text: THE FIBER DIMENSION OF A GRAPH 9 L ⊂ Z ⊂ Q Lemma 5.3. Let n be a lattice of full rank and let P d be a set such that for any ∈ ∩ Z − (cid:54)∈ L | ∩ Z | ≤ |Z L| distinct v, w P d, v w . Then P d d/ . ∩ Z { } Z → Z L L Proof. Let P d = v , . . . , v and consider the linear map φ : d d/ , φ(v) = v + . −1 (cid:54) n Z L ∈ (cid:54) By assumption, φ(v v ) = 0 in d/ for all i, j [n] with i = j. Assume that there are ∈ − (cid:54) i j − − − − − i, j [n 1] with i = j such that φ(v v ) = φ(v v ). Then φ(v v ) = φ(v v +v v ) = − − − n i n j − i(cid:54) j − i n n j φ(v v ) φ(v v ) = 0, a contradiction. Thus, φ(v v ) = φ(v v ) for all distinct i∈ n− j n |{ − ∈ − }| n− i n j i, j [n 1]. That is, φ(v v ) : i [n 1] = n 1. The proposition follows from − |{ − ∈ −n }| i≤ |Z L| − (cid:3) n 1 = φ(v v ) : i [n 1] d/ 1 n i |Z L| L Remark 5.4. Since n/ = det( ), Lemma 5.3 can be seen as a discrete analogon of Blichfeldt’s theorem [7, Theorem 2.4.1]. ∈ N (cid:100) (cid:101) Theorem 5.5. For any n , fdim(K ) = log n . n 2 (cid:100) (cid:101) Proof. Due to Proposition 5.2, the ﬁber dimension of K is bounded from above by log n . ⊂ Q n M ⊂ Z 2 Let d := ∼fdim(MKn) and P d a d-dimens∈iona∩l pZolytope and − ∈d a· ZMarkov basis such that K = P ( ). Assume there are v, w P d such that v w 2 d. Since (v + w) n ∈ ∈ ∈ Z i is even for all i [d], (v + w) is even for all i [d] and thus v + w 2 d. In particular, ∈ Z i ∈ ∩ Z then 1(v + w) d and since P is normal, 1(v + w) P d. This, however, implies that − 2∈ M − ∈ M − 2 (cid:54)∈ Z | ∩ Z | ≤ v w and 1(v w) . Thus, v w 2 d. Due to Lemma 5.3, n = P d 2d ≥ (cid:100) 2 (cid:101) (cid:3) and thus d log n . 2 Q Figure 4. Fiber graph embeddings of K , K , and K in 3. 5 6 7 Remark 5.6. Proposition 3.3 yields the trivial upper bound for complete graphs, that is ≤ − ≥ fdim(K ) n 1 for n 2. This bound is strict for the ﬁrst time for n = 4. n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: FIBER

text: DIMENSION

text: OF

text: A

text: GRAPH

text: L

text: Z

text: Q

text: Lemma

text: Let

text: n

text: be

text: a

text: lattice

text: of

text: full

text: rank

text: and

text: let

text: P

text: d

text: be

text: a

text: set

text: such

text: that

text: for

text: any

text: Z

text: L

text: Z

text: |Z

text: L|

text: distinct

text: v,

text: w

text: P

text: d,

text: v

text: w

text: Then

text: P

text: d

text: d/

text: Z

text: Z

text: Z

text: L

text: L

text: Proof.

text: Let

text: P

text: d

text: v

text: v

text: and

text: consider

text: the

text: linear

text: map

text: φ

text: d

text: d/

text: φ(v)

text: v

text: −1

text: n

text: Z

text: L

text: By

text: assumption,

text: φ(v

text: v

text: in

text: d/

text: for

text: all

text: i,

text: j

text: [n]

text: with

text: i

text: j.

text: Assume

text: that

text: there

text: are

text: i

text: j

text: i,

text: j

text: [n

text: with

text: i

text: j

text: such

text: that

text: φ(v

text: v

text: φ(v

text: v

text: Then

text: φ(v

text: v

text: φ(v

text: v

text: +v

text: v

text: n

text: i

text: n

text: j

text: j

text: i

text: n

text: n

text: j

text: φ(v

text: v

text: φ(v

text: v

text: a

text: contradiction.

text: Thus,

text: φ(v

text: v

text: φ(v

text: v

text: for

text: all

text: distinct

text: i∈

text: n−

text: j

text: n

text: n−

text: i

text: n

text: j

text: i,

text: j

text: [n

text: That

text: is,

text: φ(v

text: v

text: i

text: [n

text: n

text: The

text: proposition

text: follows

text: from

text: −n

text: i≤

text: |Z

text: L|

text: n

text: φ(v

text: v

text: i

text: [n

text: d/

text: n

text: i

text: |Z

text: L|

text: L

text: Remark

text: Since

text: n/

text: det(

text: Lemma

text: can

text: be

text: seen

text: as

text: a

text: discrete

text: analogon

text: of

text: Blichfeldt’s

text: theorem

text: Theorem

text: N

text: Theorem

text: For

text: any

text: n

text: fdim(K

text: log

text: n

text: n

text: Proof.

text: Due

text: to

text: Proposition

text: the

text: ﬁber

text: dimension

text: of

text: K

text: is

text: bounded

text: from

text: above

text: by

text: log

text: n

text: Q

text: n

text: M

text: Z

text: Let

text: d

text: ∼fdim(MKn)

text: and

text: P

text: d

text: a

text: d-dimens∈iona∩l

text: pZolytope

text: and

text: ∈d

text: a·

text: ZMarkov

text: basis

text: such

text: that

text: K

text: P

text: Assume

text: there

text: are

text: v,

text: w

text: P

text: d

text: such

text: that

text: v

text: w

text: d.

text: Since

text: (v

text: w)

text: n

text: Z

text: i

text: is

text: even

text: for

text: all

text: i

text: [d],

text: (v

text: w)

text: is

text: even

text: for

text: all

text: i

text: [d]

text: and

text: thus

text: v

text: w

text: d.

text: In

text: particular,

text: Z

text: i

text: Z

text: then

text: w)

text: d

text: and

text: since

text: P

text: is

text: normal,

text: w)

text: P

text: d.

text: This,

text: however,

text: implies

text: that

text: 2∈

text: M

text: M

text: Z

text: Z

text: v

text: w

text: and

text: w)

text: Thus,

text: v

text: w

text: d.

text: Due

text: to

text: Lemma

text: n

text: P

text: d

text: and

text: thus

text: d

text: log

text: n

text: Q

text: Figure

text: Fiber

text: graph

text: embeddings

text: of

text: K

text: K

text: and

text: K

text: in

text: Remark

text: Proposition

text: yields

text: the

text: trivial

text: upper

text: bound

text: for

text: complete

text: graphs,

text: that

text: is

text: fdim(K

text: n

text: for

text: n

text: This

text: bound

text: is

text: strict

text: for

text: the

text: ﬁrst

text: time

text: for

text: n

text: n

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: GAUSS-BONNET FOR MULTI-LINEAR VALUATIONS 17 Figure 3. The Wu curvatures of Adenine,Guanine, Cytosine and Thymine for which the Wu characteristic are 15,17,12 and 18. For graphs without triangles, the Wu characteristic gives an interac- tion energy, where bond-bond and vertex-vertex interactions count positive and vertex-bond interactions are negative. As bonds are mostly occupied by electrons and vertices by nuclei, this interpre- tation Wu functional has some merit. The Wu functional is maybe a too simple functional on molecules to be considered useful as it is even simpler than Hu¨ckel theory, which involves the eigenfunc- tions of the Laplacian. However, it could be important on a more fundamental level when looking at the fabric of space. 3. Linear valuations The f-vector v(G) of a ﬁnite simple graph G is deﬁned as v(G) = (v , v , . . . , v ) , 0 1 d where d is the maximal dimension of G. This means that d+1 is the clique number and v the volume, counting the number of facets, maximal cliques in G. All the d entries v (G) are invariant valuations. Hadwiger’s theorem shows that the list k v , . . . , v is a basis for the linear space of invariant valuations in G. While one can 0 d see the v as functionals on graphs, we look at it as a valuation, a functional on the k set of subgraphs of G. It naturally deﬁnes a functional on the set of simplicial sub complexes of the Whitney complex of G, which is the traditional way to look at valuations. The simplices in G form the analogue of convex sets in integral geometry or geometric probability and subgraphs of G are the analogue of ﬁnite union of ##LTFigure## ##LTFigure##

text: GAUSS-BONNET

label: paragraph

text: FOR

text: MULTI-LINEAR

text: VALUATIONS

text: Figure

text: The

label: caption

text: Wu

text: curvatures

text: of

text: Adenine,Guanine,

text: Cytosine

text: and

text: Thymine

text: for

text: which

text: the

text: Wu

text: characteristic

text: are

text: and

text: For

text: graphs

text: without

text: triangles,

text: the

text: Wu

text: characteristic

text: gives

text: an

text: interac-

text: tion

text: energy,

text: where

text: bond-bond

text: and

text: vertex-vertex

text: interactions

text: count

text: positive

text: and

text: vertex-bond

text: interactions

text: are

text: negative.

text: As

text: bonds

text: are

text: mostly

text: occupied

text: by

text: electrons

text: and

text: vertices

text: by

text: nuclei,

text: this

text: interpre-

text: tation

text: Wu

text: functional

text: has

text: some

text: merit.

text: The

text: Wu

text: functional

text: is

text: maybe

text: a

text: too

text: simple

text: functional

text: on

text: molecules

text: to

text: be

text: considered

text: useful

text: as

text: it

text: is

text: even

text: simpler

text: than

text: Hu¨ckel

text: theory,

text: which

text: involves

text: the

text: eigenfunc-

text: tions

text: of

text: the

text: Laplacian.

text: However,

text: it

text: could

text: be

text: important

text: on

text: a

text: more

text: fundamental

text: level

text: when

text: looking

text: at

text: the

text: fabric

text: of

text: space.

text: Linear

text: valuations

text: The

text: f-vector

text: v(G)

text: of

text: a

text: ﬁnite

text: simple

text: graph

text: G

text: is

text: deﬁned

text: as

text: v(G)

text: (v

text: v

text: v

text: d

text: where

text: d

text: is

text: the

text: maximal

text: dimension

text: of

text: G.

text: This

text: means

text: that

text: is

text: the

text: clique

text: number

text: and

text: v

text: the

text: volume,

text: counting

text: the

text: number

text: of

text: facets,

text: maximal

text: cliques

text: in

text: G.

text: All

text: the

text: d

text: entries

text: v

text: (G)

text: are

text: invariant

text: valuations.

text: Hadwiger’s

text: theorem

text: shows

text: that

text: the

text: list

text: k

text: v

text: v

text: is

text: a

text: basis

text: for

text: the

text: linear

text: space

text: of

text: invariant

text: valuations

text: in

text: G.

text: While

text: one

text: can

text: d

text: see

text: the

text: v

text: as

text: functionals

text: on

text: graphs,

text: we

text: look

text: at

text: it

text: as

text: a

text: valuation,

text: a

text: functional

text: on

text: the

text: k

text: set

text: of

text: subgraphs

text: of

text: G.

