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text: and

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text: Wellner

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text: Wooldridge, J. M. (2010). Econometric analysis of cross section and panel data. MIT press. Yuan, M. (2010). High dimensional inverse covariance matrix estimation via linear program- ming. Journal of Machine Learning Research. Zhang, C.-H. and S. S. Zhang (2014). Conﬁdence intervals for low dimensional parameters in high dimensional linear models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76(1), 217–242. 55

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text: Learning

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text: C.-H.

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text: (Statistical

text: Methodology)

label: paragraph


text: Assumption 3. There exist positive constants C and K such that P | | ≥ ≤ − ≥ (a) ε are uniformly subgaussian; that is, ( ε (cid:15)) 1Ke C(cid:15)2 for every (cid:15) 0, i = 1, . . . , N i,t i,t 2 and t = 1, . . . , T . P | | ≥ ≤ − ≥ (b) z are uniformly subgaussian; that is, ( z (cid:15)) 1Ke C(cid:15)2 for every (cid:15) 0, i = i,t,l i,t,l 2 1, . . . , N, t = 1, . . . , T and l = 1, . . . , p. In the context of the plain static regression model it is common practice to assume the error terms as well as the covariates to be subgaussian. However, this assumption is not as innocent in the context of the dynamic panel data model (1.1) as y is generated by the model and i,t its properties are thus completely determined by those of x , ε as well as the parameters of i,t i,t the model. Lemma 2 in Appendix A shows that y is subgaussian if x and ε satisfy this i,t i,t i,t property and the parameters are well-behaved. In particular, a wide class of (causal) stationary processes are included. Note also, that Assumption 3 imposes subgaussianity of the initial values y , ..., y − for all i = 1, ..., N. Caner and Kock (2014) have derived results similar to ours i,0 i,1 L in a cross sectional setting without the sub-gaussianity assumption. However, the dimension of their model can not increase as fast as here. 2.4 The Or(cid:8)acle Inequalities(cid:9) (cid:8) (cid:80) (cid:9) With the above assumptions in place we are ready to state our ﬁrst result. Deﬁning F(cid:112) ∈ R (cid:107) (cid:107) ≤ × ∈ R | | ≤ (s , ν, E) := α p : α s η N : N η ν E , one has 1 0 1 i=1 i Theorem 1 (Oracle inequalities). Let Assumptions 1 - 3 hold. Then, choosing λ = ∨ N 4MNT (log(p N))3 for some M > 0, the follow(cid:32)ing inequalities are valid with proba(cid:33)bility (cid:26) (cid:27) (cid:104) (cid:16) (cid:17) (cid:105) at least − − − − − − √λ −ν 2 1/3 1 Ap1 BM1/3 AN1 BM1/3 A(p2 +(cid:16)pN) ex(cid:17)p B N/ s1 + E N NT √ − for positive consta(cid:13)nts A and(cid:13)B and s1 + E √λ(cid:16)N ν (cid:46) (cid:17) N, (cid:18) (cid:19) (cid:13) (cid:13) NT − 1 Π(γˆ − γ) 2 ≤ 120(cid:16)λ2Ns1 + 1(cid:17)20 + 20(cid:18)√ λN (cid:19) E √λN 1 ν NT κ2(NT )2 κ2 NNT NT 2 2 − (cid:107) − (cid:107) ≤ 120λNs1 (cid:16) 120 (cid:17) √1(cid:18) √(cid:19)λN 1 ν αˆ α + + 20 E 1 κ2NT κ2 N NT 2 2 − (cid:107) − (cid:107) ≤ 120√λ s 120 √λ 1 ν N 1 N ηˆ η + + 20 E . 1 κ2 NT κ2 NT 2 2 F Moreover, the above bounds are valid uniformly over (s , ν, E). 1 Theorem 1 provides oracle inequalities for the prediction error as well as the estimation error of the parameter vectors. While these bounds are of independent interest, we primarily use them as means towards our ultimate end of conducting (joint) inference on α and η. We stress that the bounds in Theorem 1 are ﬁnite sample bounds; they hold for any ﬁxed values of N and T . The novel feature of our oracle inequalities is that E, the ”size” of η, is allowed (cid:107) − (cid:107) to grow even when we want the upper bound of ηˆ η go to zero. The special case of exact 1 sparsity of η corresponds to ν = 0 and E being the sparsity index, say s , of η. 2 We also note that the oracle inequalities are not obtained in an entirely standard manner as the mixture of exact and weak sparsity in dynamic panel data models calls for a diﬀerent proof technique which yields the upper bounds as solutions to certain quadratic equations. Furthermore, we remark that in analogy to oracle inequalities in the plain linear regression 7 ##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: Assumption

text: There

text: exist

text: positive

text: constants

text: C

text: and

text: K

text: such

text: that

text: P

label: list

text: (a)

text: ε

text: are

text: uniformly

text: subgaussian;

text: that

text: is,

text: ε

text: for

text: every

text: i

text: N

text: i,t

text: i,t

text: and

text: t

text: T

text: P

text: (b)

text: z

text: are

text: uniformly

text: subgaussian;

text: that

text: is,

text: z

text: for

text: every

text: i

text: i,t,l

text: i,t,l

text: N,

text: t

text: T

text: and

text: l

text: p.

text: In

text: the

text: context

text: of

text: the

text: plain

text: static

text: regression

text: model

text: it

text: is

text: common

text: practice

text: to

text: assume

text: the

text: error

text: terms

text: as

text: well

text: as

text: the

text: covariates

text: to

text: be

text: subgaussian.

text: However,

text: this

text: assumption

text: is

text: not

text: as

text: innocent

text: in

text: the

text: context

text: of

text: the

text: dynamic

text: panel

text: data

text: model

text: as

text: y

text: is

text: generated

text: by

text: the

text: model

text: and

text: i,t

text: its

text: properties

text: are

text: thus

text: completely

text: determined

text: by

text: those

text: of

text: x

text: ε

text: as

text: well

text: as

text: the

text: parameters

text: of

text: i,t

text: i,t

text: the

text: model.

text: Lemma

text: in

text: Appendix

text: A

text: shows

text: that

text: y

text: is

text: subgaussian

text: if

text: x

text: and

text: ε

text: satisfy

text: this

text: i,t

text: i,t

text: i,t

text: property

text: and

text: the

text: parameters

text: are

text: well-behaved.

text: In

text: particular,

text: a

text: wide

text: class

text: of

text: (causal)

text: stationary

text: processes

text: are

text: included.

text: Note

text: also,

text: that

text: Assumption

text: imposes

text: subgaussianity

text: of

text: the

text: initial

text: values

text: y

text: y

text: for

text: all

text: i

text: N.

text: Caner

text: and

text: Kock

text: have

text: derived

text: results

text: similar

text: to

text: ours

text: L

text: in

text: a

text: cross

text: sectional

text: setting

text: without

text: the

text: sub-gaussianity

text: assumption.

text: However,

text: the

text: dimension

text: of

text: their

text: model

text: can

text: not

text: increase

text: as

text: fast

text: as

text: here.

text: The

text: With

text: the

text: above

text: assumptions

text: in

text: place

text: we

text: are

text: ready

text: to

text: state

text: our

text: ﬁrst

text: result.

text: Deﬁning

text: R

text: R

text: (s

text: ν,

text: E)

text: α

text: p

text: α

text: s

text: η

text: N

text: N

text: η

text: ν

text: E

text: one

text: has

text: i

text: Theorem

text: (Oracle

text: inequalities).

text: Let

text: Assumptions

text: hold.

text: Then,

text: choosing

text: λ

text: N

text: (log(p

text: for

text: some

text: M

text: the

text: inequalities

text: are

text: valid

text: with

text: at

text: least

text: √λ

text: −ν

text: B

text: N/

text: E

text: N

text: NT

text: for

text: positive

text: A

text: and

text: E

text: ν

text: N,

text: NT

text: Π(γˆ

text: γ)

text: λN

text: E

text: √λN

text: ν

text: NT

text: κ2

text: NNT

text: NT

text: ν

text: αˆ

text: α

text: E

text: κ2

text: N

text: NT

text: s

text: √λ

text: ν

text: N

text: N

text: ηˆ

text: η

text: E

text: κ2

text: NT

text: κ2

text: NT

text: F

text: Moreover,

text: the

text: above

text: bounds

text: are

text: valid

text: uniformly

text: over

text: (s

text: ν,

text: E).

text: Theorem

text: provides

text: oracle

text: inequalities

text: for

text: the

text: prediction

text: error

text: as

text: well

text: as

text: the

text: estimation

text: error

text: of

text: the

text: parameter

text: vectors.

text: While

text: these

text: bounds

text: are

text: of

text: independent

text: interest,

text: we

text: primarily

text: use

text: them

text: as

text: means

text: towards

text: our

text: ultimate

text: end

text: of

text: conducting

text: (joint)

text: inference

text: on

text: α

text: and

text: η.

text: We

text: stress

text: that

text: the

text: bounds

text: in

text: Theorem

text: are

text: ﬁnite

text: sample

text: bounds;

text: they

text: hold

text: for

text: any

text: ﬁxed

text: values

text: of

text: N

text: and

text: T

text: The

text: novel

text: feature

text: of

text: our

text: oracle

text: inequalities

text: is

text: that

text: E,

text: the

text: ”size”

text: of

text: η,

text: is

text: allowed

text: to

text: grow

text: even

text: when

text: we

text: want

text: the

text: upper

text: bound

text: of

text: ηˆ

text: η

text: go

text: to

text: zero.

text: The

text: special

text: case

text: of

text: exact

text: sparsity

text: of

text: η

text: corresponds

text: to

text: ν

text: and

text: E

text: being

text: the

text: sparsity

text: index,

text: say

text: s

text: of

text: η.

