text: k

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 3.0 Divergence angle 16 Number at = 00 2.5 12 2.0 8 1.5 4 1.0 0 0.5 0 5 10 15 20 25 Capillary length ( ) 0 FIG. 9: (Color online) The divergence angle θ (black curve) and the number density locating the peak (red curve) as the function of the length d of the capillary, respectively. 0 such as d = 20λ and 25λ , the monoenergetic peak are approximately coincident at early 0 0 0 time, however, at later time , see Fig. 10(d), comparing the cases of d = 20λ with the 0 0 length d = 25λ , the monoenergetic peak in both of energy value and number density are 0 0 better for the former case. This is not surprising because that the longer capillary would make the longitudinal electric ﬁeld E decreasing, which would result in the decrease of x the total conversion eﬃciency from the laser to protons. Overall, the extensive simulation results indicate an optimal proper capillary length existence, e.g. d = 20λ shown in Fig. 0 0 10, by which one can achieve a high quality proton source even in longer time. IV. SUMMARY In summary, we have investigated the enhanced proton acceleration in the cone-capillary by using the 2D3V PIC simulations performance. Compared with the cone without the capillary, our results show that the protons can be accelerated and guided to a farther distance by using the cone-capillary. This enhancement has been analyzed and discussed 12 )eerged( elgna ecnegreviD Number at = 0 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Divergence

text: angle

text: Number

text: at

text: Capillary

text: length

text: FIG.

text: (Color

text: online)

text: The

text: divergence

text: angle

text: θ

text: (black

text: curve)

text: and

text: the

text: number

text: density

text: locating

text: the

text: peak

text: (red

text: curve)

text: as

text: the

text: function

text: of

text: the

text: length

text: d

text: of

text: the

text: capillary,

text: respectively.

text: such

text: as

text: d

text: and

text: the

text: monoenergetic

text: peak

text: are

text: approximately

text: coincident

text: at

text: early

text: time,

text: however,

text: at

text: later

text: time

text: see

text: Fig.

text: comparing

text: the

text: cases

text: of

text: d

text: with

text: the

text: length

text: d

text: the

text: monoenergetic

text: peak

text: in

text: both

text: of

text: energy

text: value

text: and

text: number

text: density

text: are

text: better

text: for

text: the

text: former

text: case.

text: This

text: is

text: not

text: surprising

text: because

text: that

text: the

text: longer

text: capillary

text: would

text: make

text: the

text: longitudinal

text: electric

text: ﬁeld

text: E

text: decreasing,

text: which

text: would

text: result

text: in

text: the

text: decrease

text: of

text: x

text: the

text: total

text: conversion

text: eﬃciency

text: from

text: the

text: laser

text: to

text: protons.

text: Overall,

text: the

text: extensive

text: simulation

text: results

text: indicate

text: an

text: optimal

text: proper

text: capillary

text: length

text: existence,

text: e.g.

text: d

text: shown

text: in

text: Fig.

text: by

text: which

text: one

text: can

text: achieve

text: a

text: high

text: quality

text: proton

text: source

text: even

text: in

text: longer

text: time.

text: IV.

text: SUMMARY

text: In

text: summary,

text: we

text: have

text: investigated

text: the

text: enhanced

text: proton

text: acceleration

text: in

text: the

text: cone-capillary

text: by

text: using

text: the

text: PIC

text: simulations

text: performance.

text: Compared

text: with

text: the

text: cone

text: without

text: the

text: capillary,

text: our

text: results

text: show

text: that

text: the

text: protons

text: can

text: be

text: accelerated

text: and

text: guided

text: to

text: a

text: farther

text: distance

text: by

text: using

text: the

text: cone-capillary.

text: This

text: enhancement

text: has

text: been

text: analyzed

text: and

text: discussed

text: )eerged(

text: elgna

text: ecnegreviD

text: Number

text: at

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##