The Tutte polynomial, its applications and generalizations. Sergei Chmutov The Ohio State University, Mansfield

Size: px

Start display at page:

Download "The Tutte polynomial, its applications and generalizations. Sergei Chmutov The Ohio State University, Mansfield"

Magnus Holt
5 years ago
Views:

1 University of South Alabama The Tutte polynomial, its applications and generalizations Sergei Chmutov The Ohio State University, Mansfield Thursday, April 5, 00 :0 4:0 p.m.

2 Chromatic polynomial C Γ (q) := # of proper colorings of V (Γ) in q colors Example. C (q) = q(q )(q q + ) q = : Properties: C Γ = C Γe C Γ/e, C Γ Γ = C Γ C Γ, C = q. S := C Γ (q) = σ S { } V (Γ) {,..., q} (a,b) E(Γ) ( ) δ(σ(a), σ(b)) Dichromatic polynomial Z Γ (q, v) := ( + vδ(σ(a), σ(b))) σ S C Γ (q) = Z Γ (q, ). (a,b) E(Γ) Z Γ (q, v) = F E(Γ) q k(f) v e(f). Properties: Z Γ = Z Γe +vz Γ/e, Z Γ Γ = Z Γ Z Γ, Z = q.

Let Γ be a graph; The Tutte polynomial v(γ) be the number of its vertices; e(γ) be the number of its edges; k(γ) be the number of components of Γ; r(γ) := v(γ) k(γ) be the rank of Γ; n(γ) := e(γ)

3 Let Γ be a graph; The Tutte polynomial v(γ) be the number of its vertices; e(γ) be the number of its edges; k(γ) be the number of components of Γ; r(γ) := v(γ) k(γ) be the rank of Γ; n(γ) := e(γ) r(γ) be the nullity of Γ; T Γ (x, y) := Properties. F E(Γ) (x ) r(γ)r(f) (y ) n(f) Z Γ (q, v) = q k(γ) v r(γ) T Γ ( + qv, + v) T Γ = T Γe + T Γ/e if e is neither a bridge nor a loop ; T Γ = xt Γ/e if e is a bridge ; T Γ = yt Γe if e is a loop ; T Γ Γ = T Γ Γ = T Γ T Γ for a disjoint union, Γ Γ T =. and a one-point join, Γ Γ ; T Γ (, ) is the number of spanning trees of Γ ; T Γ (, ) is the number of spanning forests of Γ ; T Γ (, ) is the number of spanning connected subgraphs of Γ ; T Γ (, ) = E(Γ) is the number of spanning subgraphs of Γ.

4 The Potts model C.Domb (95). q = the Ising model; W.Lenz (90). Atoms are located at the sites of vertices V (Γ). Nearest neighbors are indicated by edges E(Γ). An atom exists in one of q different states (spins). A state, σ S, is an assignments of spins to all vertices V (Γ). Neighboring atoms interact with each other only is their spins are the same. The energy of the interaction is J (coupling constant). The model is called ferromagnetic if J > 0 and antiferromagnetic if J < 0. Energy of a state σ (Hamiltonian), H(σ) = J δ(σ(a), σ(b)). (a,b) E(Γ) Boltzmann weight of σ: e βh(σ) = e Jβδ(σ(a),σ(b)) = (a,b) E(Γ) (a,b) E(Γ) ( ) +(e Jβ )δ(σ(a), σ(b)), where the inverse temperature β = κ T, T is the temperature, κ =.8 0 joules/kelvin is the Boltzmann constant. The Potts partition function Z Potts Γ := σ S e βh(σ) = Z Γ (q, e Jβ ) unphysical region antiferromagnetic 0 ferromagnetic region v = e Jβ

5 Probability of a state σ: P(σ) := e βh(σ) /Z Γ. Expected value of a function f(σ): f := σ f(σ)p(σ) = σ f(σ)e βh(σ) /Z Γ. Expected energy: H = σ H(σ)e βh(σ) /Z Γ = d dβ lnz Γ. Example. Γ = } {{. } n vertices T Γ = x n, Z Γ = qv n ( + qv ) n = q(q + v) n = q(q + e βj ) n. Je βj Expected energy: H = (n ) q + e. βj Expected energy per atom as n : H lim n n = JeβJ q + e βj. T (β 0): The energy per atom J/q. T 0 (β ). J < 0 (antiferromagnetic): The energy per atom 0. In general, e βj 0 and the partition function Z Γ (q, ) = C Γ (q). J > 0 (ferromagnetic): The energy per atom J.