text: We

text: also

text: note

text: that

text: the

text: oracle

text: inequalities

text: are

text: not

text: obtained

text: in

text: an

text: entirely

text: standard

text: manner

text: as

text: the

text: mixture

text: of

text: exact

text: and

text: weak

text: sparsity

text: in

text: dynamic

text: panel

text: data

text: models

text: calls

text: for

text: a

text: diﬀerent

text: proof

text: technique

text: which

text: yields

text: the

text: upper

text: bounds

text: as

text: solutions

text: to

text: certain

text: quadratic

text: equations.

text: Furthermore,

text: we

text: remark

text: that

text: in

text: analogy

text: to

text: oracle

text: inequalities

text: in

text: the

text: plain

text: linear

text: regression

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: The new estimator γ˜ is no longer sparse as it has added a bias correction terms to the sparse Lasso estimator γˆ. Therefore, we will also refer to it as the desparsiﬁed Lasso estimator in the dynamic panel context. − − (cid:48) − S (γ˜ γ) = Θˆ S 1Π ε ∆, (3.2) × (cid:107) (cid:107) For any (p + N) 1 vector ρ with ρ = 1 we shall study the asymptotic behaviour of (cid:48) − (cid:48) − (cid:48) − (cid:48) ρ S (γ˜ γ) = ρ Θˆ S 1Π ε ρ ∆. (3.3) (cid:48) − (cid:48) (cid:48) A central limit theorem for ρ Θˆ S 1Π ε as well as asymptotic negligibility of ρ ∆ will yield asymptotically gaussian inference. Furthermore, we shall provide a uniformly consistent estima- (cid:48) − (cid:48) tor of the asymptotic variance of ρ Θˆ S 1Π ε even in the presence of conditional heteroskedasic- ity. A leading special case of (3.3) is when one is only interested in the asymptotic distribution R of γ˜ corresponding to ρ = e being the jth basis vector of p+N. In general, we will be in- j j ⊆ { } terested in the asymptotic distribution of a subset H 1, ..., p + N of the indices of γ with cardinality h and shall show that asymptotically honest (uniformly valid) gaussian inference is → ∞ possible in the presence of heteroskedasticity even for h and H simultaneously involving elements of α and η. ˆ 3.2 Construction of Θ (cid:32) (cid:33) As is clear from the discussion above we need a good choice for Θˆ . In particular we shall show that Θˆ 0 Θˆ = Z 0 I N works well. Here Θˆ will be constructed using nodewise regressions as in van de Geer et al. Z (2014) and we show that this is possible even when the rows of Z are not independent and identically distributed. The construction of Θˆ parallels the one in van de Geer et al. (2014) Z − to a high extent but importantly for our context we do not need the rows of Ψ 1 to be sparse Z for the nodewise regressions to work well. We will discuss the importance of this, once we have (cid:26) (cid:27) properly constructed Θˆ . First, deﬁne Z 1 (cid:107) − (cid:107) (cid:107) (cid:107) φˆ = argmin z Z− δ 2 + 2λ δ , j = 1, ..., p, (3.4) j ∈R − NT j j node 1 δ p 1 × − where z is the jth column of Z, Z− is the NT (p 1) submatrix of Z with Z’s jth j − × j { (cid:54) } column removed, and the (p 1) 1 vector φˆ = φˆ : l = 1, . . . , p, l = j . Thus, φˆ is the  j j,l  j Lasso estimator resulting from regressing z on Z− . Next, deﬁne j j    − · · · −   −φˆ1 φˆ11,2 · · · −φφˆˆ1,p  Cˆ = ...2,1 ... ... ...2,p − − · · · φˆ φˆ 1 p,1 p,2 (cid:107) − (cid:107) (cid:107) (cid:107) and τˆ2 = 1 z Z− φˆ 2 + λ φˆ as well as Tˆ2 = diag(τˆ2, . . . , τˆ2). Finally, we set j − NT j j j node j 1 1 p Θˆ = Tˆ 2Cˆ. Let Cˆ denote the jth row of Cˆ and let Θˆ denote the jth row of Θˆ but both Z × j Z,j Z written as a p 1 vectors. Then, Θˆ = Cˆ /τˆ2. For any j = 1, ..., p, the KKT condition for a Z,j j j minimum in (3.4) are 9 ##LTLine## ##LTLine##

text: The

text: new

text: estimator

text: γ˜

text: is

text: no

text: longer

text: sparse

text: as

text: it

text: has

text: added

text: a

text: bias

text: correction

text: terms

text: to

text: the

text: sparse

text: Lasso

text: estimator

text: γˆ.

text: Therefore,

text: we

text: will

text: also

text: refer

text: to

text: it

text: as

text: the

text: desparsiﬁed

text: Lasso

text: estimator

text: in

text: the

text: dynamic

text: panel

text: context.

label: equation

text: S

text: (γ˜

text: γ)

text: Θˆ

text: S

text: 1Π

text: ε

text: For

text: any

text: (p

text: N)

text: vector

text: ρ

text: with

text: ρ

text: we

text: shall

text: study

text: the

text: asymptotic

text: behaviour

text: of

text: ρ

text: S

text: (γ˜

text: γ)

text: ρ

text: Θˆ

text: S

text: 1Π

text: ε

text: ρ

text: A

text: central

text: limit

text: theorem

text: for

text: ρ

text: Θˆ

text: S

text: 1Π

text: ε

text: as

text: well

text: as

text: asymptotic

text: negligibility

text: of

text: ρ

text: will

text: yield

text: asymptotically

text: gaussian

text: inference.

text: Furthermore,

text: we

text: shall

text: provide

text: a

text: uniformly

text: consistent

text: estima-

text: tor

text: of

text: the

text: asymptotic

text: variance

text: of

text: ρ

text: Θˆ

text: S

text: 1Π

text: ε

text: even

text: in

text: the

text: presence

text: of

text: conditional

text: heteroskedasic-

text: ity.

text: A

text: leading

text: special

text: case

text: of

text: is

text: when

text: one

text: is

text: only

text: interested

text: in

text: the

text: asymptotic

text: distribution

text: R

text: of

text: γ˜

text: corresponding

text: to

text: ρ

text: e

text: being

text: the

text: jth

text: basis

text: vector

text: of

text: p+N.

text: In

text: general,

text: we

text: will

text: be

text: in-

text: j

text: j

text: terested

text: in

text: the

text: asymptotic

text: distribution

text: of

text: a

text: subset

text: H

text: p

text: N

text: of

text: the

text: indices

text: of

text: γ

text: with

text: cardinality

text: h

text: and

text: shall

text: show

text: that

text: asymptotically

text: honest

text: (uniformly

text: valid)

text: gaussian

text: inference

text: is

text: possible

text: in

text: the

text: presence

text: of

text: heteroskedasticity

text: even

text: for

text: h

text: and

text: H

text: simultaneously

text: involving

text: elements

text: of

text: α

text: and

text: η.

text: ˆ

text: Construction

text: of

text: Θ

text: As

text: is

text: clear

text: from

text: the

text: discussion

text: above

text: we

text: need

text: a

text: good

text: choice

text: for

text: Θˆ

text: In

text: particular

text: we

text: shall

text: show

text: that

text: Θˆ

text: Θˆ

text: Z

text: I

text: N

text: works

text: well.

text: Here

text: Θˆ

text: will

text: be

text: constructed

text: using

text: nodewise

text: regressions

text: as

text: in

text: van

text: de

text: Geer

text: et

text: al.

text: Z

text: and

text: we

text: show

text: that

text: this

text: is

text: possible

text: even

text: when

text: the

text: rows

text: of

text: Z

text: are

text: not

text: independent

text: and

text: identically

text: distributed.

text: The

text: construction

text: of

text: Θˆ

text: parallels

text: the

text: one

text: in

text: van

text: de

text: Geer

text: et

text: al.

text: Z

text: to

text: a

text: high

text: extent

text: but

text: importantly

text: for

text: our

text: context

text: we

text: do

text: not

text: need

text: the

text: rows

text: of

text: Ψ

text: to

text: be

text: sparse

text: Z

text: for

text: the

text: nodewise

text: regressions

text: to

text: work

text: well.

text: We

text: will

text: discuss

text: the

text: importance

text: of

text: this,

text: once

text: we

text: have

text: properly

text: constructed

text: Θˆ

text: First,

text: deﬁne

text: Z

text: φˆ

text: argmin

text: z

text: Z−

text: δ

text: 2λ

text: δ

text: j

text: p,

text: j

text: ∈R

text: NT

text: j

text: j

text: node

text: δ

text: p

text: where

text: z

text: is

text: the

text: jth

text: column

text: of

text: Z,

text: Z−

text: is

text: the

text: NT

text: (p

text: submatrix

text: of

text: Z

text: with

text: Z’s

text: jth

text: j

text: j

text: column

text: removed,

text: and

text: the

text: (p

text: vector

text: φˆ

text: φˆ

text: l

text: p,

text: l

text: j

text: Thus,

text: φˆ

text: is

text: the

text: j

text: j,l

text: j

text: Lasso

text: estimator

text: resulting

text: from

text: regressing

text: z

text: on

text: Z−

text: Next,

text: deﬁne

text: j

text: j

text: −φˆ1

text: −φφˆˆ1,p

text: Cˆ

text: φˆ

text: φˆ

text: and

text: τˆ2

text: z

text: Z−

text: φˆ

text: λ

text: φˆ

text: as

text: well

text: as

text: Finally,

text: we

text: set

text: j

text: NT

text: j

text: j

text: j

text: node

text: j

text: p

text: Θˆ

text: Tˆ

text: Let

text: Cˆ

text: denote

text: the

text: jth

text: row

text: of

text: Cˆ

text: and

text: let

text: Θˆ

text: denote

text: the

text: jth

text: row

text: of

text: Θˆ

text: but

text: both

text: Z

text: j

text: Z,j

text: Z

text: written

text: as

text: a

text: p

text: vectors.