text: It

text: naturally

text: deﬁnes

text: a

text: functional

text: on

text: the

text: set

text: of

text: simplicial

text: sub

text: complexes

text: of

text: the

text: Whitney

text: complex

text: of

text: G,

text: which

text: is

text: the

text: traditional

text: way

text: to

text: look

text: at

text: valuations.

text: The

text: simplices

text: in

text: G

text: form

text: the

text: analogue

text: of

text: convex

text: sets

text: in

text: integral

text: geometry

text: or

text: geometric

text: probability

text: and

text: subgraphs

text: of

text: G

text: are

text: the

text: analogue

text: of

text: ﬁnite

text: union

text: of

text: ##LTFigure##

text: ##LTFigure##


text: 30 OLIVER KNILL − { } − ∅ Proof. Start with a single vertex x. Now B (x) = x and S (x) = . Now { } − i (x) = X( x ) 0. Now add recursively a new vertex to get a growing set of X,f graphs G which covers eventually G. By the properties of valuations, we have k ∪ − − − − (cid:3) X(G ) = X(G − B (x)) = X(G − ) + X(B (x)) X(S (x)). n n 1 n 1 See [85] for our ﬁrst proof in the case of Euler characteristic, and for [73] for a Morse theoretical inductive proof. We extended the result to valuations in Septem- ber 2015 while getting interested Barycentric characteristic numbers, a topic which emerged from from the construction of a graph product. Let Ω(G) denote the set of colorings of G, locally injective function f on V (G). R · Let P be a Borel probability measure on Ω(G) = v0(G) and let E[ ] its ex- pectation. Let c(G) be the chromatic number of G. Assume either that P is ≥ the counting measure on the ﬁnite set of colorings of G with c c(G) real colors → ∈ or that P is a product measure on Ω for which functions f f(y) with y V are independent identically distributed random variables with continuous probability ∈ G ∈ V density function. For all G and X : Theorem 6 (Index expectation). For any ﬁnite simple graph G and Euler char- acteristic X, we have E[i (x)] = K (x) . X,f X − Proof. Let V denote the number of k-dimensional simplices in S(x) and let V k − ∈ k the number of k-dimensional simplices in S (x). Given a vertex x V (G) and a f k-dimensional simplex K in S(x), the event k { | ∀ ∈ } A = f f(x) > f(y), y V (K ) k has probability 1/(k + 2). The reason is that the symmetric group of color per- mutations acts as measure preserving automorphisms on the probability space of functions, implying that for any f which is in A there are k +1 functions which are in the complement so that A has probability 1/(k + 2). This implies − V (x) E[V (x)] = k . k (k + 2) The same identity holds for co(cid:88)ntinuous probability space(cid:88)s. Therefore, ∞ ∞ − − − − − − − E[1 χ(S (x))] = 1 (cid:88)( 1)kE[V (x)] =(cid:88)1 + ( 1)kE[V − (x)] k k 1 k=0 k=1 ∞ ∞ − V − (x) − V − (x) = 1 + ( 1)k k 1 = ( 1)k k 1 = K(x) . (k + 1) (k + 1) k=1 k=0 (cid:3) See [87] for a proof in the case of continuous distributions. In [96] we adapted the result to ﬁnite probability spaces after getting interested in graph colorings. Assume now X(A, B) is a quadratic valuation like the Wu intersection number. → For a ﬁxed subgraph B the map A X(A, B) is a valuation and has a curvature K (x) as well as an index i (x). The Gauss-Bonnet and Poincar´e-Hopf results B f,B ##LTLine## ##LTLine## ##LTLine##

text: OLIVER

text: KNILL

text: Proof.

text: Start

text: with

text: a

text: single

text: vertex

text: x.

text: Now

text: B

text: (x)

text: x

text: and

text: S

text: (x)

text: Now

text: i

text: (x)

text: X(

text: x

text: Now

text: add

text: recursively

text: a

text: new

text: vertex

text: to

text: get

text: a

text: growing

text: set

text: of

text: X,f

text: graphs

text: G

text: which

text: covers

text: eventually

text: G.

text: By

text: the

text: properties

text: of

text: valuations,

text: we

text: have

text: k

text: X(G

text: X(G

text: B

text: (x))

text: X(G

text: X(B

text: (x))

text: X(S

text: (x)).

text: n

text: n

text: n

text: See

text: for

text: our

text: ﬁrst

text: proof

text: in

text: the

text: case

text: of

text: Euler

text: characteristic,

text: and

text: for

text: for

text: a

text: Morse

text: theoretical

text: inductive

text: proof.

text: We

text: extended

text: the

text: result

text: to

text: valuations

text: in

text: Septem-

text: ber

text: while

text: getting

text: interested

text: Barycentric

text: characteristic

text: numbers,

text: a

text: topic

text: which

text: emerged

text: from

text: from

text: the

text: construction

text: of

text: a

text: graph

text: product.

text: Let

text: Ω(G)

text: denote

text: the

text: set

text: of

text: colorings

text: of

text: G,

text: locally

text: injective

text: function

text: f

text: on

text: V

text: (G).

text: R

text: Let

text: P

text: be

text: a

text: Borel

text: probability

text: measure

text: on

text: Ω(G)

text: and

text: let

text: E[

text: its

text: ex-

text: pectation.

text: Let

text: c(G)

text: be

text: the

text: chromatic

text: number

text: of

text: G.

text: Assume

text: either

text: that

text: P

text: is

text: the

text: counting

text: measure

text: on

text: the

text: ﬁnite

text: set

text: of

text: colorings

text: of

text: G

text: with

text: c

text: c(G)

text: real

text: colors

text: or

text: that

text: P

text: is

text: a

text: product

text: measure

text: on

text: Ω

text: for

text: which

text: functions

text: f

text: f(y)

text: with

text: y

text: V

text: are

text: independent

text: identically

text: distributed

text: random

text: variables

text: with

text: continuous

text: probability

text: G

text: V

text: density

text: function.

text: For

text: all

text: G

text: and

text: X

text: Theorem

text: (Index

text: expectation).

text: For

text: any

text: ﬁnite

text: simple

text: graph

text: G

text: and

text: Euler

text: char-

text: acteristic

text: X,

text: we

text: have

text: E[i

text: (x)]

text: K

text: (x)

text: X,f

text: X

text: Proof.

text: Let

text: V

text: denote

text: the

text: number

text: of

text: k-dimensional

text: simplices

text: in

text: S(x)

text: and

text: let

text: V

text: k

text: k

text: the

text: number

text: of

text: k-dimensional

text: simplices

text: in

text: S

text: (x).

text: Given

text: a

text: vertex

text: x

text: V

text: (G)

text: and

text: a

text: f

text: k-dimensional

text: simplex

text: K

text: in

text: S(x),

text: the

text: event

text: k

text: A

text: f

text: f(x)

text: f(y),

text: y

text: V

text: (K

text: k

text: has

text: probability

text: The

text: reason

text: is

text: that

text: the

text: symmetric

text: group

text: of

text: color

text: per-

text: mutations

text: acts

text: as

text: measure

text: preserving

text: automorphisms

text: on

text: the

text: probability

text: space

text: of

text: functions,

text: implying

text: that

text: for

text: any

text: f

text: which

text: is

text: in

text: A

text: there

text: are

text: k

text: functions

text: which

text: are

text: in

text: the

text: complement

text: so

text: that

text: A

text: has

text: probability

text: This

text: implies

text: V

text: (x)

text: E[V

text: (x)]

text: k

text: k

text: (k

text: The

text: same

text: identity

text: holds

text: for

text: probability

text: Therefore,

text: χ(S

text: (x))]

text: (x)]

text: (x)]

text: k

text: k

text: V

text: (x)

text: V

text: (x)

text: k

text: k

text: K(x)

text: (k

text: (k

text: See

text: for

text: a

text: proof

text: in

text: the

text: case

text: of

text: continuous

text: distributions.

text: In

text: we

text: adapted

text: the

text: result

text: to

text: ﬁnite

text: probability

text: spaces

text: after

text: getting

text: interested

text: in

text: graph

text: colorings.

text: Assume

text: now

text: X(A,

text: B)

text: is

text: a

text: quadratic

text: valuation

text: like

text: the

text: Wu

text: intersection

text: number.

text: For

text: a

text: ﬁxed

text: subgraph

text: B

text: the

text: map

text: A

text: X(A,

text: B)

text: is

text: a

text: valuation

text: and

text: has

text: a

text: curvature

text: K

text: (x)

text: as

text: well

text: as

text: an

text: index

text: i

text: (x).

text: The

text: Gauss-Bonnet

text: and

text: Poincar´e-Hopf

text: results

text: B

text: f,B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 34 OLIVER KNILL Proof. We will see later that all Wu characteristics of a sphere are all the same and (cid:3) agree with the Euler characteristic. Poincar´e-Hopf generalizes to multi-linear valuations of higher degree: there is a function i (x , . . . , x ) which is symmetric in x , . . . , x ) and nonzero only if x , . . . , x f 1 k 1 k 1 k are in a simplex. The values of i (x , . . . , x ) can all be chip-ﬁred to vertices. But f 1 k now, for quadratic valuations, the index on vertices is a rational number of the form p/2. For reﬁned graphs, it is an integer. For cubic valuations, when chip-ﬁred onto vertices, the charges are now fractions of the form k/3. (cid:88) Theorem 10. The Wu characteristic ω satisﬁes k ω (G) = i (v , . . . , v ) k f 1 k × ···× V V V for an index function i . f 8. Product property The Cartesian product of two graphs G, H is deﬁned as the incidence graph of the · product f f in the Stanley-Reisner ring. We have introduced this product in G H [101] for graphs. (We had at that time not been aware of the Stanley ring [141]. There are various products already known in graph theory. The ring theoretically deﬁned product does the right thing on cohomology, with respect to dimension and also with respect to valuations like Euler characteristic. It has been known for simplicial complexes, but was so far unused in graph theory. It has the properties known from the continuum: in full generality, for any ﬁnite simple graph, it satisﬁes the Kuenneth formula [101] × ⊕ ⊗ Hk(G H) = Hi(G) Hj(G) . i+j=k It also satisﬁes the dimension inequality × ≥ dim(G H) dim(G) + dim(H) . These two results hold for general ﬁnite simple graphs which need not necessarily have to be related to any geometric setup. Like cohomology, homotopy or calculus, the product works for any network. × A special case of a graph product is G K which is the Barycentric reﬁne- − − 1 ment. As χ(G) = f ( 1), the Euler characteristic is multiplicative in general G on the class of ﬁnite simple graphs. One can see the multiplicity of the Euler (cid:80) characteristic also from the general Euler-Poincar´e formula equating cohomological and combinatorial Euler characteristic and the fact that the Poincar´e polynomial − p (x) = dim(Hk(G))xk satisﬁes p × (x) = p (x)p (y) and χ(G) = p ( 1) G k=0 × G H G H G which implies also χ(G H) = χ(G)χ(H). There is a also a direct algebraic way to see that the χ is preserved when taking → a Barycentric reﬁnement G G . Given a graph G, the vertices of G are the 1 − {− } 1 simplices of G. The function I(x) = ( 1)dim(x) is a 1, 1 - valued function on the vertices of G . If f is the polynomial in the Stanley ring and the mono- 1 G mials are ordered alphabetically, then this deﬁnes a function f on the vertex set − − of G . It turns out that i (x) = I(x). Proof: i (x) = 1 χ(S (x)). If x is 1 f f

text: OLIVER

text: KNILL

text: Proof.

