AN INTRODUCTION TO TRANSITION POLYNOMIALS Jo Ellis-Monaghan The - - PowerPoint PPT Presentation

AN INTRODUCTION TO TRANSITION POLYNOMIALS

Jo Ellis-Monaghan

The story…

• There are deletion-contraction polynomials.
• The Tutte polynomial assimilates them all.
• There are many generalizations of the Tutte polynomial to adapt it

to other settings and applications.

• There are skein-type polynomials.
• The generalized transition polynomial assimilates them all.
• There are many general settings to which the GTP also applies.
• Deletion-contraction and skein-type polynomials are

related to via a medial graph construction.

• Can generalize these various skein-type polynomials to

embedded graphs, and gain further insights.

• But these are only one large family of polynomials—they

don’t tell the whole story….

Transitions or Vertex States

• r Skein Relations

v → … etc.

= x v → 1 + 1 + 1 v → 1 + 0 + 1

The Circuit Partition Polynomial for Oriented Graphs: (The weight is

1 for coherent states, 0 else)

The Circuit Partition Polynomial

= x

Martin, or circuit partition polynomial

Let G be an Eulerian graph (all vertices of even degree). Let G be an Eulerian digraph (directed edges and Kirchhoff laws).

(Martin ’77, cf Las Vergnas)

Martin polynomial example

3 2

( ; ) 2 j G x x x x = + + 

, get Then for G = For an oriented graph, the recursion is:

= x v → 1 + 1 + 0 + + + +

( )

( )

( )

( )

( ) ( )

; , ;

c S c S S states G S states G

J G x x j G x x

∈ ∈

= =

∑ ∑ 



A graph state with 2 components

State Models as well:

The Penrose Polynomial

v → + 1

• 1

Defined for planar graphs and computed via the medial graph with the following skein-type recursion relation:

“Applications of Negative Dimensional Tensors”—R. Penrose, 1969

= x v

e

Penrose polynomial example

( )

3 2

; 3 2 = − + P G x x x x

State Model Formulation:

( )

( ) ( ) Penrose states

; ( 1) = −

∑

cr s c s s

P G x x

Kauffman Bracket

where

A2= t -1/2 The Jones polynomial is a knot invariant:

Unify these

• We can unify all these skein-type polynomials

with the transition polynomial, just as we were able to unify the deletion-contraction polynomials with the Tutte polynomial.

v → a

+ b + c

The general recursive form:

The Transition Polynomial

“On Transition Polynomials of 4-Regular Graphs”

• -F. Jaeger, 1987

( )

( ) ( ) ( ) ( ) a state

• f

; ,

α β γ

= ∑

s s s c s s G

A a b c q G x x

= x

State sum form: A is the weight system specifying the coefficient for each transition. G is a four regular graph, often the medial graph of a plane graph.

v → a + b + c

The general form:

Example

( )

2 3 2 2 2

; , 2( ) (2 ) = + + + + + q G x a x ab ac x bc b c A x

= x

, get

+ + +

G = For example, using the plane embedding to determine the weights:

a + b + c aa ab + ba + … ac

In an even more general setting, these weights can depend on the vertex.

Martin polynomials are specializations of the Transition polynomial

Transition Polynomials for 4-Regular Graphs The general form: The Martin Polynomial (families of cycles in Eulerian graphs): The Martin Polynomial for Oriented Graphs: (The weight is 1 for coherent states, 0 else)

v → a + b + c = x v → 1 + 1 + 1 v → 1 + 0 + 1

Set a = b = c = 1 Set the weight to one if the edges correspond to an in-out pair, 0 else.

(Jaeger)

These are polynomials of abstract graphs—the assignment of weights does not depend on any embedding.

The Penrose Polynomial is a specialization of the Transition polynomial

( )

; , ( ; ) =

m

q G A x P G x

v → + 1

• 1

Where A is the weight system with pair weight of 0 if pair bounds a white face 1 if pair bounds a black face, -1 if changes face boundary.