text: Then,

text: Θˆ

text: Cˆ

text: For

text: any

text: j

text: p,

text: the

text: KKT

text: condition

text: for

text: a

text: Z,j

text: j

text: j

text: minimum

text: in

text: are

text: ##LTLine##

text: ##LTLine##


text: − 1 (cid:48) − Z− (z Z− φˆ ) + λ w = 0, (3.5) NT j j j j node j (cid:107) (cid:107) where w is the subdiﬀerential of x evaluated at φˆ . Using this, the deﬁnition of τˆ , and (cid:48) (cid:107) (cid:107)j 1 j j φˆ w = φˆ yields j j j 1 1 − (cid:48) − (cid:107) (cid:107) 1 − (cid:48) τˆ2 = (z Z− φˆ ) (z Z− φˆ ) + λ φˆ = (z Z− φˆ ) z . (3.6) j NT j j j j j j node j 1 NT j j j j Thus, by the deﬁnition of Θˆ , and as τˆ2 is bounded away from zero (we shall later argue Z,j j rigorously for this) 1 (cid:48) z ZΘˆ = 1. (3.7) NT j Z,j Furthermore, the KKT conditions (3.5) can also be written as 1 (cid:48) − Z− (z Z− φˆ ) = λ w , (3.8) NT j j j j node j (cid:13) (cid:13) (cid:48) which implies N1T Z−jZΘˆ Z,j =(cid:13)(cid:13)(cid:13) λnodewj/τˆj2. Com(cid:13)(cid:13)(cid:13)bining with (3.7) yields 1 (cid:48) − ≤ λ Z ZΘˆ e node , (3.9) NT Z,j j ∞ τˆ2 j (cid:107) − (cid:107) which together with an oracle inequality for γˆ γ provides an upper bound on the 1 jth entry of ∆ in (3.3). In other words, (3.9) will be used to show the required asymptotic (cid:48) negligibility of ρ ∆ in (3.3) by arguments made rigorous in the appendix. 3.3 Asymptotic Properties of the Approximate Inverse (cid:48) − (cid:48) In order to show that ρ Θˆ S 1(cid:32)Π ε is asym(cid:33)ptotic(cid:32)ally gaussian(cid:33)one needs to understand the limiting behaviour of Θˆ constructed above. We show that Θˆ is close to − Θ 0 Ψ 1 0 Θ = Z := Z 0 I 0 I N N in an appro(cid:104)priate sense. To this end, note (cid:105)that by Yuan (2010) − − − 1 − − Θ = Ψ Ψ − Ψ 1− − Ψ − and Θ − = Θ Ψ − Ψ 1− − , (3.10) Z,j,j Z,j,j Z,j, j Z, j, j Z, j,j Z,j, j Z,j,j Z,j, j Z, j, j × − where Θ is the jth diagonal entry of Θ , Θ − is the 1 (p 1) vector obtained by Z,j,j Z Z,j, j removing the jth entry of the jth row of Θ , Ψ − − is the submatrix of Ψ with the jth row Z Z, j, j Z and column removed, Ψ − is the jth row of Ψ with its jth entry removed, Ψ − is the jth Z,j, j Z Z, j,j column of Ψ with its jth entry removed. Next, let z be the jth element of z and z − Z − × i,t,j i,t i,t, j be all elements except the jth. Deﬁne the (p(cid:88) 1(cid:88)) 1 vector N T 1 E − (cid:48) φ := argmin [z z − δ]2 j NT i,t,j i,t, j δ i=1 t=1 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Z−

text: (z

text: Z−

text: φˆ

text: λ

text: w

text: NT

text: j

text: j

text: j

text: j

text: node

text: j

text: where

text: w

text: is

text: the

text: subdiﬀerential

text: of

text: x

text: evaluated

text: at

text: φˆ

text: Using

text: this,

text: the

text: deﬁnition

text: of

text: τˆ

text: and

text: j

text: j

text: φˆ

text: w

text: φˆ

text: yields

text: j

text: j

text: j

text: τˆ2

text: (z

text: Z−

text: φˆ

text: (z

text: Z−

text: φˆ

text: λ

text: φˆ

text: (z

text: Z−

text: φˆ

text: z

text: j

text: NT

text: j

text: j

text: j

text: j

text: j

text: j

text: node

text: j

text: NT

text: j

text: j

text: j

text: j

text: Thus,

text: by

text: the

text: deﬁnition

text: of

text: Θˆ

text: and

text: as

text: τˆ2

text: is

text: bounded

text: away

text: from

text: zero

text: (we

text: shall

text: later

text: argue

text: Z,j

text: j

text: rigorously

text: for

text: this)

text: z

text: ZΘˆ

text: NT

text: j

text: Z,j

text: Furthermore,

text: the

text: KKT

text: conditions

text: can

text: also

text: be

text: written

text: as

text: Z−

text: (z

text: Z−

text: φˆ

text: λ

text: w

text: NT

text: j

text: j

text: j

text: j

text: node

text: j

text: which

text: implies

text: Z−jZΘˆ

text: Z,j

text: with

text: yields

text: λ

text: Z

text: ZΘˆ

text: e

text: node

text: NT

text: Z,j

text: j

text: τˆ2

text: j

text: which

text: together

text: with

text: an

text: oracle

text: inequality

text: for

text: γˆ

text: γ

text: provides

text: an

text: upper

text: bound

text: on

text: the

text: jth

text: entry

text: of

text: in

text: In

text: other

text: words,

text: will

text: be

text: used

text: to

text: show

text: the

text: required

text: asymptotic

text: negligibility

text: of

text: ρ

text: in

text: by

text: arguments

text: made

text: rigorous

text: in

text: the

text: appendix.

text: Asymptotic

text: Properties

text: of

text: the

text: Approximate

text: Inverse

text: In

text: order

text: to

text: show

text: that

text: ρ

text: Θˆ

text: S

text: ε

text: is

text: needs

text: to

text: understand

text: the

text: limiting

text: behaviour

text: of

text: Θˆ

text: constructed

text: above.

text: We

text: show

text: that

text: Θˆ

text: is

text: close

text: to

text: Θ

text: Ψ

text: Θ

text: Z

text: Z

text: I

text: I

text: N

text: N

text: in

text: an

text: sense.

text: To

text: this

text: end,

text: note

text: by

text: Yuan

text: Θ

text: Ψ

text: Ψ

text: Ψ

text: 1−

text: Ψ

text: and

text: Θ

text: Θ

text: Ψ

text: Ψ

text: 1−

text: Z,j,j

text: Z,j,j

text: Z,j,

text: j

text: Z,

text: j,

text: j

text: Z,

text: j,j

text: Z,j,

text: j

text: Z,j,j

text: Z,j,

text: j

text: Z,

text: j,

text: j

text: where

text: Θ

text: is

text: the

text: jth

text: diagonal

text: entry

text: of

text: Θ

text: Θ

text: is

text: the

text: (p

text: vector

text: obtained

text: by

text: Z,j,j

text: Z

text: Z,j,

text: j

text: removing

text: the

text: jth

text: entry

text: of

text: the

text: jth

text: row

text: of

text: Θ

text: Ψ

text: is

text: the

text: submatrix

text: of

text: Ψ

text: with

text: the

text: jth

text: row

text: Z

text: Z,

text: j,

text: j

text: Z

text: and

text: column

text: removed,

text: Ψ

text: is

text: the

text: jth

text: row

text: of

text: Ψ

text: with

text: its

text: jth

text: entry

text: removed,

text: Ψ

text: is

text: the

text: jth

text: Z,j,

text: j

text: Z

text: Z,

text: j,j

text: column

text: of

text: Ψ

text: with

text: its

text: jth

text: entry

text: removed.

text: Next,

text: let

text: z

text: be

text: the

text: jth

text: element

text: of

text: z

text: and

text: z

text: Z

text: i,t,j

text: i,t

text: i,t,

text: j

text: be

text: all

text: elements

text: except

text: the

text: jth.

text: Deﬁne

text: the

text: vector

text: N

text: T

text: E

text: φ

text: argmin

text: [z

text: z

text: j

text: NT

text: i,t,j

text: i,t,

text: j

text: δ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: standard deviation σ of each curve ﬁtting parameter was then estimated from the 95% conﬁdence interval CI as σ ≈ CI/4. Finally, these standard deviations were used with standard error propagation formulas to estimate the uncertainty of the motility parameters and behavioral mode. Calculation of Eﬀective Diﬀusivity, D eﬀ To estimate the eﬀective diﬀusivity D , we ﬁt the mean-squared displace- eﬀ ment (cid:104)[∆x(τ)]2(cid:105) over the diﬀusive regime. For this purpose, we deﬁned the diﬀusive regime as the time-lag interval after which the normalized velocity autocorrelation C (τ)/C (0) decayed to 0.1. We note that in some cases (es- (cid:126)v (cid:126)v pecially for fast-moving strains such as CB4856, JU775 and sjh2) the ﬁt to (cid:104)[∆x(τ)]2(cid:105) = 4D τ in this regime was poor due to boundary eﬀects arising eﬀ from the ﬁnite size of the behavioral arena. For these strains, D should be eﬀ regarded as a lower bound for the true diﬀusivity. Reversal Analysis The reversal state was assigned as described in the main text using the ∆ψ(t) variable. Runs were identiﬁed from contiguous segments of forward or reverse motion. Frames with missing postural data were marked as ambiguous ends of runs. Assuming a stochastic telegraph process that generates states ∆ψ = 0 (forward) and ∆ψ = π (reverse) with probabilities 1−f and f , respectively, rev rev the autocorrelation at long time lags is C (τ → ∞) = (1 − 2f )2. For the ∆ψ rev proposed telegraph process, each state has an exponen(cid:0)tially distri(cid:1)buted lifetime τ rev (τ , τ ) and therefore f = . The expected correlation timescale fwd rev rev τ + τ for the mixture of the two statersevis g(fτ(cid:20)wredv, τfwd) = τ(cid:21)f−w1d + τr−ev1 −1. The ∆ψ autocorrelation function was therefore ﬁt to τ C (τ) = [1 − f(τ , τ )] exp − + f(τ , τ ) (7) ∆ψ rev fwd rev fwd g(τ , τ ) rev fwd (cid:112) τ − τ where f(τ , τ ) = ( fwd rev )2. The fraction of time spent reversing rev fwd τ + τ fwd rev is: f = 0.5 − f(τ , τ )/4, where f ∈ [0, 0.5]. The transition time rev rev fwd rev g(τ , τ ) g(τ , τ ) rev fwd rev fwd constants are then τ = and τ = . rev 1 − f fwd f rev rev Speed Analysis Transitions between forward and reverse runs were ignored in the speed data by excluding data points within 1 s of the start or end of a run. The speed set point µ was ﬁt by taking the mean. The remaining parameters s of the speed dynamics (3) were ﬁt by its analytical autocorrelation function: C (τ) = D τ exp (−τ/τ ). s s s s Orientation Analysis Drift in the body bearing (ψ(t)) was identiﬁed by ﬁtting the linear trend in the unwrapped bearing over 100 s windows, ψ(t) = k t + ψ . The average ψ 0 absolute value of the drift rate, (cid:104)|k |(cid:105), across the ensemble of windows was used ψ 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: standard

