Where do topological Tutte polynomial come from? Iain Moffatt - - PowerPoint PPT Presentation

Where do topological Tutte polynomial come from?

Iain Moffatt

Carolyn Chun, Jo Ellis-Monaghan, Thomas Krajewski, Steve Noble, Ralf Rueckriemen, Ben Smith, Adrian Tanasa

Royal Holloway, University of London

Dagstuhl, 13th June 2016

16 1

Background Minors Constructing the polynomials Matroids Remarks

A review of the T utte polynomial

The T utte polynomial, T(G; x, y) ∈ Z[x, y]

T(G) =          1 if G edgeless xT(G/e) if e a bridge yT(G\e) if e a loop T(G\e) + T(G/e)

• therwise

E.g., T

• = x2 + x + y

Theorem

◮ T(G) is well-defined. ◮ T(G) = A⊆E

(x − 1)r(G)−r(A)(y − 1)|A|−r(A)

◮ r(A) = #verts. − #cpts. of (V, A) . ◮ Defined for matroids - take r to to be rank function. ◮ T(C(G)) = T(G), where C(G) is cycle matroid

16 2

Background Minors Constructing the polynomials Matroids Remarks

Embedded graphs = graphs in surfaces

◮ Plane graph - drawn on a sphere, edges don’t

meet, faces are disks.

◮ embedded graph = graph in surface - drawn on

surface, edges don’t meet.

◮ cellularly embedded graph - drawn on surface,

faces are disks.

16 3

Background Minors Constructing the polynomials Matroids Remarks

Three T

• pological T

utte polynomials

◮ M. Las Vergnas’ 1978 polynomial, L(G; x, y, z)

(y − 1)z2 · L(G; x, y, 1/(y − 1)z2) :=

• A⊆E

(x − 1)r(G)−r(A)(y − 1)|A|−r(A)zγG(A)−γG∗(E\A)

◮ B. Bollobás and O. Riordan’s 2002 polynomial

R(G; x, y, z) :=

• A⊆E

(x − 1)r(G)−r(A)y|A|−r(A)zγ(A)

◮ V. Kruskal’s 2011 polynomial

K(G; x, y, a, b) :=

• A⊆E(G)

xr(G)−r(A)yκ(A)a

1 2 s(A)b 1 2 s⊥(A)

16 4

Background Minors Constructing the polynomials Matroids Remarks

The plan

◮ Explain how all three polynomials arise as the

“T utte polynomial” of embedded graphs. (Big picture.)

◮ unified / canonical approach ◮ Want:

◮ deletion-contraction relation ◮ terminates in edgeless graph

◮ Problems:

◮ The definition of deletion and contraction ◮ Cases for the relation (analogues of loop, bridge,

• rdinary)

16

Background

5

Minors Constructing the polynomials Matroids Remarks

Contraction for graphs in surfaces

16

Background

6

Minors Constructing the polynomials Matroids Remarks

Deletion for graphs in surfaces

16

Background

7

Minors Constructing the polynomials Matroids Remarks

Deletion-contraction relations

◮ 2 deletion and 2 contractions 4 domains

• 1. Cellularly embedded graphs in surfaces
• 2. graphs in surfaces
• 3. Cellularly embedded graphs in pseudo-surfaces
• 4. graphs in pseudo-surfaces

◮ four “T

utte polynomials”

◮ Need to recognise these four polynomials.

16

Background

8

Minors Constructing the polynomials Matroids Remarks

A ribbon graph framework

Ribbon graph

A “topological graph” with

◮ discs for vertices, ◮ ribbons for edges.

(Up to homeos. that preserve vertex-edge structure and cyclic order at vertices.)

◮ Ribbon graphs describe exactly cellularly

embedded graphs.

◮ Ribbon graph deletion / contraction

← → cell. embed. graph deletion / contraction

16

Background

9

Minors Constructing the polynomials Matroids Remarks

Coloured ribbon graph

vertex coloured boundary coloured

◮ ribbon graph ←

→ cell. embed. in surface

◮ boundary coloured r.g. ←

→ graph in surface

◮ vertex coloured r.g. ←

→ cell. embed. in pseudo-surface

◮ vertex and boundary coloured r.g. ←

→ graph in pseudo-surface

16

Background Minors

10 Constructing the

polynomials Matroids Remarks

Defining a T utte polynomial

T(G) =          1 if G edgeless xT(G/e) if e a bridge yT(G\e) if e a loop T(G\e) + T(G/e)

• therwise

◮ We have

◮ deletion and contraction ◮ objects closed under the operations ◮ reduce to edgeless ribbon graph

◮ We need

◮ to identify the cases for the recursive definition ◮ i.e., find analogues of loops and bridges

16

Background Minors

11 Constructing the

polynomials Matroids Remarks

The classical case

◮ 2 graphs on 1 edge:

,

◮ For ec = E\e, look at pair

(G\ec, G/ec)

◮

• ,
• ⇐

⇒ e bridge

• ,
• ⇐

⇒ e loop

• ,
• ⇐

⇒ e ordinary

• ,
• is impossible

◮ Define

U(G) =

• 1

if G edgeless aiU(G\e) + bjU(G/e) if e is (i, j)

◮ Then U(G) is T

utte polynomial: U(G) = a|E|−r(G)

l

br(G)

b

T(G; ab

bb + 1, bl al + 1)

16

Background Minors

12 Constructing the

polynomials Matroids Remarks

The topological case

◮ Apply canonical construction to topological graphs. ◮ 5 vertex and boundary coloured ribbon graphs: ◮ Define a T

utte polynomial U(G) =

• 1

if G edgeless aiU(G\e) + bjU(G/e) if e is (i, j)

◮ 4 variables (for well-definedness) ◮ 6 term deletion-contraction definition ◮ Krushkal’s polynomial K(G; x, y, a, b).

16

Background Minors

13 Constructing the

polynomials Matroids Remarks

The topological case continued...

◮ The deletion-contraction invariants for the other

• bjects are:

graph T(G, x, y) ribbon graph R(G, x, y, 1/√xy) vertex col. r.g. R(G, x, y, z) boundary col. r.g. R(G∗, x, y, z)

• vert. & bound. col. r.g.

K(G; x, y, a, b)

• vert. & bound. col. without r.g.

L(G; x, y, z)

• r

graph T(G, x, y)

• cell. emb. surface

R(G, x, y, 1/√xy)

• cell. emb. pseudo-surface

R(G, x, y, z) surface R(G∗, x, y, z) pseudo-surface K(G; x, y, a, b) from pseudo-surface L(G; x, y, z)

c.f., Krajewski, Moffatt, & T anasa, arXiv:1508.00814.

16

Background Minors Constructing the polynomials

14 Matroids

Remarks

A matroid framework

Moffatt & Smith, ask; Chun, Moffatt, Noble & Rueckriemen, arXiv:1403.0920, arXiv:1602.01306; Ellis-Monaghan & Moffatt, arXiv:1311.3762; Krajewski, Moffatt, & T anasa, arXiv:1508.00814.

16

Background Minors Constructing the polynomials

15 Matroids

Remarks

Summary

◮ Canonical approach to the T

utte polynomial.

◮ 4 types of deletion and contraction for embedded

graphs.

◮ 4 (or 5 or 3) topological T

utte polynomials

◮ “full” recursive definition ◮ recovers Bollobás-Riordan, Las Vergnas, and

Krushkal polynomials

◮ All polys recovered from Krushkal by forgetting

information.

◮ matroid framework.

16

Background Minors Constructing the polynomials

15 Matroids

Remarks

Thank You!

16

Background Minors Constructing the polynomials

15 Matroids

Remarks