text: We

text: will

text: see

text: later

text: that

text: all

text: Wu

text: characteristics

text: of

text: a

text: sphere

text: are

text: all

text: the

text: same

text: and

text: agree

text: with

text: the

text: Euler

text: characteristic.

text: Poincar´e-Hopf

text: generalizes

text: to

text: multi-linear

text: valuations

text: of

text: higher

text: degree:

text: there

text: is

text: a

text: function

text: i

text: (x

text: x

text: which

text: is

text: symmetric

text: in

text: x

text: x

text: and

text: nonzero

text: only

text: if

text: x

text: x

text: f

text: k

text: k

text: k

text: are

text: in

text: a

text: simplex.

text: The

text: values

text: of

text: i

text: (x

text: x

text: can

text: all

text: be

text: chip-ﬁred

text: to

text: vertices.

text: But

text: f

text: k

text: now,

text: for

text: quadratic

text: valuations,

text: the

text: index

text: on

text: vertices

text: is

text: a

text: rational

text: number

text: of

text: the

text: form

text: For

text: reﬁned

text: graphs,

text: it

text: is

text: an

text: integer.

text: For

text: cubic

text: valuations,

text: when

text: chip-ﬁred

text: onto

text: vertices,

text: the

text: charges

text: are

text: now

text: fractions

text: of

text: the

text: form

text: Theorem

text: The

text: Wu

text: characteristic

text: ω

text: satisﬁes

text: k

text: ω

text: (G)

text: i

text: (v

text: v

text: k

text: f

text: k

text: V

text: V

text: V

text: for

text: an

text: index

text: function

text: i

text: f

text: Product

text: property

text: The

text: Cartesian

text: product

text: of

text: two

text: graphs

text: G,

text: H

text: is

text: deﬁned

text: as

text: the

text: incidence

text: graph

text: of

text: the

text: product

text: f

text: f

text: in

text: the

text: Stanley-Reisner

text: ring.

text: We

text: have

text: introduced

text: this

text: product

text: in

text: G

text: H

text: for

text: graphs.

text: (We

text: had

text: at

text: that

text: time

text: not

text: been

text: aware

text: of

text: the

text: Stanley

text: ring

text: There

text: are

text: various

text: products

text: already

text: known

text: in

text: graph

text: theory.

text: The

text: ring

text: theoretically

text: deﬁned

text: product

text: does

text: the

text: right

text: thing

text: on

text: cohomology,

text: with

text: respect

text: to

text: dimension

text: and

text: also

text: with

text: respect

text: to

text: valuations

text: like

text: Euler

text: characteristic.

text: It

text: has

text: been

text: known

text: for

text: simplicial

text: complexes,

text: but

text: was

text: so

text: far

text: unused

text: in

text: graph

text: theory.

text: It

text: has

text: the

text: properties

text: known

text: from

text: the

text: continuum:

text: in

text: full

text: generality,

text: for

text: any

text: ﬁnite

text: simple

text: graph,

text: it

text: satisﬁes

text: the

text: Kuenneth

text: formula

text: Hk(G

text: H)

text: Hi(G)

text: Hj(G)

text: i+j=k

text: It

text: also

text: satisﬁes

text: the

text: dimension

text: inequality

text: dim(G

text: H)

text: dim(G)

text: dim(H)

text: These

text: two

text: results

text: hold

text: for

text: general

text: ﬁnite

text: simple

text: graphs

text: which

text: need

text: not

text: necessarily

text: have

text: to

text: be

text: related

text: to

text: any

text: geometric

text: setup.

text: Like

text: cohomology,

text: homotopy

text: or

text: calculus,

text: the

text: product

text: works

text: for

text: any

text: network.

text: A

text: special

text: case

text: of

text: a

text: graph

text: product

text: is

text: G

text: K

text: which

text: is

text: the

text: Barycentric

text: reﬁne-

text: ment.

text: As

text: χ(G)

text: f

text: the

text: Euler

text: characteristic

text: is

text: multiplicative

text: in

text: general

text: G

text: on

text: the

text: class

text: of

text: ﬁnite

text: simple

text: graphs.

text: One

text: can

text: see

text: the

text: multiplicity

text: of

text: the

text: Euler

text: characteristic

text: also

text: from

text: the

text: general

text: Euler-Poincar´e

text: formula

text: equating

text: cohomological

text: and

text: combinatorial

text: Euler

text: characteristic

text: and

text: the

text: fact

text: that

text: the

text: Poincar´e

text: polynomial

text: p

text: (x)

text: dim(Hk(G))xk

text: satisﬁes

text: p

text: (x)

text: p

text: (x)p

text: (y)

text: and

text: χ(G)

text: p

text: G

text: G

text: H

text: G

text: H

text: G

text: which

text: implies

text: also

text: χ(G

text: H)

text: χ(G)χ(H).

text: There

text: is

text: a

text: also

text: a

text: direct

text: algebraic

text: way

text: to

text: see

text: that

text: the

text: χ

text: is

text: preserved

text: when

text: taking

text: a

text: Barycentric

text: reﬁnement

text: G

text: G

text: Given

text: a

text: graph

text: G,

text: the

text: vertices

text: of

text: G

text: are

text: the

text: simplices

text: of

text: G.

text: The

text: function

text: I(x)

text: is

text: a

text: valued

text: function

text: on

text: the

text: vertices

text: of

text: G

text: If

text: f

text: is

text: the

text: polynomial

text: in

text: the

text: Stanley

text: ring

text: and

text: the

text: mono-

text: G

text: mials

text: are

text: ordered

text: alphabetically,

text: then

text: this

text: deﬁnes

text: a

text: function

text: f

text: on

text: the

text: vertex

text: set

text: of

text: G

text: It

text: turns

text: out

text: that

text: i

text: (x)

text: I(x).

text: Proof:

text: i

text: (x)

text: χ(S

text: (x)).

text: If

text: x

text: is

text: f

text: f


text: GAUSS-BONNET FOR MULTI-LINEAR VALUATIONS 5 a cubic f-form Vijk(G) counting the−(cid:88)numb−er of i, j or k-simplices in G which simultaneously intersect: with χ = (1, 1, 1, 1, . . . ), 1 ω (G) = V (G)χ χ χ = V (G)χ (i)χ (j)χ (k) . 3 1 1 1 ijk 1 1 1 ijk The higher order Wu characteristics ω are deﬁned similarly using k-linear f-forms k − V (G) or f-tensors. We have ω (K ) = χ(K ) = 1 and ω (K ) = ω (K ) = ( 1)d. 3 d d 4 d 2 d All these characteristics turn out to be invariant under Barycentric reﬁnement as well as have the multiplicative property and agree up to a sign with Euler charac- − teristic χ for d-graphs, graphs for which every unit sphere is a (d 1)-sphere. − Euler proved χ(G) = v e + f = 2 with v = v , e = v , f = v for planar graphs, 0 1 2 where f counts not only triangles but any region deﬁned by an embedding of G into a 2-sphere. In our terminology, graphs like the dodecahedron or cube graph are 1-dimensional, as they have no triangles. Among the ﬁve Platonic solids, only the octahedron and icosahedron are as 2-spheres, the tetrahedron is a 3-dimensional simplex, has Euler characteristic 1 and is contractible. We can reformulate Euler’s result then that for a discrete 2-sphere G, the Euler characteristic is 2. We know that χ is a homotopy invariant, that it is the ﬁrst Dehn-Sommerville invariant [81] and also that it is a Barycentric characteristic number to the eigenvalue 1, obtained by looking at the linear map A on f-vector v of the graph and taking an eigenvec- (cid:80) · tor a of AT and deﬁning X(G) = a v. As χ is the number of even-dimensional simplices minus the number of odd-dimensional simplices, it is the super trace of the identity operator on the v (G)-dimensional Hilbert space of discrete diﬀer- i i ential forms. It is robust under heat deformation: if d is the exterior derivative, (cid:80) − the discrete McKean-Singer formula [113, 88] tells that χ(G) = str(exp( tL)), ∗ where L = (d + d )2 is the Hodge Laplacian on discrete diﬀerential forms and − (cid:80) str(L) = ( 1)ktr(L ) if L is the part of L acting on k-forms. k k By the way, the total dimension X(G) = v (G) of all discrete diﬀerential forms i i is an example of an invariant valuation. It is believed that for d-graphs it is bounded − below by 3d 1 by a conjecture of Kalai of 1989 [143] (formulated for polytopes) as one can learn in [14]. This number is also interesting as it is hard to compute for general graphs, that it is the number of vertices of the Barycentric reﬁnement of G or the value f (1, 1, . . . , 1) if f is the Stanley-Reisner polynomial of G. The small- G (cid:80) G est d-sphere is believed to be the sharp lower bound like 8 attained for the C graph, 4 (cid:80) 26 attained for the octahedron, 80 for the 3-cross polytop or 242 for the 4-cross poly- (cid:80) top; the reason is that 1+ v xk+1 = (1+2x)d+1 for the smallest d-sphere. As the (cid:80) k k Kalai valuation X has the curvature K(x) = d+1 V − (x)/(k +1) which is always k=0 k 1 ∞ ≥ − positive, one could try to use Gauss-Bonnet and induction, as V (x) 3d 1 ≥ − k=0 k implying V − (x)/(k + 1) 3d 1/(d + 1). But since the number of vertices of ≥k k 1 ≥ − G is only 2d we have X 3d 1(2d/(d + 1)) = 3d(2/3)(d/(d + 1)). If one could ≤ verify the conjecture for all graphs G with 3d + 3 vertices, then the induction ≥ − assumption would give X (3d + 3)3d 1/(d + 1) = 3d. The Kalai conjecture is already by deﬁnition veriﬁable in ﬁnitely many cases for each d (in principle) but the just given argument reduces the veriﬁcation range from 3d to 3d+d. The argu- ≥ ≥ ment still gives inductively X(G) ad for any a < 2 so that X(G) ad holds for any graph with clique number d + 1. But since for G = K we have X(G) = 2d d+1

text: GAUSS-BONNET

text: FOR

text: MULTI-LINEAR

text: VALUATIONS

text: a

text: cubic

text: f-form

text: Vijk(G)

text: counting

text: of

text: i,

text: j

text: or

text: k-simplices

text: in

text: G

text: which

text: simultaneously

text: intersect:

text: with

text: χ

text: ω

text: (G)

text: V

text: (G)χ

text: χ

text: χ

text: V

text: (G)χ

text: (i)χ

text: (j)χ

text: (k)

text: ijk

text: ijk

text: The

text: higher

text: order

text: Wu

text: characteristics

text: ω

text: are

text: deﬁned

text: similarly

text: using

text: k-linear

text: f-forms

text: k

text: V

text: (G)

text: or

text: f-tensors.