= x v

Need a construction for the connection to the Kauffman bracket/Jones polynomial.

+

• 1. Start by face 2-

coloring the diagram

• 2. + if rotating top

strand counterclockwise sweeps out colored reason,

• if it sweeps out

white region +

• +
• 3. Replace

crossings by vertices.

GL, the signed, face 2- colored universe of a link L

The Kauffman bracket is a specialization of the Transition polynomial (hence so is the Jones polynomial, with a prefactor)

( )

2 2 2 2

; , ( ) [ ]

− −

+ = +

L

q G W A A A A K L

v+ → 1/A + A + 0 v- → A + 1/ A + 0

The Kauffman bracket of a link L: Let GL be the signed, face 2-colored universe of a link L, and let the weight system W be given as follows:

How do they fit?

• How do these polynomials interrelate with others, e.g.

deletion-contraction polynomials?

• For plane graphs, some of them coincide with the Tutte

polynomial along

y x = ±

The Tutte-Martin connection means that the Tutte and transition polynomials are related via a medial graph construction.

A Planar graph G Gm with the vertex faces colored black Orient Gm so that black faces are to the left of each edge.

e

delete contract

Then, with this orientation of Gm,

( ) ( )

( )

; , ; ( ; 1, 1) = = + +  

k G m m

q G A x j G x x t G x x

Thistlethwaite’s Theorem

• If L is an alternating link and GL is the blackface graph of

the face two-colored knot universe, then

1 1 2 2

(3 ( ) ( ) ( ))/4 ( ) 1 ( ) 1

( ; ) ( 1) ( ) ( ; , )

L L L

w L r G n G k G w L L

V L x x x x T G x x

− − + − −

= − − − − −

GL

All these polynomials extend to embedded graphs

• Kauffman/Jones
• Penrose
• Martin
• Classical Tutte

(via medial graph restricted to a special line)

Topological Transition Polynomial Transition Polynomial

• Virtual Kauffman/Jones
• Topo Penrose
• Martin
• Bollobás-Riordan

polynomial (via medial graph restricted to a special surface)

Insights from transition poly properties

The transition-Tutte relation lifts, now restricted to a surface instead of a line: The Penrose polynomial is captured by the Tutte (Bollobás-Riordan) polynomial: The Penrose polynomial has a (twisted) deletion-contraction reduction Combine specialization to R and P for further insights:

so…

E-M & Moffatt

4/10/08

, where G is an arbitrary Eulerian graph, and W is a weight system which assigns a value in R (a ring with unit) to every pair of adjacent half edges in G. Then,

( )

; , q G W x

+ def abc + … = lmn x (New edge pairs have weight 1)

The generalized transition polynomial

Underlying Algebraic Structure

This is a Hopf algebra map from the binomial bialgebra to a Hopf algebra

• f Eulerian graph (multiplication is disjoint union, comultiplication is

summing over disjoint pairs of Eulerian subgraphs).

A: The generalized transition polynomial. B: Coloured Tutte polynomial. C: Jaeger’s transition polynomials. D: The classic Tutte polynomial. E: The Martin polynomial. F: The interlace polynomial G: The Potts and Ising models. H: The Penrose polynomial. I: Knot and link invariants. J: Chromatic polynomial. K: Flow polynomial. L: Reliability polynomial. M: External field Potts N: The U/V/W polynomials

N L J K M

Open questions

• How do the Characteristic and other polynomials fit? E.g. can

fit the interlace polynomial into this, but need to go through several derived graphs to get there

• Is there any information from a ‘dual’ to e.g. penrose like

flow/char?

• Other curves of coincidence other than y=x and the one for

topo Transition-B-R connection?

• What other kinds of reductions besides deletion/contraction,

transitions, pivots, vertex nbhrds, etc. can we handle --a comprehensive theory should encompass not only existing polys, but putative future polys….

• Delta matroids generalize and capture embedded graphs. Are

they an informative domain for transition polynomials?