text: deviation

text: σ

text: of

text: each

text: curve

text: ﬁtting

text: parameter

text: was

text: then

text: estimated

text: from

text: the

text: conﬁdence

text: interval

text: CI

text: as

text: σ

text: Finally,

text: these

text: standard

text: deviations

text: were

text: used

text: with

text: standard

text: error

text: propagation

text: formulas

text: to

text: estimate

text: the

text: uncertainty

text: of

text: the

text: motility

text: parameters

text: and

text: behavioral

text: mode.

text: Calculation

label: section

text: of

text: Eﬀective

text: Diﬀusivity,

text: D

text: eﬀ

text: To

text: estimate

text: the

text: eﬀective

text: diﬀusivity

text: D

text: we

text: ﬁt

text: the

text: mean-squared

text: displace-

text: eﬀ

text: ment

text: over

text: the

text: diﬀusive

text: regime.

text: For

text: this

text: purpose,

text: we

text: deﬁned

text: the

text: diﬀusive

text: regime

text: as

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text: [34] Culotti, J. G & Russell, R. L. (1978) Osmotic avoidance defective mutants of the nematode Caenorhabditis elegans. Genetics 90, 243–56. [35] Green, M. S. (1954) Markoﬀ Random Processes and the Statistical Me- chanics of Time-Dependent Phenomena. II. Irreversible Processes in Fluids. J. Chem. Phys. 22, 398. [36] Kubo, R. (1957) Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. J. Phys. Soc. Japan 12, 570–586. [37] P. Murphy, K. (2012) Machine Learning: A Probabilistic Perspective. (The MIT Press, Cambridge, MA). [38] Osborne, K. a, Robichon, A, Burgess, E, Butland, S, Shaw, R. A, Coulthard, A, Pereira, H. S, Greenspan, R. J, & Sokolowski, M. B. (1997) Natural behavior polymorphism due to a cGMP-dependent protein kinase of Drosophila. Science 277, 834–6. [39] Jordan, D, Kuehn, S, Katifori, E, & Leibler, S. (2013) Behavioral diversity in microbes and low-dimensional phenotypic spaces. Proc. Natl. Acad. Sci. U. S. A. 110, 14018–23. [40] Slatkin, M. (1974) Hedging one’s evolutionary bets. Nature 250, 704–705. [41] Philippi, T & Seger, J. (1989) Hedging one’s evolutionary bets, revisited. Trends Ecol. Evol. 4, 41–44. [42] Kiontke, K & Fitch, D. H. (2013) Nematodes. Curr. Biol. 23, R862–R864. [43] Flavell, S. W, Pokala, N, Macosko, E. Z, Albrecht, D. R, Larsch, J, & Bargmann, C. I. (2013) Serotonin and the neuropeptide PDF initiate and extend opposing behavioral states in C. elegans. Cell 154, 1023–1035. [44] Stephens, G. J, Johnson-Kerner, B, Bialek, W, & Ryu, W. S. (2008) Di- mensionality and dynamics in the behavior of C. elegans. PLoS Comput. Biol. 4, e1000028. [45] Brown, A. E. X, Yemini, E. I, Grundy, L. J, Jucikas, T, & Schafer, W. R. (2013) A dictionary of behavioral motifs reveals clusters of genes aﬀecting Caenorhabditis elegans locomotion. Proc. Natl. Acad. Sci. U. S. A. 110, 791–6. [46] Davies, N. B, Krebs, J. R, & West, S. A. (2012) An Introduction to Be- havioural Ecology. (Wiley-Blackwell), 4th edition. [47] Reaume, C. J & Sokolowski, M. B. (2009) cGMP-dependent protein kinase as a modiﬁer of behaviour. Handb. Exp. Pharmacol. 191, 423–43. [48] Kaun, K. R & Sokolowski, M. B. (2009) cGMP-dependent protein kinase: linking foraging to energy homeostasis. Genome 52, 1–7. 21

text: Culotti,

text: J.

text: G

text: Russell,

text: R.

text: L.

text: Osmotic

text: avoidance

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text: mutants

text: of

text: the

text: nematode

text: Caenorhabditis

text: elegans.

text: Genetics

text: Green,

text: M.

text: S.

text: Markoﬀ

text: Random

text: Processes

text: and

text: the

text: Statistical

text: Me-

text: chanics

text: of

text: Time-Dependent

text: Phenomena.

text: II.

text: Irreversible

text: Processes

text: in

text: Fluids.

text: J.

text: Chem.

text: Phys.

text: Kubo,

text: R.

text: Statistical-Mechanical

text: Theory

text: of

text: Irreversible

text: Processes.

text: I.

text: General

text: Theory

text: and

text: Simple

text: Applications

text: to

text: Magnetic

text: and

text: Conduction

text: Problems.

text: J.

text: Phys.

text: Soc.

text: Japan

text: P.

text: Murphy,

text: K.

text: Machine

text: Learning:

text: A

text: Probabilistic

text: Perspective.

text: (The

text: MIT

text: Press,

text: Cambridge,

text: MA).

text: Osborne,

text: K.

text: a,

text: Robichon,

text: A,

text: Burgess,

text: E,

text: Butland,

text: S,

text: Shaw,

text: R.

text: A,

text: Coulthard,

text: A,

text: Pereira,

text: H.

text: S,

text: Greenspan,

text: R.

text: J,

text: Sokolowski,

text: M.

text: B.

text: Natural

text: behavior

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text: cGMP-dependent

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text: kinase

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text: Drosophila.

text: Science

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text: Katifori,

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text: elegans.

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text: elegans.

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text: Comput.

text: Biol.

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text: Genome


text: deepS gniraeB yticoleV ytilibaborP )s/mµ( )ged( tnemngilA ytisneD A B N2 CB4856 PS312 300 s(t) 200 100 0 180 90 ψ(t) 0 −90 −180 1 cos ∆ψ(t) 0 −1 10 s Time C 2.0 Forward Reverse N2 1.6 CB4856 1.2 PS312 0.8 0.4 0.0 −90 0 90 180 270 ∆ψ (°) Figure 2: The random walk of nematodes is composed of speed, turning, and reversal dynamics. (A) The motility of the worm is described by a time-varying centroid position (cid:126)x(t) and physical orientation ψ(t). From this, the speed s(t) and a measure of the alignment of the velocity with the orientation ∆ψ(t) are calculated. (B) One minute of speed, orientation, and velocity alignment data from the 30 minute experiment is shown for individuals from three strains. (C) The probability distribution of the velocity alignment reveals a bimodal structure corresponding to forward and reverse motion states. Shaded areas represent 95% conﬁdence intervals. 25 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: ytisneD ytilibaborP “Dwelling” “Roaming” 1.0 N2 - diversified 0.5 0.0 2.0 CB4856 - roaming specialized 1.0 0.0 2.0 PS312 - dwelling specialized 1.0 0.0 1.0 Interspecies 0.5 0.0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Behavioral Mode Projection Figure 6: Specialized and diversiﬁed behavioral strategies across strains. Phe- notypic distributions showing the variability among individual trajectories sam- pled from a single strain (N2, CB4856, and PS312) and (bottom) the variability in the mean phenotype of each species are shown. Red dashes indicate individ- ual observations, the distributions shown are kernel density estimates. The 95% conﬁdence interval for a single, randomly chosen data point is shaded in gray for reference. 29 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