text: We

text: have

text: ω

text: (K

text: χ(K

text: and

text: ω

text: (K

text: ω

text: (K

text: d

text: d

text: d

text: d

text: All

text: these

text: characteristics

text: turn

text: out

text: to

text: be

text: invariant

text: under

text: Barycentric

text: reﬁnement

text: as

text: well

text: as

text: have

text: the

text: multiplicative

text: property

text: and

text: agree

text: up

text: to

text: a

text: sign

text: with

text: Euler

text: charac-

text: teristic

text: χ

text: for

text: d-graphs,

text: graphs

text: for

text: which

text: every

text: unit

text: sphere

text: is

text: a

text: (d

text: Euler

text: proved

text: χ(G)

text: v

text: e

text: f

text: with

text: v

text: v

text: e

text: v

text: f

text: v

text: for

text: planar

text: graphs,

text: where

text: f

text: counts

text: not

text: only

text: triangles

text: but

text: any

text: region

text: deﬁned

text: by

text: an

text: embedding

text: of

text: G

text: into

text: a

text: In

text: our

text: terminology,

text: graphs

text: like

text: the

text: dodecahedron

text: or

text: cube

text: graph

text: are

text: as

text: they

text: have

text: no

text: triangles.

text: Among

text: the

text: ﬁve

text: Platonic

text: solids,

text: only

text: the

text: octahedron

text: and

text: icosahedron

text: are

text: as

text: the

text: tetrahedron

text: is

text: a

text: simplex,

text: has

text: Euler

text: characteristic

text: and

text: is

text: contractible.

text: We

text: can

text: reformulate

text: Euler’s

text: result

text: then

text: that

text: for

text: a

text: discrete

text: G,

text: the

text: Euler

text: characteristic

text: is

text: We

text: know

text: that

text: χ

text: is

text: a

text: homotopy

text: invariant,

text: that

text: it

text: is

text: the

text: ﬁrst

text: Dehn-Sommerville

text: invariant

text: and

text: also

text: that

text: it

text: is

text: a

text: Barycentric

text: characteristic

text: number

text: to

text: the

text: eigenvalue

text: obtained

text: by

text: looking

text: at

text: the

text: linear

text: map

text: A

text: on

text: f-vector

text: v

text: of

text: the

text: graph

text: and

text: taking

text: an

text: eigenvec-

text: tor

text: a

text: of

text: AT

text: and

text: deﬁning

text: X(G)

text: a

text: v.

text: As

text: χ

text: is

text: the

text: number

text: of

text: even-dimensional

text: simplices

text: minus

text: the

text: number

text: of

text: odd-dimensional

text: simplices,

text: it

text: is

text: the

text: super

text: trace

text: of

text: the

text: identity

text: operator

text: on

text: the

text: v

text: (G)-dimensional

text: Hilbert

text: space

text: of

text: discrete

text: diﬀer-

text: i

text: i

text: ential

text: forms.

text: It

text: is

text: robust

text: under

text: heat

text: deformation:

text: if

text: d

text: is

text: the

text: exterior

text: derivative,

text: the

text: discrete

text: McKean-Singer

text: formula

text: tells

text: that

text: χ(G)

text: str(exp(

text: tL)),

text: where

text: L

text: (d

text: d

text: is

text: the

text: Hodge

text: Laplacian

text: on

text: discrete

text: diﬀerential

text: forms

text: and

text: str(L)

text: if

text: L

text: is

text: the

text: part

text: of

text: L

text: acting

text: on

text: k-forms.

text: k

text: k

text: By

text: the

text: way,

text: the

text: total

text: dimension

text: X(G)

text: v

text: (G)

text: of

text: all

text: discrete

text: diﬀerential

text: forms

text: i

text: i

text: is

text: an

text: example

text: of

text: an

text: invariant

text: valuation.

text: It

text: is

text: believed

text: that

text: for

text: d-graphs

text: it

text: is

text: bounded

text: below

text: by

text: by

text: a

text: conjecture

text: of

text: Kalai

text: of

text: (formulated

text: for

text: polytopes)

text: as

text: one

text: can

text: learn

text: in

text: This

text: number

text: is

text: also

text: interesting

text: as

text: it

text: is

text: hard

text: to

text: compute

text: for

text: general

text: graphs,

text: that

text: it

text: is

text: the

text: number

text: of

text: vertices

text: of

text: the

text: Barycentric

text: reﬁnement

text: of

text: G

text: or

text: the

text: value

text: f

text: if

text: f

text: is

text: the

text: Stanley-Reisner

text: polynomial

text: of

text: G.

text: The

text: small-

text: G

text: G

text: est

text: d-sphere

text: is

text: believed

text: to

text: be

text: the

text: sharp

text: lower

text: bound

text: like

text: attained

text: for

text: the

text: C

text: graph,

text: attained

text: for

text: the

text: octahedron,

text: for

text: the

text: polytop

text: or

text: for

text: the

text: poly-

text: top;

text: the

text: reason

text: is

text: that

text: v

text: for

text: the

text: smallest

text: d-sphere.

text: As

text: the

text: k

text: k

text: Kalai

text: valuation

text: X

text: has

text: the

text: curvature

text: K(x)

text: V

text: (x)/(k

text: which

text: is

text: always

text: k

text: positive,

text: one

text: could

text: try

text: to

text: use

text: Gauss-Bonnet

text: and

text: induction,

text: as

text: V

text: (x)

text: k

text: implying

text: V

text: (x)/(k

text: But

text: since

text: the

text: number

text: of

text: vertices

text: of

text: ≥k

text: k

text: G

text: is

text: only

text: we

text: have

text: X

text: If

text: one

text: could

text: verify

text: the

text: conjecture

text: for

text: all

text: graphs

text: G

text: with

text: vertices,

text: then

text: the

text: induction

text: assumption

text: would

text: give

text: X

text: The

text: Kalai

text: conjecture

text: is

text: already

text: by

text: deﬁnition

text: veriﬁable

text: in

text: ﬁnitely

text: many

text: cases

text: for

text: each

text: d

text: (in

text: principle)

text: but

text: the

text: just

text: given

text: argument

text: reduces

text: the

text: veriﬁcation

text: range

text: from

text: to

text: The

text: argu-

text: ment

text: still

text: gives

text: inductively

text: X(G)

text: ad

text: for

text: any

text: a

text: so

text: that

text: X(G)

text: ad

text: holds

text: for

text: any

text: graph

text: with

text: clique

text: number

text: d

text: But

text: since

text: for

text: G

text: K

text: we

text: have

text: X(G)


text: 42 OLIVER KNILL 0 0 0 0 0 0 4 4 0 - 0 4 4 0 4 0 4 36 4 0 0 0 0 0 4 4 0 0 0 4 4 0 0 Figure 6. The Euler and Wu curvatures of a 1-dimensional − sphere bouquet of 5 spheres. The Euler characteristic is 4, the − Wu characteristic is 76 which is 4k2 5k + 1 for k = 5. 1 - 2 0 0 1 2 0 0 - 0 1 2 1 2 0 Figure 7. The Euler and Wu curvatures of a line graph, a 1- dimensional disk. The curvatures start to be diﬀerent in distance 1 to the boundary. The Euler characteristic is 1, the Wu characteris- − tic of the disk is 1. The one dimensional line graphs are the only connected graphs without triangle for which the Wu characteristic is negative. 3) adding a two dimensional sphere to a two dimensional graph increases the Wu characteristic by 1. This means that it decreases the curvature by 1. 4) Gluing a two and three dimensional sphere along a point produces a graph of Wu characteristic 1. − It follows from χ(G) χ(δG) that for 2-graphs with boundary, the Wu curvature is − 1 d(x)/6 in distance 2 to the boundary. At the boundary, the Wu curvature is 0. - 1 1 -1616 130 0 136-13-1 0 0136 -116 00 1616 1 016-131 16 16160 0 1 31 6 6 6 0 6 6 0 0 Figure 8. The Euler curvatures and Wu curvatures of a 2-disk. Both, Euler characteristic and Wu characteristic are 1. The later follows because the boundary, a circular graph, has zero Euler characteristic. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: OLIVER

text: KNILL

font: default

text: Figure

text: The

text: Euler

text: and

text: Wu

text: curvatures

text: of

text: a

text: sphere

text: bouquet

text: of

text: spheres.

text: The

text: Euler

text: characteristic

text: is

text: the

text: Wu

text: characteristic

text: is

text: which

text: is

text: for

text: k

text: Figure

text: The

text: Euler

text: and

text: Wu

text: curvatures

text: of

text: a

text: line

text: graph,

text: a

text: dimensional

text: disk.

text: The

text: curvatures

text: start

text: to

text: be

text: diﬀerent

text: in

text: distance

text: to

text: the

text: boundary.

text: The

text: Euler

text: characteristic

text: is

text: the

text: Wu

text: characteris-

text: tic

text: of

text: the

text: disk

text: is

text: The

text: one

text: dimensional

text: line

text: graphs

text: are

text: the

text: only

text: connected

text: graphs

text: without

text: triangle

text: for

text: which

text: the

text: Wu

text: characteristic

text: is

text: negative.

text: adding

text: a

text: two

text: dimensional

text: sphere

text: to

text: a

text: two

text: dimensional

text: graph

text: increases

text: the

text: Wu

text: characteristic

text: by

text: This

text: means

text: that

text: it

text: decreases

text: the

text: curvature

text: by

text: Gluing

text: a

text: two

text: and

text: three

text: dimensional

text: sphere

text: along

text: a

text: point

text: produces

text: a

text: graph

text: of

text: Wu

text: characteristic

text: It

text: follows

text: from

text: χ(G)

text: χ(δG)

text: that

text: for

text: with

text: boundary,

text: the

text: Wu

text: curvature

text: is

text: in

text: distance

text: to

text: the

text: boundary.

text: At

text: the

text: boundary,

text: the

text: Wu

text: curvature

text: is

text: Figure

text: The

text: Euler

text: curvatures

text: and

text: Wu

text: curvatures

text: of

text: a

text: Both,

text: Euler

text: characteristic

text: and

text: Wu

text: characteristic

text: are

text: The

text: later

text: follows

text: because

text: the

text: boundary,

text: a

text: circular

text: graph,

text: has

text: zero

text: Euler

text: characteristic.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: (cid:80) GAUSS-BONNET FOR MULTI-LINEAR VALUATIONS 49 − the Wu characteristic is K(x) = 1 5d/2 + d2/2 + d /2, where d is the ver- i i tex degree at x and d the neighboring vertex degrees. Quadratic valuations also i produce intersection numbers and are interesting for algebraic sets, where they pro- duces combinatorial invariants for varieties which go beyond Euler characteristic as the quadratic Dehn-Sommerville curvatures at isolated singularities report on their internal structure.