6 Morwen Thistlethwaite (987) Up to a sign and a power of t the Jones polynomial V L (t) of an alternating link L is equal to the Tutte polynomial T ΓL (t, t ). L Γ L V L (t) = t + t t 4 T ΓL (x, y) = y + x + x = t (t t + t ) T ΓL (t, t ) = t t + t

7 The Kauffman bracket Let L be a virtual link diagram. A-splitting A state S is a choice of either A- or B-splitting at every classical crossing. B-splitting α(s) = #(of A-splittings in S) β(s) = #(of B-splittings in S) δ(s) = #(of circles in S) [L](A, B, d) := S A α(s) B β(s) d δ(s) J L (t) := () w(l) t w(l)/4 [L](t /4, t /4, t / t / ) Example (α, β, δ) (, 0, ) (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) (0,, ) [L] = A + A Bd + AB + AB d + B d ; J L (t) =

8 Graphs on surfaces Ribbon graphs A ribbon graph G is a surface represented as a union of verticesdiscs and edges-ribbons discs and ribbons intersect by disjoint line segments, each such line segment lies on the boundary of precisely one vertex and precisely one edge; every edge contains exactly two such line segments. + = +

9 The Bollobás-Riordan polynomial Let F be a ribbon graph; v(f) be the number of its vertices; e(f) be the number of its edges; k(f) be the number of components of F; r(f) := v(f) k(f) be the rank of F; n(f) := e(f) r(f) be the nullity of F; bc(f) be the number of boundary components of F; s(f) := e (F) e (F). R G (x, y, z) := x r(g)r(f)+s(f) y n(f)s(f) z k(f)bc(f)+n(f) F Relations to the Tutte polynomial. R G (x, y, ) = T G (x, y) If G is planar (genus zero): R G (x, y, z) = T G (x, y)

10 Example. + (k, r, n, bc, s) (,,,, ) (,, 0,, 0) (,, 0,, 0) (, 0, 0,, ) (,,,, ) (,,,, 0) (,,,, 0) (, 0,,, ) r(f) := v(f) k(f); n(f) := e(g) r(f); bc(f) is the number of boundary components; s(f) := e (F) e (F). R G (x, y, z) = x + + y + xyz + yz + y z.

11 Construction of a ribbon graph from a virtual link diagram L A B B Diagram State s + Attaching planar bands Replacing bands by arrows Untwisting state circles + ; ; Pulling state circles apart + Forming the ribbon graph G s L

12 Theorem Let L be a virtual link diagram with e classical crossings, G s L be the signed ribbon graph corresponding to a state s, and v := v(g s L ), k := k(gs L ). Then e = e(gs L ) and [L](A, B, d) = A (x e k y v z v+ R G s L (x, y, z) x= Ad B, y= Bd A, z= d ). Idea of the proof. One-to-one correspondence between states s of L and spanning subgraphs F of G s L : An edge e of G s L belongs to the spanning subgraph F if and only if the corresponding crossing was split in s differently comparably with s.

13 Iain Moffatt: Further developments Knot invariants and the Bollobás-Riordan polynomial of embedded graphs, European Journal of Combinatorics, 9 (008) arxiv:math/ Partial duality and Bollobás and Riordan s ribbon graph polynomial, Discrete Mathematics, 0 (00) arxiv: A characterization of partially dual graphs. arxiv: Fabien Vignes-Tourneret: The multivariate signed Bollobás-Riordan polynomial, Discrete Mathematics, 09 (009) arxiv: (joint with T. Krajewski, V. Rivasseau) Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories. arxiv: (non-commutative Grosse- Wulkenhaar quantum field theory)

SERGEI CHMUTOV AND IGOR PAK

Dedicated to Askold Khovanskii on the occasion of his 60th birthday THE KAUFFMAN BRACKET OF VIRTUAL LINKS AND THE BOLLOBÁS-RIORDAN POLYNOMIAL SERGEI CHMUTOV AND IGOR PAK Abstract. We show that the Kauffman