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text: those for the product of component-wise measured correlation functions (i.e., (cid:46) C − C C − C ). We therefore conclude that the relatively (cid:126)v,obs (cid:126)v,model (cid:126)v,obs (cid:126)v,indep subtle diﬀerences between the data and model arise primarily in instances where this assumption of independence between the three motility components (speed, turning, and reversals) breaks down. Low-Dimensionality in Behavioral Variation across Species The results presented in the previous sections demonstrate that a random- walk model with a total of seven parameters describing independent speed, turning and reversal dynamics, provides a good approximation of the worms’ motile behavior. The model parameters thus deﬁne a seven-dimensional space of motility phenotypes in which behavioral variation across strains and species can be examined. If components of behavior were physiologically regulated or evolutionarily selected for in a coordinated manner, we would expect to ﬁnd correlated patterns in the variation of these traits. We ﬁt our model to the trajectory statistics of each individual worm and built a phenotypic matrix of 139 worms x 7 behavioral parameters (for a sum- mary of all parameters for each strain, see Tables S2-S4). To characterize the pattern of variation within the phenotype space, we computed the covariance matrix for the seven parameters (Figure 5A). The parameters with the largest variance (diagonal elements in Fig. 5A) were the forward and reverse state life- times (τ , τ ), followed by those describing speed dynamics (τ , D ). More fwd rev s s interestingly, there was extensive covariation among the model parameters, not only within the parameters of each motility component (speed, orientation, re- versals) but also between those of diﬀerent components. We looked for the dominant patterns in the correlations using principal com- ponent analysis (Figure 5B), a technique which identiﬁes linear combinations of parameters that represent orthogonal axes (modes) capturing the most variance in the data [37]. The distribution of variance captured by these modes uncovered a single dominant mode of correlated variation (Figure 5B, left). This principal mode, capturing about 60% of the total variation, described signiﬁcant correla- tions among all the parameters except for D (Figure 5, right, Table S5). We s did not attempt to interpret higher modes since, individually, none signiﬁcantly exceeded the captured variance under a randomization test (see Methods, and Figure 5B, left). A non-linear technique for describing the behavioral manifold did not reveal any distinct additional structure (Figure S11). We used numerical simulations to determine the eﬀects on motile behavior of varying parameters along the principal mode. The measured trajectory pheno- types projected onto this mode in the range {−2, 2} centered about the average phenotype located at the origin, and we performed simulations for parameter sets evenly sampled along this range. As the projection on the principal mode increased, the simulated trajectories became more expansive and the eﬀective diﬀusivity increased by more than two orders of magnitude (Figure 5C). This largely reproduced the observed variation in the measured diﬀusivities D as eﬀ a function of the projection along the ﬁrst mode. The signiﬁcant loadings on nearly all of the model parameters indicated that 9

text: those

text: for

text: the

text: product

text: of

text: component-wise

text: measured

text: correlation

text: functions

text: (i.e.,

text: C

text: C

text: C

text: C

text: We

text: therefore

text: conclude

text: that

text: the

text: relatively

text: subtle

text: diﬀerences

text: between

text: the

text: data

text: and

text: model

text: arise

text: primarily

text: in

text: instances

text: where

text: this

text: assumption

text: of

text: independence

text: between

text: the

text: three

text: motility

text: components

text: (speed,

text: turning,

text: and

text: reversals)

text: breaks

text: down.

text: Low-Dimensionality

text: in

text: Behavioral

text: Variation

text: across

text: Species

text: The

text: results

text: presented

text: in

text: the

text: previous

text: sections

text: demonstrate

text: that

text: a

text: random-

text: walk

text: model

text: with

text: a

text: total

text: of

text: seven

text: parameters

text: describing

text: independent

text: speed,

text: turning

text: and

text: reversal

text: dynamics,

text: provides

text: a

text: good

text: approximation

text: of

text: the

text: worms’

text: motile

text: behavior.

text: The

text: model

text: parameters

text: thus

text: deﬁne

text: a

text: seven-dimensional

text: space

text: of

text: motility

text: phenotypes

text: in

text: which

text: behavioral

text: variation

text: across

text: strains

text: and

text: species

text: can

text: be

text: examined.

text: If

text: components

text: of

text: behavior

text: were

text: physiologically

text: regulated

text: or

text: evolutionarily

text: selected

text: for

text: in

text: a

text: coordinated

text: manner,

text: we

text: would

text: expect

text: to

text: ﬁnd

text: correlated

text: patterns

text: in

text: the

text: variation

text: of

text: these

text: traits.

text: We

text: ﬁt

text: our

text: model

text: to

text: the

text: trajectory

text: statistics

text: of

text: each

text: individual

text: worm

text: and

text: built

text: a

text: phenotypic

text: matrix

text: of

text: worms

text: x

text: behavioral

text: parameters

text: (for

text: a

text: sum-

text: mary

text: of

text: all

text: parameters

text: for

text: each

text: strain,

text: see

text: Tables

text: To

text: characterize

text: the

text: pattern

text: of

text: variation

text: within

text: the

text: phenotype

text: space,

text: we

text: computed

text: the

text: covariance

text: matrix

text: for

text: the

text: seven

text: parameters

text: (Figure

text: The

text: parameters

text: with

text: the

text: largest

text: variance

text: (diagonal

text: elements

text: in

text: Fig.

text: were

text: the

text: forward

text: and

text: reverse

text: state

text: life-

text: times

text: (τ

text: τ

text: followed

text: by

text: those

text: describing

text: speed

text: dynamics

text: (τ

text: D

text: More

text: fwd

text: rev

text: s

text: s

text: interestingly,

text: there

text: was

text: extensive

text: covariation

text: among

text: the

text: model

text: parameters,

text: not

text: only

text: within

text: the

text: parameters

text: of

text: each

text: motility

text: component

text: (speed,

text: orientation,

text: re-

text: versals)

text: but

text: also

text: between

text: those

text: of

text: diﬀerent

text: components.

text: We

text: looked

text: for

text: the

text: dominant

text: patterns

text: in

text: the

text: correlations

text: using

text: principal

text: com-

text: ponent

text: analysis

text: (Figure

text: a

text: technique

text: which

text: identiﬁes

text: linear

text: combinations

text: of

text: parameters

text: that

text: represent

text: orthogonal

text: axes

text: (modes)

text: capturing

text: the

text: most

text: variance

text: in

text: the

text: data

text: The

text: distribution

text: of

text: variance

text: captured

text: by

text: these

text: modes

text: uncovered

text: a

text: single

text: dominant

text: mode

text: of

text: correlated

text: variation

text: (Figure

text: left).

text: This

text: principal

text: mode,

text: capturing

text: about

text: of

text: the

text: total

text: variation,

text: described

text: signiﬁcant

text: correla-

text: tions

text: among

text: all

text: the

text: parameters

text: except

text: for

text: D

text: (Figure

text: right,

text: Table

text: We

text: s

text: did

text: not

text: attempt

text: to

text: interpret

text: higher

text: modes

text: since,

text: individually,

text: signiﬁcantly

text: exceeded

text: the

text: captured

text: variance

text: under

text: a

text: randomization

text: test

text: (see

text: Methods,

text: and

text: Figure

text: left).

text: A

text: non-linear

text: technique

text: for

text: describing

text: the

text: behavioral

text: manifold

text: did

text: not

text: reveal

text: any

text: distinct

text: additional

text: structure

text: (Figure

text: We

text: used

text: numerical

text: simulations

text: to

text: determine

text: the

text: eﬀects

text: on

text: motile

text: behavior

text: of

text: varying

text: parameters

text: along

text: the

text: principal

text: mode.

text: The

text: measured

text: trajectory

text: pheno-

text: types

text: projected

text: onto

text: this

text: mode

text: in

text: the

text: range

text: centered

text: about

text: the

text: average

text: phenotype

text: located

text: at

text: the

text: origin,

text: and

text: we

text: performed

text: simulations

text: for

text: parameter

text: sets

text: evenly

text: sampled

text: along

text: this

text: range.

text: As

text: the

text: projection

text: on

text: the

text: principal

text: mode

text: increased,

text: the

text: simulated

text: trajectories

text: became

text: more

text: expansive

text: and

text: the

text: eﬀective

text: diﬀusivity

text: increased

text: by

text: more

text: than

text: two

text: orders

text: of

text: magnitude

text: (Figure

text: This

text: largely

text: reproduced

text: the

text: observed

text: variation

text: in

text: the

text: measured

text: diﬀusivities

text: D

text: as

text: eﬀ

text: a

text: function

text: of

text: the

text: projection

text: along

text: the

text: ﬁrst

text: mode.

text: The

text: signiﬁcant

text: loadings

text: on

text: nearly

text: all

text: of

text: the

text: model

text: parameters

text: indicated

text: that


text: OPTIMAL COBORDISMS BETWEEN TORUS KNOTS PETER FELLER Abstract. We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection C of a smooth algebraic curve in 2 with the unit 4-ball from which a 4-ball of smaller radius is removed. Connections to the realization problem of An- singularities on algebraic plane curves and the adjacency problem for plane curve singularities are discussed. To obstruct the existence of cobordisms, we use Ozsv´ath, Stipsicz, and Szab´o’s Υ-invariant, which we provide explicitly for torus knots of braid index 3 and 4. 1. Introduction For a knot K—a smooth and oriented embedding of the unit circle S1 into the unit 3-sphere S3—the slice genus g (K) is the minimal genus of smooth, oriented s ⊂ surfaces F in the closed unit 4-ball B4 with oriented boundary K ∂B4 = S3. For torus knots, the slice genus is equal to their genus g; i.e., for coprime positive integers p and q, one has − − (p 1)(q 1) (1) g (T ) = g(T ) = . s p,q p,q 2 More generally, the slice genus is known for knots K that arise as the transversal C intersection of S3 with a smooth algebraic curve V in 2, by Kronheimer and f Mrowka’s resolution of the Thom conjecture [KM93, Corollary 1.3]: the surface ⊂ C in B4 2 given as the intersection of B4 with V has genus g (K); see also f s Rudolph’s slice-Bennequin inequality [Rud93]. We determine the slice genus for connected sums of torus knots of small braid index; see Corollary 3. The cobordism distance d (K, T) between two knots K and T is the minimal c × genus of smoothly embedded and oriented surfaces C in S3 [0, 1] with boundary ×{ }∪ ×{ } K 0 T 1 such that the induced orientation agrees with the orientation of T and disagrees with the orientation of K. Such a C is called a cobordism between K and T. Equivalently, d (K, T) can be deﬁned as the slice genus of the connected sum − − c K(cid:93) T of K and T—the mirror image of T with reversed orientation. Cobordism ≥ | − | distance satisﬁes the triangle inequality; in particular, d (K, T) g (T) g (K) c s s for all knots K and T. We call a cobordism betwCeen two kn\ots a∼lgebra×ic if it arises as the intersection of a smooth algebraic curve in 2 with B4 B4 = S3 [0, 1], where 2 1 2010 Mathematics Subject Classiﬁcation. 57M25, 57M27, 14B07, 32S55. Key words and phrases. Slice genus, torus knots, cobordism distance, positive braids, quasi- positive braids and links, Υ-invariant. The author gratefully acknowledges support by the Swiss National Science Foundation Grant 155477. 1 ##LTLine## ##LTLine## ##LTLine##

text: OPTIMAL

text: COBORDISMS

text: BETWEEN

text: TORUS

text: KNOTS

text: PETER

text: FELLER

text: Abstract.