text: GAUSS-BONNET

text: FOR

text: MULTI-LINEAR

text: VALUATIONS

text: the

text: Wu

text: characteristic

text: is

text: K(x)

text: d

text: where

text: d

text: is

text: the

text: ver-

text: i

text: i

text: tex

text: degree

text: at

text: x

text: and

text: d

text: the

text: neighboring

text: vertex

text: degrees.

text: Quadratic

text: valuations

text: also

text: i

text: produce

text: intersection

text: numbers

text: and

text: are

text: interesting

text: for

text: algebraic

text: sets,

text: where

text: they

text: pro-

text: duces

text: combinatorial

text: invariants

text: for

text: varieties

text: which

text: go

text: beyond

text: Euler

text: characteristic

text: as

text: the

text: quadratic

text: Dehn-Sommerville

text: curvatures

text: at

text: isolated

text: singularities

text: report

text: on

text: their

text: internal

text: structure.


text: GAUSS-BONNET FOR MULTI-LINEAR VALUATIONS 55 Appendix: Code More detailed code and examples can be obtained by downloading the LaTeX source ﬁle of this preprint on the ArXiv, then copy paste the Mathematica part from the ﬁle. We compressed the code as much as possible as shortness of code beneﬁts both communication and veriﬁcation. The Wolfram language is suitable to serve as pseudo code. Even if the language might evolve in the future, the following code should remain readable. Since Mathematica has not yet built in a procedure to list of all complete subgraphs in a graph, we do that ﬁrst by hand. One can also use the iGraph library which uses compiled code and is faster. We don’t use it by default (cid:7) as this is a complex third party library. But the general problem is NP complete, so that one can not expect a very eﬃcient algorithm. CliqueNumber [ s ]:=Length[ Firs{t [ FindClique [ s ] ] ] ; {} } ListOfCliques [ s , k ]:=Module[ n, t ,m,u, r ,V,W,U, l = ,L ,L=Length; VL=VertexLis{t ;EL=}EdgeList ;V={VL[ s ] ;W=EL[ s ] ; m=}L[W{ ] ; n=L}[V] ; r=Subsets [V, k,k ];U=Table[ W[ [ j ,1]] ,W[ [ j ,2]] , j ,L[W] ]; If [k==1,l=V, If [k=−=2,l=U,Do[ t=Subgraph [ s , r{[ [ j ] ] ] ; } If [L[EL[ t]]==k(k 1{)/2, l=Append[ l ,VL[ t ]]] , j},L[ r ] ] ] ] ; l ] ; Whitney [ s ]:=Mo−dule[ F,a ,u,v,d,V,LC,L=Length ,V=VertexList [ s ] ; d=If [L[V]==0, 1,Clique{Number}[ s{] ] ;LC=L}istOfCliques ; (cid:6) If [d>=0,a [ x ]:=Table[ x [ [ k]] , k,L[x ] {]}; (cid:5) F[ t , l ]:= If [ l==1,a[LC[ t ,1]]{, If [ l =}=0, ,LC[ t ,{l}] ] ] ; u=Delete [Union[Table[F[ s , l ]{, l ,0 ,d ]] ,1}]; {v= ; } {} ∗Do[Do[v=Append[v,u [ [m, l ]]] , l ,L[u [ [m]]] ] , ∗m,L[u] ] ,v= ];v ] ; ( Import [”IGraphM ‘”]; Whitney=IGCliques ; ) (cid:7) Here is example code to compute the Euler characteristic χ Euler curvature entering Gauss Bonnet and Euler indices i (x) entering Poincar´e-Hopf: f { } Euler [ s ]{:=}Module[−c=Whitney [ s ] , { } If [ c== ,0,Sum[( 1)ˆ{(Len}gth[ c [ [ k]]]+1) , k,Length[ c ] ] ] ] ; EulerC [ s , v ]:=Module[ a ,n , a=Whitney [ VertexDelete [ N−eighborhoodG{raph [ s ,v] ,v ] }] ; 1+Sum[n=Length[ a [ [ k]]];( 1{)ˆn/(n+1), k,Len}gth[ a ] ] ] ; EulerCurvatures [ s ]:=Modu{le[ V=Vertex}List [ s ] , Table[ EulerC [ s ,V[ [ k ]]] , k,L{engt{h}[V] }] ] ; EulerIndex [ f , s , v ]:=Module[ a= ,t ,V , V=VertexList [ NeighborhoodGraph [ s ,v ] ] ; { } Do[ If [ f [ [V[ [ k]]]] <− f [ [ v ]] , a=Append[a ,V[ [ k ]]]] , k,Length[V] ]; (cid:6) t=Subgraph [ s , a];1 Euler [ t ]{] ; } (cid:5) EulerIndices [ f , s ]:=Module[ V={VertexList [ s }] , Table[ EulerIn{dex [ f , s},V[ [ k ]]] , k,Length[V] ] ] ; {s=RandomGraph[ 40 ,100 ]; } Euler [ s ] , Total [ EulerIndices [Range[40] , s ]] (cid:7) The following procedures compute the Wu characteristic and the cubic Wu Char- acteristic as well as the intersection number ω(G, H) { } {} Wu[ s ]:=Module[ c=Whitney [ s ] ,v=0,k , If [ c== ,1, Do[Do−[ If [Length[ Intersection [ c [ [ k ]] , c [ [ l ]]]] >0 , v{+=( 1)ˆ(Len}gth{[ c [ [ k]]]+Len}gth[ c [ [ l ] ] ] ) ; ] , k,Length[ c ] ]{, l ,Length[ c ] ]; v ] ] ; } {} Wu3[ s ]:=Module[ c=Whitney [ s ] ,v=0,k, l ,m , If [ c== ,v=1, Do[Do−[Do[ If [Length[ Intersection [ c [ [ k ]] , c [ [ l ]] , c [ [m]]]] >0 , v{+=( 1)ˆ(Len}gth{[ c [ [ k]]]+Len}gth{[ c [ [ l ]]]+Leng}th[ c [ [m]]]+1);] , k,Length[ c ] ] , l ,{Length[ c ] ] , m,Length[ c ] ] ] ; v ] ; } Wu[ s1 , s2 ]:=Module[ c1=Whitney [ s1 ] , c2=Whitney [ s2 ] ,v=0 , (cid:6) Do[Do[−If [Length[ Intersection [ c1 [ [ k ]] , c2 [ [ l ]]]] >0 , (cid:5) {v+=( 1)ˆ(Len}gth{[ c1 [ [ k]]]+Len}gth[ c2 [ [ l ] ] ] ) ] , k,Length[ c1 ] ] , l ,Length[ c2 ] ]; v ] ; { } Wu[RandomGraph[ 40 ,100 ]] Here are procedures to test the Gauss-Bonnet theorem for the Wu characteristic as well as the Poincar´e-Hopf theorem for the Wu characteristic, two results proven in this paper: ##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## ##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: GAUSS-BONNET

text: FOR

text: MULTI-LINEAR

text: VALUATIONS

text: Appendix:

label: section

text: Code

text: More

text: detailed

text: code

text: and

text: examples

text: can

text: be

text: obtained

text: by

text: downloading

text: the

text: LaTeX

text: source

text: ﬁle

text: of

text: this

text: preprint

text: on

text: the

text: ArXiv,

text: then

text: copy

text: paste

text: the

text: Mathematica

text: part

text: from

text: the

text: ﬁle.

text: We

text: compressed

text: the

text: code

text: as

text: much

text: as

text: possible

text: as

text: shortness

text: of

text: code

text: beneﬁts

text: both

text: communication

text: and

text: veriﬁcation.

text: The

text: Wolfram

text: language

text: is

text: suitable

text: to

text: serve

text: as

text: pseudo

text: code.

text: Even

text: if

text: the

text: language

text: might

text: evolve

text: in

text: the

text: future,

text: the

text: following

text: code

text: should

text: remain

text: readable.

text: Since

text: Mathematica

text: has

text: not

text: yet

text: built

text: in

text: a

text: procedure

text: to

text: list

text: of

text: all

text: complete

text: subgraphs

text: in

text: a

text: graph,

text: we

text: do

text: that

text: ﬁrst

text: by

text: hand.

text: One

text: can

text: also

text: use

text: the

text: iGraph

text: library

text: which

text: uses

text: compiled

text: code

text: and

text: is

text: faster.

text: We

text: don’t

text: use

text: it

text: by

text: default

text: as

text: this

text: is

text: a

text: complex

text: third

text: party

text: library.

text: But

text: the

text: general

text: problem

text: is

text: NP

text: complete,

text: so

text: that

text: one

text: can

text: not

text: expect

text: a

text: very

text: eﬃcient

text: algorithm.

text: CliqueNumber

text: s

text: ]:=Length[

text: Firs{t

text: FindClique

text: s

text: ListOfCliques

text: s

text: k

text: ]:=Module[

text: n,

text: t

text: ,m,u,

text: r

text: ,V,W,U,

text: l

text: ,L

text: ,L=Length;

text: VL=VertexLis{t

text: ;EL=}EdgeList

text: ;V={VL[

text: s

text: ;W=EL[

text: s

text: m=}L[W{

text: n=L}[V]

text: r=Subsets

text: [V,

text: k,k

text: ];U=Table[

text: W[

text: j

text: ,W[

text: j

text: j

text: ,L[W]

text: If

text: If

text: t=Subgraph

text: s

text: r{[

text: j

text: If

text: [L[EL[

text: t]]==k(k

text: l=Append[

text: l

text: ,VL[

text: t

text: j},L[

text: r

text: l

text: Whitney

text: s

text: ]:=Mo−dule[

text: F,a

text: ,u,v,d,V,LC,L=Length

text: ,V=VertexList

text: s

text: d=If

text: s{]

text: ;LC=L}istOfCliques

text: If

text: x

text: ]:=Table[

text: x

text: k]]

text: k,L[x

text: F[

text: t

text: l

text: If

text: t

text: If

text: l

text: ,LC[

text: t

text: ,{l}]

text: u=Delete

text: [Union[Table[F[

text: s

text: l

text: l

text: ,d

text: {v=

text: ∗Do[Do[v=Append[v,u

text: [m,

text: l

text: l

text: ,L[u

text: [m]]]

text: ∗m,L[u]

text: ,v=

text: ];v

text: Import

text: [”IGraphM

text: Whitney=IGCliques

text: Here

text: is

text: example

text: code

text: to

text: compute

text: the

text: Euler

text: characteristic

text: χ

text: Euler

text: curvature

text: entering

text: Gauss

text: Bonnet

text: and

text: Euler

text: indices

text: i

text: (x)

text: entering

text: Poincar´e-Hopf:

text: f

text: Euler

text: s

text: ]{:=}Module[−c=Whitney

text: s

text: If

text: c==

text: c

text: k,Length[

text: c

text: EulerC

text: s

text: v

text: ]:=Module[

text: a

text: ,n

text: a=Whitney

text: VertexDelete

text: N−eighborhoodG{raph

text: s

text: ,v]