text: We

label: abstract

text: construct

text: cobordisms

text: of

text: small

text: genus

text: between

text: torus

text: knots

text: and

text: use

text: them

text: to

text: determine

text: the

text: cobordism

text: distance

text: between

text: torus

text: knots

text: of

text: small

text: braid

text: index.

text: In

text: fact,

text: the

text: cobordisms

text: we

text: construct

text: arise

text: as

text: the

text: intersection

text: C

text: of

text: a

text: smooth

text: algebraic

text: curve

text: in

text: with

text: the

text: unit

text: from

text: which

text: a

text: of

text: smaller

text: radius

text: is

text: removed.

text: Connections

text: to

text: the

text: realization

text: problem

text: of

text: An-

text: singularities

text: on

text: algebraic

text: plane

text: curves

text: and

text: the

text: adjacency

text: problem

text: for

text: plane

text: curve

text: singularities

text: are

text: discussed.

text: To

text: obstruct

text: the

text: existence

text: of

text: cobordisms,

text: we

text: use

text: Ozsv´ath,

text: Stipsicz,

text: and

text: Szab´o’s

text: Υ-invariant,

text: which

text: we

text: provide

text: explicitly

text: for

text: torus

text: knots

text: of

text: braid

text: index

text: and

text: Introduction

text: For

text: a

text: knot

text: K—a

text: smooth

text: and

text: oriented

text: embedding

text: of

text: the

text: unit

text: circle

text: into

text: the

text: unit

text: slice

text: genus

text: g

text: (K)

text: is

text: the

text: minimal

text: genus

text: of

text: smooth,

text: oriented

text: s

text: surfaces

text: F

text: in

text: the

text: closed

text: unit

text: with

text: oriented

text: boundary

text: K

text: For

text: torus

text: knots,

text: the

text: slice

text: genus

text: is

text: equal

text: to

text: their

text: genus

text: g;

text: i.e.,

text: for

text: coprime

text: positive

text: integers

text: p

text: and

text: q,

text: one

text: has

text: (p

text: g

text: (T

text: g(T

text: s

text: p,q

text: p,q

text: More

text: generally,

text: the

text: slice

text: genus

text: is

text: known

text: for

text: knots

text: K

text: that

text: arise

text: as

text: the

text: transversal

text: C

text: intersection

text: of

text: with

text: a

text: smooth

text: algebraic

text: curve

text: V

text: in

text: by

text: Kronheimer

text: and

text: f

text: Mrowka’s

text: resolution

text: of

text: the

text: Thom

text: conjecture

text: Corollary

text: the

text: surface

text: C

text: in

text: given

text: as

text: the

text: intersection

text: of

text: with

text: V

text: has

text: genus

text: g

text: (K);

text: see

text: also

text: f

text: s

text: Rudolph’s

text: slice-Bennequin

text: inequality

text: We

text: determine

text: the

text: slice

text: genus

text: for

text: connected

text: sums

text: of

text: torus

text: knots

text: of

text: small

text: braid

text: index;

text: see

text: Corollary

text: The

text: cobordism

text: distance

text: d

text: (K,

text: T)

text: between

text: two

text: knots

text: K

text: and

text: T

text: is

text: the

text: minimal

text: c

text: genus

text: of

text: smoothly

text: embedded

text: and

text: oriented

text: surfaces

text: C

text: in

text: with

text: boundary

text: K

text: T

text: such

text: that

text: the

text: induced

text: orientation

text: agrees

text: with

text: the

text: orientation

text: of

text: T

text: and

text: disagrees

text: with

text: the

text: orientation

text: of

text: K.

text: Such

text: a

text: C

text: is

text: called

text: a

text: cobordism

text: between

text: K

text: and

text: T.

text: Equivalently,

text: d

text: (K,

text: T)

text: can

text: be

text: deﬁned

text: as

text: the

text: slice

text: genus

text: of

text: the

text: connected

text: sum

text: c

text: T

text: of

text: K

text: and

text: T—the

text: mirror

text: image

text: of

text: T

text: with

text: reversed

text: orientation.

text: Cobordism

text: distance

text: satisﬁes

text: the

text: triangle

text: inequality;

text: in

text: particular,

text: d

text: (K,

text: T)

text: g

text: (T)

text: g

text: (K)

text: c

text: s

text: s

text: for

text: all

text: knots

text: K

text: and

text: T.

text: We

text: call

text: a

text: cobordism

text: betwCeen

text: two

text: kn\ots

text: a∼lgebra×ic

text: if

text: it

text: arises

text: as

text: the

text: intersection

text: of

text: a

text: smooth

text: algebraic

text: curve

text: in

text: with

text: where

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Key

text: words

text: and

text: phrases.

text: Slice

text: genus,

text: torus

text: knots,

text: cobordism

text: distance,

text: positive

text: braids,

text: quasi-

text: positive

text: braids

text: and

text: links,

text: Υ-invariant.

text: The

text: author

text: gratefully

text: acknowledges

text: support

text: by

text: the

text: Swiss

text: National

text: Science

text: Foundation

text: Grant

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: OPTIMAL COBORDISMS BETWEEN TORUS KNOTS 17 − and inductively we get ∆2l = (a a a a a )l(a )l. The subword a a a occurs l 1 1 2 1 1−2 1 2 1 2 times in (a a a a a )l(a )l. Applying l 1 times the braid relation a a a = a a a 1 2 1 1 2 1 2 1 2 1 2 1 gives the 3-braid word − w = a a a a (a a a a a )l 1a (a )l. 1 2 1 1 1 2 1 1 1 2 1 − Deleting all but the ﬁrst a in w yiel(cid:0)ds a a a5l 2, which has T − a(cid:1)s closure. 2 1 2 1 2,5l 1 If m is 3l + 1 for some positive integer l, we write T as the closure of 3,3l+1 − a a ∆2l = a a w = a a a a a a (a a a a a )l 1a (a )l 1 2 1 2 1 2 1 2 1 1 1 2 1 1 1 2 1 − = a a a a a a (a a a a a )l 1a (a )l. 1 1 2 1 1 1 1 2 1 1 1 2 1 − Deleting all but the ﬁrst a yields a a a (a )5l 1, which has closure T . 2 1 1 2 1 2,5l+1 Finally, if m is 3l + 2 for some positive integer l, view T as the closure of (cid:0) 3,3l+2 (cid:1) a a a a ∆2l = a a a a ∆2l = a (a a ∆2l)a 1 2 1 2 1 1 2 1 1 1 2 1 − = a a a a a a a (a a a a a )l 1a (a )l a . 1 1 1 2 1 1 1 1 2 1 1 1 2 1 1 (cid:3) Deleting all but the ﬁrst a yields a a a a (a )5l, which has T as closure. 2 1 1 1 2 1 2,5l+3 Proof of Proposition 23. We view T as the closure of the 4-braid (a a a )2l+1. 4,2l+1 1 3 2 Using the fact that the half twist on 4 strands ∆ = a a a a a a = a a a a a a 1 3 2 1 3 2 1 2 1 3 2 1 anti-commutes with every other 4-braid, i.e. a ∆ = ∆a , a ∆ = ∆a , and a ∆ = 1 3 3 1 2 ∆a2, we have that (cid:16) (cid:17) − − ∆k = a a a a a a ∆k 1 = a a a a a ∆k 1a − − 1 2 1 3 2 1 1 2 1 3 2 2+( 1)k 1 − = a a a a a a a a a a ∆k 2a − − a − − 1 2 1 3 2 1 2 1 3 2 2+( 1)k 2 2+( 1)k 1 = (a a a a a )2∆k−2a a = · · · = (a a a a a )ka(cid:98)k+21(cid:99)a(cid:98)k2(cid:99). 1 2 1 3 2 1 3 1 2 1 3 2 1 3 With this we can write (a a a )2l+1 as follows. 1 3 2 − (a a a )2l+1 = (a a a )2a a a (a a a )2(l 1) 1 3 2 1 3 2 1 3 2 1 3 2 = (a a a )2a a a (a a a a a )l−1a(cid:98)2l (cid:99)a(cid:98)l−21(cid:99) 1 3 2 1 3 2 1 2 1 3 2 1 3 = (a a a )2a a (a a a a a )l−1a a(cid:98)2l (cid:99)a(cid:98)l−21(cid:99) 1 3 2 1 3 2 1 2 1 3 2 1 3 = (a a a )2a a (a a a a a )l−1a a(cid:98)2l (cid:99)a(cid:98)l−21(cid:99). 1 3 2 1 3 1 2 1 1 3 2 1 3 Deleting the last l occurrences of a in this braid word gives 2 (a a a )2a a (a a a a )l−1a(cid:98)2l (cid:99)a(cid:98)l−21(cid:99) = (a a a )2a3l−2+(cid:98)2l (cid:99)al+(cid:98)l−21(cid:99), 1 3 2 1 3 1 1 1 3 1 3 1 3 2 1 3 which has closure T − (cid:98) (cid:99) (cid:98) − (cid:99) = T . 2,4+(3l 2+ l )+(l+ l 1 ) 2,5l+1 2 2 Similarly, the torus link T is the closure of the 4-braid 4,2l+2 (a a a )2l+2 = (a a a )2a a a a a (a a a a a )l−1a a(cid:98)2l (cid:99)a(cid:98)l−21(cid:99). 1 3 2 1 3 2 1 3 2 1 3 1 2 1 1 3 2 1 3 (cid:3) Deleting the last l+1 occurrences of a yields a 4-braid that has closure T . 2 2,5l+3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: OPTIMAL

text: COBORDISMS

text: BETWEEN

text: TORUS

text: KNOTS

text: and

text: inductively

text: we

text: get

text: (a

text: a

text: a

text: a

text: a

text: )l(a

text: )l.

text: The

text: subword

text: a

text: a

text: a

text: occurs

text: l

text: times

text: in

text: (a

text: a

text: a

text: a

text: a

text: )l(a

text: )l.

text: Applying

text: l

text: times

text: the

text: braid

text: relation

text: a

text: a

text: a

text: a

text: a

text: a

text: gives

text: the

text: word

text: w

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: )l

text: (a

text: )l.

text: Deleting

text: all

text: but

text: the

text: ﬁrst

text: a

text: in

text: w

text: a

text: a

text: which

text: has

text: T

text: closure.

text: If

text: m

text: is

text: for

text: some

text: positive

text: integer

text: l,

text: we

text: write

text: T

text: as

text: the

text: closure

text: of

text: a

text: a

text: a

text: a

text: w

text: a

text: a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: )l

text: (a

text: )l

text: a

text: a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: )l

text: (a

text: )l.

text: Deleting

text: all

text: but

text: the

text: ﬁrst

text: a

text: yields

text: a

text: a

text: a

text: (a

text: which

text: has

text: closure

text: T

text: Finally,

text: if

text: m

text: is

text: for

text: some

text: positive

text: integer

text: l,

text: view

text: T

text: as

text: the

text: closure

text: of

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: )l

text: (a

text: )l

text: a

text: Deleting

text: all

text: but

text: the

text: ﬁrst

text: a

text: yields

text: a

text: a

text: a

text: a

text: (a

text: which

text: has

text: T

text: as

text: closure.

text: Proof

text: of

text: Proposition

text: We

text: view

text: T

text: as

text: the

text: closure

text: of

text: the

text: (a

text: a

text: a

text: Using

text: the

text: fact

text: that

text: the

text: half

text: twist

text: on

text: strands

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: anti-commutes

text: with

text: every

text: other

text: i.e.

text: a

text: ∆a

text: a

text: ∆a

text: and

text: a

text: we

text: have

text: that

text: ∆k

text: a

text: a

text: a

text: a

text: a

text: a

text: ∆k

text: a

text: a

text: a

text: a

text: a

text: ∆k

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: ∆k

text: a

text: (a

text: a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: With

text: this

text: we

text: can

text: write

text: (a

text: a

text: a

text: as

text: follows.

text: (a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: Deleting

text: the

text: last

text: l

text: occurrences

text: of

text: a

text: in

text: this

text: braid

text: word

text: gives

text: (a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: (a

text: a

text: a

text: which

text: has

text: closure

text: T

text: T

text: l

text: )+(l+

text: l

text: Similarly,

text: the

text: torus

text: link

text: T

text: is

text: the

text: closure

text: of

text: the

text: (a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: a

text: a

text: (a

text: a

text: a

text: a

text: a

text: Deleting

text: the

text: last

text: occurrences

text: of

text: a

text: yields

text: a

text: that

text: has

text: closure

text: T

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: OPTIMAL COBORDISMS BETWEEN TORUS KNOTS 3 are obtained by taking the mirror image, the sign changes), one has − − − −(p 1)(q 1) τ(T ) = g(T ) = [OS03, Corollary 1.7], p,q p,q 2 for coprime positive integers p and q, and − − − − (2) υ(T ) = k, υ(T ) = υ(T ) = 2k, υ(T ) = 2k 1, 2,2k+1 3,3k+1 4,2k+1 3,3k+2 for positive integers k. The υ-values for torus knots of braid index 2 follow, for example, from the fact that υ equals half the signature for alternating knots [OSS14, Theorem 1.14]. The other υ-values in (2) can be derived from the inductive formula for υ provided by Ozsv´ath, Stipsticz and Szab´o [OSS14, Theorem 1.15]. We do this in Section . Of course, part of the statement of Corollary 3 was known before; for example in the cases covered by the following remark. Remark 4. If K and T are positive and negative torus knots, respectively, one has − d (T, K) = g (K(cid:93) T) = g (K) + g (T) = g(K) + g(T) c s s s by the Thom conjecture. If K and T are both positive (both negative) torus knots of the same braid index, then there exists an optimal cobordism between them; compare Example 20. Determining the cobordism distance between all torus knots seems out of reach. However, a coarse estimation of the cobordism distance between torus knots was provided by Baader [Baa12]. The study of optimal and algebraic cobordisms between torus knots seems nat- ural from a knot theoretic point of view. We discuss additional motivation from algebraic geometry questions in Section . Section recalls the notions of positive and quasi-positive braids. In Section we show that optimal cobordisms given by quasi-positive braid sequences are algebraic. In Section optimal cobordisms are constructed and used to prove Theorem 1, Theorem 2, and Corollary 3. Section provides the Υ-values for torus knots of braid index 3 and 4. Acknowledgements: I thank Sebastian Baader for sharing his insight into cobordisms and for his ongoing support. Thanks also to Adam Levine for point- ing me towards the Υ-invariant, and Immanuel Stampﬂi and Andrew Yarmola for valuable suggestions. Finally, I wish to thank the referee for helpful suggestions and corrections. C 2. Algebraic motivation: Plane curve singularities over In this section, we discuss motivations to study algebraic and optimal cobordisms between torus knots coming from singularity theory. Mathematically, the rest of the paper is independent of this. C We consider isolated singularities on algebraic curves in 2 and we denote sin- gularities by the function germs that deﬁne them. A general question asks, what C (topological) type of singularities can occur on an algebraic curve V in 2 given C f as the zero-set of a square-free polynomial f in [x, y] of some ﬁxed degree d; see e.g. Greuel, Lossen, and Shustin’s work [GLS98]. Even for simple singulari- ties—those corresponding to Dynkin diagrams [Arn72]—a lot is unknown. For the − A -singularities—the simple singularities given by y2 xk+1—the following bounds k were provided by Guse˘ın-Zade and Nekhoroshev ##LTLine##

text: OPTIMAL

text: COBORDISMS

text: BETWEEN

text: TORUS

text: KNOTS

text: are

text: obtained

text: by

text: taking

text: the

text: mirror

text: image,

text: the

text: sign

text: changes),

text: one

text: has

text: −(p

text: τ(T

text: g(T

text: Corollary

text: p,q

text: p,q

text: for

text: coprime

text: positive

text: integers

text: p

text: and

text: q,

text: and

text: υ(T

text: k,

text: υ(T

text: υ(T

text: υ(T

text: for

text: positive

text: integers

text: k.

text: The

text: υ-values

text: for

text: torus

text: knots

text: of

text: braid

text: index

text: follow,

text: for

text: example,

text: from

text: the

text: fact

text: that

text: υ

text: equals

text: half

text: the

text: signature

text: for

text: alternating

text: knots

text: Theorem

text: The

text: other

text: υ-values

text: in

text: can

text: be

text: derived

text: from

text: the

text: inductive

text: formula

text: for

text: υ

text: provided

text: by

text: Ozsv´ath,

text: Stipsticz

text: and

text: Szab´o

text: Theorem

text: We

text: do

text: this

text: in

text: Section

text: Of

text: course,

text: part

text: of

text: the

text: statement

text: of

text: Corollary

text: was

text: known

text: before;

text: for

text: example

text: in

text: the

text: cases

text: covered

text: by

text: the

text: following

text: remark.

text: Remark

text: If

text: K

text: and

text: T

text: are

text: positive

text: and

text: negative

text: torus

text: knots,

text: respectively,

text: one

text: has

text: d

text: (T,

text: K)

text: g

text: T)

text: g

text: (K)

text: g

text: (T)

text: g(K)

text: g(T)

text: c

text: s

text: s

text: s

text: by

text: the

text: Thom

text: conjecture.

text: If

text: K

text: and

text: T

text: are

text: both

text: positive

text: (both

text: negative)

text: torus

text: knots

text: of

text: the

text: same

text: braid

text: index,

text: then

text: there

text: exists

text: an

text: optimal

text: cobordism

text: between

text: them;

text: compare

text: Example

text: Determining

text: the

text: cobordism

text: distance

text: between

text: all

text: torus

text: knots

text: seems

text: out

text: of

text: reach.

text: However,

text: a

text: coarse

text: estimation

text: of

text: the

text: cobordism

text: distance

text: between

text: torus

text: knots

text: was

text: provided

text: by

text: Baader

text: The

text: study

text: of

text: optimal

text: and

text: algebraic

text: cobordisms

text: between

text: torus

text: knots

text: seems

text: nat-

text: ural

text: from

text: a

text: knot

text: theoretic

text: point

text: of

text: view.

text: We

text: discuss

text: additional

text: motivation

text: from

text: algebraic

text: geometry

text: questions

text: in

text: Section

text: Section

text: recalls

text: the

text: notions

text: of

text: positive

text: and

text: quasi-positive

text: braids.

text: In

text: Section

text: we

text: show

text: that

text: optimal

text: cobordisms

text: given

text: by

text: quasi-positive

text: braid

text: sequences

text: are

text: algebraic.

text: In

text: Section

text: optimal

text: cobordisms

text: are

text: constructed

text: and

text: used

text: to

text: prove

text: Theorem

text: Theorem

text: and

text: Corollary

text: Section

text: provides

text: the

text: Υ-values

text: for

text: torus

text: knots

text: of

text: braid

text: index

text: and

text: Acknowledgements:

text: I

text: thank

text: Sebastian

text: Baader

text: for

text: sharing

text: his

text: insight

text: into

text: cobordisms

text: and

text: for

text: his

text: ongoing

text: support.