text: ,v

text: a

text: k]]];(

text: k,Len}gth[

text: a

text: EulerCurvatures

text: s

text: ]:=Modu{le[

text: V=Vertex}List

text: s

text: Table[

text: EulerC

text: s

text: ,V[

text: k

text: k,L{engt{h}[V]

text: EulerIndex

text: f

text: s

text: v

text: ]:=Module[

text: a=

text: ,t

text: ,V

text: V=VertexList

text: NeighborhoodGraph

text: s

text: ,v

text: Do[

text: If

text: f

text: [V[

text: k]]]]

text: f

text: v

text: a=Append[a

text: ,V[

text: k

text: k,Length[V]

text: t=Subgraph

text: s

text: Euler

text: t

text: EulerIndices

text: f

text: s

text: ]:=Module[

text: V={VertexList

text: s

text: Table[

text: EulerIn{dex

text: f

text: s},V[

text: k

text: k,Length[V]

text: {s=RandomGraph[

text: Euler

text: s

text: Total

text: EulerIndices

text: s

text: The

text: following

text: procedures

text: compute

text: the

text: Wu

text: characteristic

text: and

text: the

text: cubic

text: Wu

text: Char-

text: acteristic

text: as

text: well

text: as

text: the

text: intersection

text: number

text: ω(G,

text: H)

text: Wu[

text: s

text: ]:=Module[

text: c=Whitney

text: s

text: If

text: c==

text: Do[Do−[

text: If

text: [Length[

text: Intersection

text: c

text: k

text: c

text: l

text: v{+=(

text: c

text: k]]]+Len}gth[

text: c

text: l

text: k,Length[

text: c

text: l

text: ,Length[

text: c

text: v

text: s

text: ]:=Module[

text: c=Whitney

text: s

text: l

text: ,m

text: If

text: c==

text: Do[Do−[Do[

text: If

text: [Length[

text: Intersection

text: c

text: k

text: c

text: l

text: c

text: [m]]]]

text: v{+=(

text: c

text: k]]]+Len}gth{[

text: c

text: l

text: ]]]+Leng}th[

text: c

text: k,Length[

text: c

text: l

text: ,{Length[

text: c

text: m,Length[

text: c

text: v

text: Wu[

text: ]:=Module[

text: Do[Do[−If

text: [Length[

text: Intersection

text: k

text: l

text: {v+=(

text: k]]]+Len}gth[

text: l

text: k,Length[

text: l

text: ,Length[

text: v

text: Wu[RandomGraph[

text: Here

text: are

text: procedures

text: to

text: test

text: the

text: Gauss-Bonnet

text: theorem

text: for

text: the

text: Wu

text: characteristic

text: as

text: well

text: as

text: the

text: Poincar´e-Hopf

text: theorem

text: for

text: the

text: Wu

text: characteristic,

text: two

text: results

text: proven

text: in

text: this

text: paper:

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text: 60 OLIVER KNILL random linear function f with equidistant level surfaces and produce the random ∈ k variable X (γ) N counting the number of intersections with the curve. The law of k large numbers shows that the expectation of this random variable can be explored by a Monte Carlo computation. It deﬁnes a length functional on curves, the property that it is additive is related to the additivity of probability. We have now only a | | semi-metric d(A, B) = inf γ on Rd but the Kolmogorov quotient is now a metric γ space. The Kolmogorov quotient just takes equivalence classes of points for which − the semimetric is zero. For the semi metric d((x , y ), (x , y )) = x x in the 1 1 2 2 1 2 plane for exmple, the Kolmogorov quotint is the real line. If µ is a measure which is invariant under translations and rotations, then it recovers the usual metric on Rd up to a scaling factor. This is the Crofton formula which for polygons reduces to the Buﬀon needle computation. If µ is a ﬁnite point measure, then the Kolmogorov quotient is a ﬁnite graph. This setup is rather general and most conveniently described in the projective situation, where translation is part of the projective group. Any compact Riemannian manifold can be treated like that: Nash embed it into some projective space Pd, look at the Haar measure on all linear functions on Pd invariant under projective transformations. Then, as the arc length on curves in Pd is the same than arc length on M, we can look at the measure µ on the set Ω of Morse functions on M given by the push forward of the Haar measure on linear functions to the space of Morse functions on M. This deﬁnes now a probability space on the class of Morse functions on M which allows to recover the Riemannian metric integral-geometrically within the manifold M. The point is however that by choosing a diﬀerent measure µ on Morse functions, we get diﬀerent metric spaces. In particular, if we take a discrete ﬁnite point measure, we get a discrete space. Going back to the case Rd, the probability space Ω is a set of linear | − | ≤ functions, we have convex sets obtained by inequalities f k c . Why does the i i dimension of valuations in Rd agree with the dimension of valuations on graphs? For any ﬁnite measures µ we have a (d + 1)-dimensional set of valuations by discrete m Hadwiger. Now approximate the original Haar measure with ﬁnite measures leading to d-dimensional graphs by doing Barycentric subdivisions, we can get to the Haar measure. Since valuations go over to the limit and the dimension of the set of valuations is upper semi continuous, the discrete approximation argument shows that we have at least a (d + 1)-dimensional set of valuations in the continuum. We can also go backwards and approximate a given discrete ﬁnite measure by absolutely continuous measures. Assume that the vector space of valuations is k-dimensional with k > d + 1. We would get k diﬀerent valuations in the discrete limit which is not the case. On simplices we can compute the k’th valuation by adding up the k- dimensional measures of k-dimensional subsimplices. On convex subsets K, we can make a triangulation and suﬃciently many Barycentric subdivisions in Euclidean space allowing to compute the valuation numerically by adding up the valuations on subsimplices. Smooth manifolds allowing an approximation by polytopes, the valuation is still deﬁned as a limit. The setup is much more than just a uniﬁcation the theory of valuations on Euclidean space or graphs. We can see that for any measure µ, which even might be singular continuous, we get to geometric spaces which have a Hadwiger theorem. These spaces can be objects with fractal dimension. They can be seen as approximations of graphs or then as limiting cases of smooth manifolds. The more general setup can use to make sense of curvature also on more general spaces as we can deﬁne curvature integral geometrically. We

text: OLIVER

text: KNILL

text: random

text: linear

text: function

text: f

text: with

text: equidistant

text: level

text: surfaces

text: and

text: produce

text: the

text: random

text: k

text: variable

text: X

text: (γ)

text: N

text: counting

text: the

text: number

text: of

text: intersections

text: with

text: the

text: curve.

text: The

text: law

text: of

text: k

text: large

text: numbers

text: shows

text: that

text: the

text: expectation

text: of

text: this

text: random

text: variable

text: can

text: be

text: explored

text: by

text: a

text: Monte

text: Carlo

text: computation.

text: It

text: deﬁnes

text: a

text: length

text: functional

text: on

text: curves,

text: the

text: property

text: that

text: it

text: is

text: additive

text: is

text: related

text: to

text: the

text: additivity

text: of

text: probability.

text: We

text: have

text: now

text: only

text: a

text: semi-metric

text: d(A,

text: B)

text: inf

text: γ

text: on

text: Rd

text: but

text: the

text: Kolmogorov

text: quotient

text: is

text: now

text: a

text: metric

text: γ

text: space.

text: The

text: Kolmogorov

text: quotient

text: just

text: takes

text: equivalence

text: classes

text: of

text: points

text: for

text: which

text: the

text: semimetric

text: is

text: zero.

text: For

text: the

text: semi

text: metric

text: d((x

text: y

text: (x

text: y

text: x

text: x

text: in

text: the

text: plane

text: for

text: exmple,

text: the

text: Kolmogorov

text: quotint

text: is

text: the

text: real

text: line.

text: If

text: µ

text: is

text: a

text: measure

text: which

text: is

text: invariant

text: under

text: translations

text: and

text: rotations,

text: then

text: it

text: recovers

text: the

text: usual

text: metric

text: on

text: Rd

text: up

text: to

text: a

text: scaling

text: factor.

text: This

text: is

text: the

text: Crofton

text: formula

text: which

text: for

text: polygons

text: reduces

text: to

text: the

text: Buﬀon

text: needle

text: computation.

text: If

text: µ

text: is

text: a

text: ﬁnite

text: point

text: measure,

text: then

text: the

text: Kolmogorov

text: quotient

text: is

text: a

text: ﬁnite

text: graph.

text: This

text: setup

text: is

text: rather

text: general

text: and

text: most

text: conveniently

text: described

text: in

text: the

text: projective

text: situation,

text: where

text: translation

text: is

text: part

text: of

text: the

text: projective

text: group.

text: Any

text: compact

text: Riemannian

text: manifold

text: can

text: be

text: treated

text: like

text: that:

text: Nash

text: embed

text: it

text: into

text: some

text: projective

text: space

text: Pd,

text: look

text: at

text: the

text: Haar

text: measure

text: on

text: all

text: linear

text: functions

text: on

text: Pd

text: invariant

text: under

text: projective

text: transformations.

text: Then,

text: as

text: the

text: arc

text: length

text: on

text: curves

text: in

text: Pd

text: is

text: the

text: same

text: than

text: arc

text: length

text: on

text: M,

text: we

text: can

text: look

text: at

text: the

text: measure

text: µ

text: on

text: the

text: set

text: Ω

text: of

text: Morse

text: functions

text: on

text: M

text: given

text: by

text: the

text: push

text: forward

text: of

text: the

text: Haar

text: measure

text: on

text: linear

text: functions

text: to

text: the

text: space

text: of

text: Morse

text: functions

text: on

text: M.

text: This

text: deﬁnes

text: now

text: a

text: probability

text: space

text: on

text: the

text: class

text: of

text: Morse

text: functions

text: on

text: M

text: which

text: allows

text: to

text: recover

text: the

text: Riemannian

text: metric

text: integral-geometrically

text: within

text: the

text: manifold

text: M.

text: The

text: point

text: is

text: however

text: that

text: by

text: choosing

text: a

text: diﬀerent

text: measure

text: µ

text: on

text: Morse

text: functions,

text: we

text: get

text: diﬀerent

text: metric

text: spaces.

text: In

text: particular,

text: if

text: we

text: take

text: a

text: discrete

text: ﬁnite

text: point

text: measure,

text: we

text: get

text: a

text: discrete

text: space.

text: Going

text: back

text: to

text: the

text: case

text: Rd,

text: the

text: probability

text: space

text: Ω

text: is

text: a

text: set

text: of

text: linear

text: functions,

text: we

text: have

text: convex

text: sets

text: obtained

text: by

text: inequalities

text: f

text: k

text: c

text: Why

text: does

text: the

text: i

text: i

text: dimension

text: of

text: valuations

text: in

text: Rd

text: agree

text: with

text: the

text: dimension

text: of

text: valuations

text: on

text: graphs?

text: For

text: any

text: ﬁnite

text: measures

text: µ

text: we

text: have

text: a

text: (d

text: set

text: of

text: valuations

text: by

text: discrete

text: m

text: Hadwiger.

text: Now

text: approximate

text: the

text: original

text: Haar

text: measure

text: with

text: ﬁnite

text: measures

text: leading

text: to

text: d-dimensional

text: graphs

text: by

text: doing

text: Barycentric

text: subdivisions,

text: we

text: can

text: get

text: to

text: the

text: Haar

text: measure.