text: Thanks

text: also

text: to

text: Adam

text: Levine

text: for

text: point-

text: ing

text: me

text: towards

text: the

text: Υ-invariant,

text: and

text: Immanuel

text: Stampﬂi

text: and

text: Andrew

text: Yarmola

text: for

text: valuable

text: suggestions.

text: Finally,

text: I

text: wish

text: to

text: thank

text: the

text: referee

text: for

text: helpful

text: suggestions

text: and

text: corrections.

text: C

text: Algebraic

text: motivation:

text: Plane

text: curve

text: singularities

text: over

text: In

text: this

text: section,

text: we

text: discuss

text: motivations

text: to

text: study

text: algebraic

text: and

text: optimal

text: cobordisms

text: between

text: torus

text: knots

text: coming

text: from

text: singularity

text: theory.

text: Mathematically,

text: the

text: rest

text: of

text: the

text: paper

text: is

text: independent

text: of

text: this.

text: C

text: We

text: consider

text: isolated

text: singularities

text: on

text: algebraic

text: curves

text: in

text: and

text: we

text: denote

text: sin-

text: gularities

text: by

text: the

text: function

text: germs

text: that

text: deﬁne

text: them.

text: A

text: general

text: question

text: asks,

text: what

text: C

text: (topological)

text: type

text: of

text: singularities

text: can

text: occur

text: on

text: an

text: algebraic

text: curve

text: V

text: in

text: given

text: C

text: f

text: as

text: the

text: zero-set

text: of

text: a

text: square-free

text: polynomial

text: f

text: in

text: [x,

text: y]

text: of

text: some

text: ﬁxed

text: degree

text: d;

text: see

text: e.g.

text: Greuel,

text: Lossen,

text: and

text: Shustin’s

text: work

text: Even

text: for

text: simple

text: singulari-

text: ties—those

text: corresponding

text: to

text: Dynkin

text: diagrams

text: lot

text: is

text: unknown.

text: For

text: the

text: A

text: -singularities—the

text: simple

text: singularities

text: given

text: by

text: following

text: bounds

text: k

text: were

text: provided

text: by

text: Guse˘ın-Zade

text: and

text: Nekhoroshev

text: ##LTLine##


text: 8 PETER FELLER C \ ∈ a ﬁxed embedded curve in γ in B with a marked start point p / B+, one gets an explicit procedure, how to read oﬀ a braid word for the braid β corresponding to the arc starting and ending at p going counter-clockwise around γ: by a small C \ isotopy of γ in B, we may assume that γ meets G(f) transversally and in edges only. Starting at p we move counter-clockwise around γ. Whenever we cross an − edge e transversally at a point x, we write down the generator a or its inverse a 1 ke ke depending on whether the orientation at x given by G(f) and γ agrees or disagrees C with the complex orientation of . The study of the graphs G(f) motivates the following deﬁnition. Fix some surface S with boundary. In fact, we will only consider cases where S is either { ∈ C | | | ≤ } { ∈ C | ≤ | | ≤ } the unit disc D = x x 1 or the annulus A = x 1 x 2 . { · · · − } A Rudolph diagram on S is an oriented, 1, , n 1 -labeled graph G with smooth edges (we also allow smooth closed cycles) that enters and exits the boundary of S transversely and is locally modelled on graphs G(f) coming from an algebraic ∈ C function f [x, y] as above; i.e. locally around a vertex G is given as in Figure 1. ⊂ We denote the set of 1-valent vertices by B G. Of course, a huge source of exam- C ples are obtained by embedding (or immersing) S in and using this embedding to deﬁne G as the pull back of G(f) for some algebraic function f. Any closed curve γ missing 1-valent vertices deﬁnes a closed braid β by isotoping γ to meet G transversally in edges and then reading oﬀ a braid word β for that closure as described in the G(f)-case. A Rudolph diagram is said to be smooth if it contains only 1-valent vertices. (cid:81) − Example 9. Given a quasi-positive braid word β = w a w 1, there exists a l il l smooth Rudolph diagram on D such that braid word read oﬀ when following S1 = − ∂D is β. Figure 2 illustrates how one factor ωa ω 1 is realized. i 4 3 2 1 ∂D = S1 Figure 2. Piece of a Rudolph diagram that yields the braid word − − ωa ω 1 when following the boundary, for i = 4 and ω = a a 1a . i 1 2 3 Orevkov describes which smooth Rudolph diagrams on D arise as G(f). Proposition 10. [Ore96, Proposition 2.1] Let G be a smooth Rudolph diagram on D. There exists an algebraic function − · · · ∈ C f = yn + c − (x)yn 1 + + c (x) [x, y] n 1 0 ∩ such that G is isotopic to G(f) D if and only if G(f) contains no closed cycles. ##LTFigure## ##LTLine##

text: PETER

text: FELLER

text: C

text: a

text: ﬁxed

text: embedded

text: curve

text: in

text: γ

text: in

text: B

text: with

text: a

text: marked

text: start

text: point

text: p

text: B+,

text: one

text: gets

text: an

text: explicit

text: procedure,

text: how

text: to

text: read

text: oﬀ

text: a

text: braid

text: word

text: for

text: the

text: braid

text: β

text: corresponding

text: to

text: the

text: arc

text: starting

text: and

text: ending

text: at

text: p

text: going

text: counter-clockwise

text: around

text: γ:

text: by

text: a

text: small

text: C

text: isotopy

text: of

text: γ

text: in

text: B,

text: we

text: may

text: assume

text: that

text: γ

text: meets

text: G(f)

text: transversally

text: and

text: in

text: edges

text: only.

text: Starting

text: at

text: p

text: we

text: move

text: counter-clockwise

text: around

text: γ.

text: Whenever

text: we

text: cross

text: an

text: edge

text: e

text: transversally

text: at

text: a

text: point

text: x,

text: we

text: write

text: down

text: the

text: generator

text: a

text: or

text: its

text: inverse

text: a

text: ke

text: ke

text: depending

text: on

text: whether

text: the

text: orientation

text: at

text: x

text: given

text: by

text: G(f)

text: and

text: γ

text: agrees

text: or

text: disagrees

text: C

text: with

text: the

text: complex

text: orientation

text: of

text: The

text: study

text: of

text: the

text: graphs

text: G(f)

text: motivates

text: the

text: following

text: deﬁnition.

text: Fix

text: some

text: surface

text: S

text: with

text: boundary.

text: In

text: fact,

text: we

text: will

text: only

text: consider

text: cases

text: where

text: S

text: is

text: either

text: C

text: C

text: the

text: unit

text: disc

text: D

text: x

text: x

text: or

text: the

text: annulus

text: A

text: x

text: x

text: A

text: Rudolph

text: diagram

text: on

text: S

text: is

text: an

text: oriented,

text: n

text: -labeled

text: graph

text: G

text: with

text: smooth

text: edges

text: (we

text: also

text: allow

text: smooth

text: closed

text: cycles)

text: that

text: enters

text: and

text: exits

text: the

text: boundary

text: of

text: S

text: transversely

text: and

text: is

text: locally

text: modelled

text: on

text: graphs

text: G(f)

text: coming

text: from

text: an

text: algebraic

text: C

text: function

text: f

text: [x,

text: y]

text: as

text: above;

text: i.e.

text: locally

text: around

text: a

text: vertex

text: G

text: is

text: given

text: as

text: in

text: Figure

text: We

text: denote

text: the

text: set

text: of

text: vertices

text: by

text: B

text: G.

text: Of

text: course,

text: a

text: huge

text: source

text: of

text: exam-

text: C

text: ples

text: are

text: obtained

text: by

text: embedding

text: (or

text: immersing)

text: S

text: in

text: and

text: using

text: this

text: embedding

text: to

text: deﬁne

text: G

text: as

text: the

text: pull

text: back

text: of

text: G(f)

text: for

text: some

text: algebraic

text: function

text: f.

text: Any

text: closed

text: curve

text: γ

text: missing

text: vertices

text: deﬁnes

text: a

text: closed

text: braid

text: β

text: by

text: isotoping

text: γ

text: to

text: meet

text: G

text: transversally

text: in

text: edges

text: and

text: then

text: reading

text: oﬀ

text: a

text: braid

text: word

text: β

text: for

text: that

text: closure

text: as

text: described

text: in

text: the

text: G(f)-case.

text: A

text: Rudolph

text: diagram

text: is

text: said

text: to

text: be

text: smooth

text: if

text: it

text: contains

text: only

text: vertices.

text: Example

text: Given

text: a

text: quasi-positive

text: braid

text: word

text: β

text: w

text: a

text: w

text: there

text: exists

text: a

text: l

text: il

text: l

text: smooth

text: Rudolph

text: diagram

text: on

text: D

text: such

text: that

text: braid

text: word

text: read

text: oﬀ

text: when

text: following

text: ∂D

text: is

text: β.

text: Figure

text: illustrates

text: how

text: one

text: factor

text: ωa

text: ω

text: is

text: realized.

text: i

text: ∂D

text: Figure

text: Piece

text: of

text: a

text: Rudolph

text: diagram

text: that

text: yields

text: the

text: braid

text: word

text: ωa

text: ω

text: when

text: following

text: the

text: boundary,

text: for

text: i

text: and

text: ω

text: a

text: a

text: i

text: Orevkov

text: describes

text: which

text: smooth

text: Rudolph

text: diagrams

text: on

text: D

text: arise

text: as

text: G(f).

text: Proposition

text: Proposition

text: Let

text: G

text: be

text: a

text: smooth

text: Rudolph

text: diagram

text: on

text: D.

text: There

text: exists

text: an

text: algebraic

text: function

text: C

text: f

text: yn

text: c

text: (x)yn

text: c

text: (x)

text: [x,

text: y]

text: n

text: such

text: that

text: G

text: is

text: isotopic

text: to

text: G(f)

text: D

text: if

text: and

text: only

text: if

text: G(f)

text: contains

text: no

text: closed

text: cycles.

text: ##LTFigure##

text: ##LTLine##