text: Since

text: valuations

text: go

text: over

text: to

text: the

text: limit

text: and

text: the

text: dimension

text: of

text: the

text: set

text: of

text: valuations

text: is

text: upper

text: semi

text: continuous,

text: the

text: discrete

text: approximation

text: argument

text: shows

text: that

text: we

text: have

text: at

text: least

text: a

text: (d

text: set

text: of

text: valuations

text: in

text: the

text: continuum.

text: We

text: can

text: also

text: go

text: backwards

text: and

text: approximate

text: a

text: given

text: discrete

text: ﬁnite

text: measure

text: by

text: absolutely

text: continuous

text: measures.

text: Assume

text: that

text: the

text: vector

text: space

text: of

text: valuations

text: is

text: k-dimensional

text: with

text: k

text: d

text: We

text: would

text: get

text: k

text: diﬀerent

text: valuations

text: in

text: the

text: discrete

text: limit

text: which

text: is

text: not

text: the

text: case.

text: On

text: simplices

text: we

text: can

text: compute

text: the

text: k’th

text: valuation

text: by

text: adding

text: up

text: the

text: k-

text: dimensional

text: measures

text: of

text: k-dimensional

text: subsimplices.

text: On

text: convex

text: subsets

text: K,

text: we

text: can

text: make

text: a

text: triangulation

text: and

text: suﬃciently

text: many

text: Barycentric

text: subdivisions

text: in

text: Euclidean

text: space

text: allowing

text: to

text: compute

text: the

text: valuation

text: numerically

text: by

text: adding

text: up

text: the

text: valuations

text: on

text: subsimplices.

text: Smooth

text: manifolds

text: allowing

text: an

text: approximation

text: by

text: polytopes,

text: the

text: valuation

text: is

text: still

text: deﬁned

text: as

text: a

text: limit.

text: The

text: setup

text: is

text: much

text: more

text: than

text: just

text: a

text: uniﬁcation

text: the

text: theory

text: of

text: valuations

text: on

text: Euclidean

text: space

text: or

text: graphs.

text: We

text: can

text: see

text: that

text: for

text: any

text: measure

text: µ,

text: which

text: even

text: might

text: be

text: singular

text: continuous,

text: we

text: get

text: to

text: geometric

text: spaces

text: which

text: have

text: a

text: Hadwiger

text: theorem.

text: These

text: spaces

text: can

text: be

text: objects

text: with

text: fractal

text: dimension.

text: They

text: can

text: be

text: seen

text: as

text: approximations

text: of

text: graphs

text: or

text: then

text: as

text: limiting

text: cases

text: of

text: smooth

text: manifolds.

text: The

text: more

text: general

text: setup

text: can

text: use

text: to

text: make

text: sense

text: of

text: curvature

text: also

text: on

text: more

text: general

text: spaces

text: as

text: we

text: can

text: deﬁne

text: curvature

text: integral

text: geometrically.

text: We


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text: Berkeley,

text: third

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text: Staecker

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text: for

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text: maps.

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text: and

text: Commutative

text: Algebra.

text: Progress

text: in

text: Math.

text: Birkhaeuser,

text: second

text: edition,

text: R.

text: Stanley.

text: Positivity

text: problems

text: and

text: conjectures

text: in

text: algebraic

text: combinatorics.

text: In

text: P.

text: Lax

text: V.Arnold,

text: M.

text: Atiyah

text: and

text: B.

text: Mazur,

text: editors,

text: Mathematics:

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text: T.

text: Bijikoglu

text: and

text: P.

text: Stadler.

text: Laplacian

text: Eigenvectors

text: of

text: Graphs,

text: volume

text: of

text: Lecture

text: Notes

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text: Mathematics.

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text: T.

text: R´eti,

text: E.

text: Bitay,

text: Z.

text: Kosztol´anyi.

text: On

text: the

text: polyhedral

text: graphs

text: with

text: positive

text: combinatorial

text: curvature.

text: Acta

text: Polytechnica

text: Hungarica,

text: B.

text: Tng.

text: How

text: big

text: is

text: that

text: cookie?

text: the

text: integral

text: geometric

text: approach

text: to

text: geometric

text: quantitites.

text: Senior

text: thesis,


text: STRING-NODE NETS AND MESHES S.C. POWER AND B. SCHULZE Abstract. New classes of inﬁnite bond-node structures are introduced, namely string- node nets and meshes, a mesh being a string-node net for which the nodes are dense in the strings. Various construction schemes are given including the minimal extension of a (countable) line segment net by a countable scaling group. A linear mesh has strings that are straight lines and nodes given by the intersection points of these lines. Classes R R of meshes, such as the regular meshes in 2 and 3, are deﬁned and classiﬁed. String- length preserving motions are also determined for a number of fundamental examples and contrasting ﬂexing and rigidity properties are obtained with respect to noncrossing motions in the space of smooth meshes. 1. introduction A bar-joint framework is an abstract structure in a Euclidean space composed of rigid bars which are connected at joints where there is free articulation (Asimow and Roth [2],[3], Whiteley [31]). A tensegrity framework is a similar structure in which there are also cables, or strings, (and sometimes struts) which constrain the motion of the joints (Gruenbaum and Shephard [12], Roth and Whiteley [23], Snelson [26]). In what follows we move in a more mathematical direction and introduce new classes of distance-constrained structures, namely string-node nets and meshes. By a mesh we mean an inﬁnite string- node net for which the nodes are dense in the strings, which in turn may possibly be dense in the ambient space. A planar mesh could be likened to a fabric or material, made of inﬁnitesimally thin threads, except that the threads are joined pointwise rather than being interlaced, and the join points are dense on each thread. The primary constraint for ﬂexes and motions is that the internodal string lengths must be preserved. In fact in Sections 2, 3 and 4 we shall be concerned solely with string-node nets and meshes as static geometrical constructs. These structures are particularly interesting in the periodic case since their discrete counterparts provide the fundamental bond-node nets in crystallography and in reticular chemistry. In this connection Delgado-Friedrichs, O’Keeﬀe and Yaghi [7] have observed that there are exactly 5 regular discrete periodic nets in three dimensions. The subsequent papers [8], [9], [11] examine semiregular nets, quasi- regular nets and binodal networks, amongst other aspects, and indicate their appearance in material crystals. The 5-fold classiﬁcation of regular 3-periodic discrete bond-node nets is not so well-known in mathematical circles and, moreover, a simple formal self-contained proof does not seem to be ready-to-hand, and so we provide one here. In particular we give Date: SCP supported by EPSRC grant EP/J008648/1. BS supported by EPSRC grant EP/M013642/1. 2010 Mathematics Subject Classiﬁcation. 52C25, 52C22, 74N05 . Key words and phrases: periodic net, string-node mesh, rigidity, ﬂexibility, Sierpinski mesh. 1 ##LTLine##

text: STRING-NODE

text: NETS

text: AND

text: MESHES

text: S.C.

text: POWER

text: AND

text: B.

text: SCHULZE

text: Abstract.

text: New

label: abstract

text: classes

text: of

text: inﬁnite

text: bond-node

text: structures

text: are

text: introduced,

text: namely

text: string-

text: node

text: nets

text: and

text: meshes,

text: a

text: mesh

text: being

text: a

text: string-node

text: net

text: for

text: which

text: the

text: nodes

text: are

text: dense

text: in

text: the

text: strings.

text: Various

text: construction

text: schemes

text: are

text: given

text: including

text: the

text: minimal

text: extension

text: of

text: a

text: (countable)

text: line

text: segment

text: net

text: by

text: a

text: countable

text: scaling

text: group.

text: A

text: linear

text: mesh

text: has

text: strings

text: that

text: are

text: straight

text: lines

text: and

text: nodes

text: given

text: by

text: the

text: intersection

text: points

text: of

text: these

text: lines.

text: Classes

text: R

text: R

text: of

text: meshes,

text: such

text: as

text: the

text: regular

text: meshes

text: in

text: and

text: are

text: deﬁned

text: and

text: classiﬁed.

text: String-

text: length

text: preserving

text: motions

text: are

text: also

text: determined

text: for

text: a

text: number

text: of

text: fundamental

text: examples

text: and

text: contrasting

text: ﬂexing

text: and

text: rigidity

text: properties

text: are

text: obtained

text: with

text: respect

text: to

text: noncrossing

text: motions

text: in

text: the

text: space

text: of

text: smooth

text: meshes.

text: introduction

text: A

text: bar-joint

text: framework

text: is

text: an

text: abstract

text: structure

text: in

text: a

text: Euclidean

text: space

text: composed

text: of

text: rigid

text: bars

text: which

text: are

text: connected

text: at

text: joints

text: where

text: there

text: is

text: free

text: articulation

text: (Asimow

text: and

text: Roth

text: Whiteley

text: A

text: tensegrity

text: framework

text: is

text: a

text: similar

text: structure

text: in

text: which

text: there

text: are

text: also

text: cables,

text: or

text: strings,

text: (and

text: sometimes

text: struts)

text: which

text: constrain

text: the

text: motion

text: of

text: the

text: joints

text: (Gruenbaum

text: and

text: Shephard

text: Roth

text: and

text: Whiteley

text: Snelson

text: In

text: what

text: follows

text: we

text: move

text: in

text: a

text: more

text: mathematical

text: direction

text: and

text: introduce

text: new

text: classes

text: of

text: distance-constrained

text: structures,

text: namely

text: string-node

text: nets

text: and

text: meshes.

text: By

text: a

text: mesh

text: we

text: mean

text: an

text: inﬁnite

text: string-

text: node

text: net

text: for

text: which

text: the

text: nodes

text: are

text: dense

text: in

text: the

text: strings,

text: which

text: in

text: turn

text: may

text: possibly

text: be

text: dense

text: in

text: the

text: ambient

text: space.

text: A

text: planar

text: mesh

text: could

text: be

text: likened

text: to

text: a

text: fabric

text: or

text: material,

text: made

text: of

text: inﬁnitesimally

text: thin

text: threads,

text: except

text: that

text: the

text: threads

text: are

text: joined

text: pointwise

text: rather

text: than

text: being

text: interlaced,

text: and

text: the

text: join

text: points

text: are

text: dense

text: on

text: each

text: thread.

text: The

text: primary

text: constraint

text: for

text: ﬂexes

text: and

text: motions

text: is

text: that

text: the

text: internodal

text: string

text: lengths

text: must

text: be

text: preserved.

text: In

text: fact

text: in

text: Sections

text: and

text: we

text: shall

text: be

text: concerned

text: solely

text: with

text: string-node

text: nets

text: and

text: meshes

text: as

text: static

text: geometrical

text: constructs.

text: These

text: structures

text: are

text: particularly

text: interesting

text: in

text: the

text: periodic

text: case

text: since

text: their

text: discrete

text: counterparts

text: provide

text: the

text: fundamental

text: bond-node

text: nets

text: in

text: crystallography

text: and

text: in

text: reticular

text: chemistry.

text: In

text: this

text: connection

text: Delgado-Friedrichs,

text: O’Keeﬀe

text: and

text: Yaghi

text: have

text: observed

text: that

text: there

text: are

text: exactly

text: regular

text: discrete

text: periodic

text: nets

text: in

text: three

text: dimensions.

text: The

text: subsequent

text: papers

text: examine

text: semiregular

text: nets,

text: quasi-

text: regular

text: nets

text: and

text: binodal

text: networks,

text: amongst

text: other

text: aspects,

text: and

text: indicate

text: their

text: appearance

text: in

text: material

text: crystals.

text: The

text: classiﬁcation

text: of

text: regular

text: discrete

text: bond-node

text: nets

text: is

text: not

text: so

text: well-known

text: in

text: mathematical

text: circles

text: and,

text: moreover,

text: a

text: simple

text: formal

text: self-contained

text: proof

text: does

text: not

text: seem

text: to

text: be

text: ready-to-hand,

text: and

text: so

text: we

text: provide

text: one

text: here.

text: In

text: particular

text: we

text: give

text: Date:

text: SCP

text: supported

text: by

text: EPSRC

text: grant

text: BS

text: supported

text: by

text: EPSRC

text: grant

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Key

text: words

text: and

text: phrases:

text: periodic

text: net,

text: string-node

text: mesh,

text: rigidity,

text: ﬂexibility,

text: Sierpinski

text: mesh.

text: ##LTLine##


text: 2 S.C. POWER AND B. SCHULZE an elementary direct construction of the most intriguing example, namely the K -crystal 4 net (Coxeter [6], Hyde et al [13], Sunada [27]). We make use of the discrete classiﬁcation to obtain, in Theorem 4.2, a classiﬁcation of regular meshes in dimensions 2 and 3 up to conformal aﬃne isomorphism. We also indicate and comment on the correspondence between our constructs, notation and terminology and that used by reticular chemists and a summary table of this correspondence appears at the end of Section 3. The characterisation of regular linear meshes shows that they are parametrised by cer- N N tain pairs , F where is a regular discrete string-node net and F is a dense countable R abelian subgroup of . Moreover we show that certain strongly regular meshes correspond to F being a countable subﬁeld. Motivated by this relationship we also show, in Theorem 4.7, how one may construct a minimal extension mesh of any discrete line segment net by any countable ﬁeld. The regular and strongly regular linear meshes provide the most basic meshes in two and three dimensions where the space group acts with full transitivity. As we indicate in Section 3 it will be of interest to enlarge the classiﬁcations here to meshes with weaker forms of transitivity. In Section 5 we turn to some fundamental dynamical considerations and initiate a theory of rigidity and ﬂexibility for string-node meshes. We focus on continuous motions that preserve internodal string lengths and which satisfy a natural noncrossing condition, so that each motion is in fact a path of injective placements. It is shown that for a square grid mesh in the unit square such motions, if smooth, necessarily have a simple laminar form for some small ﬁnite time interval. On the other hand we show that the triadic kagome mesh is rigid for continuously diﬀerentiable motions. This contrasts with the many ways in which the discrete kagome bar-joint framework is periodically ﬂexible. We also construct highly ﬂexible line segment meshes which, unlike the square grid meshes, admit localised ﬁnite motions. In this case, to pursue the allusion above, the motions are more akin to that of a planar ﬁbred liquid, or liquid crystal, rather than that of an idealised ﬁbred material. We note that there are some similar aspects of rigidity and ﬂexibility in the analysis of ”continuous tensegrities” considered by Ashton [1], and in the very recent analysis of semidiscrete surfaces in Karpenkov [15]. Nevertheless meshes are in a quite diﬀerent category, having no rigid basic components, and analysing their string-length preserving motions require new methods. In the ﬁnal section we indicate some further directions and problem areas, including two problems which we paraphrase here as follows. 1. In three dimensions the noncrossing motion of a grid mesh cube need not have a strictly laminar form for small time values, in the sense of being axially determined. Classify these small time motions. 2. The line segment mesh associated with the Sierpinski triangle is a limit of minimally rigid bar-joint frameworks. Determine if this string-node mesh is rigid with respect to noncrossing continuous string-length-preserving motions. We would like to thank Davide Proserpio and Egon Schulte for alerting us to articles in reticular chemistry, to the Reticular Chemistry Structure Resource, and to articles on the classiﬁcation of discrete periodic nets.

text: S.C.

text: POWER

text: AND

text: B.

text: SCHULZE

text: an

text: elementary

text: direct

text: construction

text: of

text: the

text: most

text: intriguing

text: example,

text: namely

text: the

text: K

text: -crystal

text: net

text: (Coxeter

text: Hyde

text: et

text: al

text: Sunada

text: We

text: make

text: use

text: of

text: the

text: discrete

text: classiﬁcation

text: to

text: obtain,

text: in

text: Theorem

text: a

text: classiﬁcation

text: of

text: regular

text: meshes

text: in

text: dimensions

text: and

text: up

text: to

text: conformal

text: aﬃne

text: isomorphism.

text: We

text: also

text: indicate

text: and

text: comment

text: on

text: the

text: correspondence

text: between

text: our

text: constructs,

text: notation

text: and

text: terminology

text: and

text: that

text: used

text: by

text: reticular

text: chemists

text: and

text: a

text: summary

text: table

text: of

text: this

text: correspondence

text: appears

text: at

text: the

text: end

text: of

text: Section

text: The

text: characterisation

text: of

text: regular

text: linear

text: meshes

text: shows

text: that

text: they

text: are

text: parametrised

text: by

text: cer-

text: N

text: N

text: tain

text: pairs

text: F

text: where

text: is

text: a

text: regular

text: discrete

text: string-node

text: net

text: and

text: F

text: is

text: a

text: dense

text: countable

text: R

text: abelian

text: subgroup

text: of

text: Moreover

text: we

text: show

text: that

text: certain

text: strongly

text: regular

text: meshes

text: correspond

text: to

text: F

text: being

text: a

text: countable

text: subﬁeld.

text: Motivated

text: by

text: this

text: relationship

text: we

text: also

text: show,

text: in

text: Theorem

text: how

text: one

text: may

text: construct

text: a

text: minimal

text: extension

text: mesh

text: of

text: any

text: discrete

text: line

text: segment

text: net

text: by

text: any

text: countable

text: ﬁeld.

text: The

text: regular

text: and

text: strongly

text: regular

text: linear

text: meshes

text: provide

text: the

text: most

text: basic

text: meshes

text: in

text: two

text: and

text: three

text: dimensions

text: where

text: the

text: space

text: group

text: acts

text: with

text: full

text: transitivity.

text: As

text: we

text: indicate

text: in

text: Section

text: it

text: will

text: be

text: of

text: interest

text: to

text: enlarge

text: the

text: classiﬁcations

text: here

text: to

text: meshes

text: with

text: weaker

text: forms

text: of

text: transitivity.

text: In

text: Section

text: we

text: turn

text: to

text: some

text: fundamental

text: dynamical

text: considerations

text: and

text: initiate

text: a

text: theory

text: of

text: rigidity

text: and

text: ﬂexibility

text: for

text: string-node

text: meshes.

text: We

text: focus

text: on

text: continuous

text: motions

text: that

text: preserve

text: internodal

text: string

text: lengths

text: and

text: which

text: satisfy

text: a

text: natural

text: noncrossing

text: condition,

text: so

text: that

text: each

text: motion

text: is

text: in

text: fact

text: a

text: path

text: of

text: injective

text: placements.

text: It

text: is

text: shown

text: that

text: for

text: a

text: square

text: grid

text: mesh

text: in

text: the

text: unit

text: square

text: such

text: motions,

text: if

text: smooth,

text: necessarily

text: have

text: a

text: simple

text: laminar

text: form

text: for

text: some

text: small

text: ﬁnite

text: time

text: interval.

text: On

text: the

text: other

text: hand

text: we

text: show

text: that

text: the

text: triadic

text: kagome

text: mesh

text: is

text: rigid

text: for

text: continuously

text: diﬀerentiable

text: motions.

text: This

text: contrasts

text: with

text: the

text: many

text: ways

text: in

text: which

text: the

text: discrete

text: kagome

text: bar-joint

text: framework

text: is

text: periodically

text: ﬂexible.

text: We

text: also

text: construct

text: highly

text: ﬂexible

text: line

text: segment

text: meshes

text: which,

text: unlike

text: the

text: square

text: grid

text: meshes,

text: admit

text: localised

text: ﬁnite

text: motions.

text: In

text: this

text: case,

text: to

text: pursue

text: the

text: allusion

text: above,

text: the

text: motions

text: are

text: more

text: akin

text: to

text: that

text: of

text: a

text: planar

text: ﬁbred

text: liquid,

text: or

text: liquid

text: crystal,

text: rather

text: than

text: that

text: of

text: an

text: idealised

text: ﬁbred

text: material.

text: We

text: note

text: that

text: there

text: are

text: some

text: similar

text: aspects

text: of

text: rigidity

text: and

text: ﬂexibility

text: in

text: the

text: analysis

text: of

text: ”continuous

text: tensegrities”

text: considered

text: by

text: Ashton

text: and

text: in

text: the

text: very

text: recent

text: analysis

text: of

text: semidiscrete

text: surfaces

text: in

text: Karpenkov

text: Nevertheless

text: meshes

text: are

text: in

text: a

text: quite

text: diﬀerent

text: category,

text: having

text: no

text: rigid

text: basic

text: components,

text: and

text: analysing

text: their

text: string-length

text: preserving

text: motions

text: require

text: new

text: methods.

text: In

text: the

text: ﬁnal

text: section

text: we

text: indicate

text: some

text: further

text: directions

text: and

text: problem

text: areas,

text: including

text: two

text: problems

text: which

text: we

text: paraphrase

text: here

text: as

text: follows.

text: In

text: three

text: dimensions

text: the

text: noncrossing

text: motion

text: of

text: a

text: grid

text: mesh

text: cube

text: need

text: not

text: have

text: a

text: strictly

text: laminar

text: form

text: for

text: small

text: time

text: values,

text: in

text: the

text: sense

text: of

text: being

text: axially

text: determined.

text: Classify

text: these

text: small

text: time

text: motions.

text: The

text: line

text: segment

text: mesh

text: associated

text: with

text: the

text: Sierpinski

text: triangle

text: is

text: a

text: limit

text: of

text: minimally

text: rigid

text: bar-joint

text: frameworks.

text: Determine

text: if

text: this

text: string-node

text: mesh

text: is

text: rigid

text: with

text: respect

text: to

text: noncrossing

text: continuous

text: string-length-preserving

text: motions.

text: We

text: would

text: like

text: to

text: thank

text: Davide

text: Proserpio

text: and

text: Egon

text: Schulte

text: for

text: alerting

text: us

text: to

text: articles

text: in

text: reticular

text: chemistry,

text: to

text: the

text: Reticular

text: Chemistry

text: Structure

text: Resource,

text: and

text: to

text: articles

text: on

text: the

text: classiﬁcation

text: of

text: discrete

text: periodic

text: nets.