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 or Skein Relations v → … etc.

(Martin ’77, cf Martin, or circuit partition polynomial Las Vergnas) Let G be an Eulerian graph (all vertices of even degree). Let G be an Eulerian digraph (directed edges and Kirchhoff laws). The Circuit Partition Polynomial v → 1 + 1 + 1 = x The Circuit Partition Polynomial for Oriented Graphs: (The weight is 1 for coherent v → 1 + 0 + 1 = x states, 0 else)

Martin polynomial example For an oriented graph, the recursion is: → 1 v = x + 1 + 0 Then for G = , get + + + +  = 3 + 2 + j G x ( ; ) x 2 x x State Models as well:  ( ) ∑ ∑  ( ) A graph = = c S ( ) c S ( ) J G x ; x , j G x ; x state with 2 ( ) ( ) ∈ ∈ components S states G S states G

The Penrose Polynomial “Applications of Negative Dimensional Tensors”—R. Penrose, 1969 Defined for planar graphs and computed via the medial graph with the following skein-type recursion relation: e v v → 0 + 1 - 1 = x

Penrose polynomial example ( ) = − + 3 2 P G x ; x 3 x 2 x ∑ ( ) State Model Formulation: = − cr s ( ) c s ( ) P G x ; ( 1) x Penrose states s

Kauffman Bracket where The Jones polynomial is a knot invariant: A 2 = t -1/2

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.

The Transition Polynomial “On Transition Polynomials of 4-Regular Graphs” --F. Jaeger, 1987 G is a four regular graph, often the medial graph of a plane graph. The general recursive form: v → a + b + c = x State sum form: = ∑ ( ) α β γ ( ) s ( ) s ( ) s c s ( ) q G ; A , x a b c x s a state of G A is the weight system specifying the coefficient for each transition.

Example The general form: v → a + b + c = x For example, using the plane embedding to determine the weights: G = , get a + c + + b + + + ba aa ab ac + … ( ) = + + + + + 2 3 2 2 2 q G ; , A x a x 2( ab ac x ) (2 bc b c ) x 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 (Jaeger) The general form: These are polynomials of abstract v → a graphs—the assignment of weights + b + c does not depend on any embedding. = x The Martin Polynomial (families of cycles in Eulerian graphs): Set a = b = c = 1 → 1 v + 1 + 1 The Martin Polynomial for Oriented Graphs: Set the weight to one if the edges v → 1 + 0 + 1 correspond to an in-out pair, 0 else. (The weight is 1 for coherent states, 0 else)

The Penrose Polynomial is a specialization of the Transition polynomial v v → 0 + 1 - 1 = x ( ) = q G ; , A x P G x ( ; ) m 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.

Need a construction for the connection to the Kauffman bracket/Jones polynomial. 2. + if rotating top strand 1. Start by face 2- counterclockwise coloring the sweeps out + - diagram colored reason, - if it sweeps out white region + 3. Replace crossings by - + vertices. G L , 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) The Kauffman bracket of a link L : Let G L be the signed, face 2-colored universe of a link L , and let the weight system W be given as follows: v - → + 1/ A A + 0 v + → + A + 0 1/ A ( ) − − + = + 2 2 2 2 q G ; W A , A ( A A ) [ ] K L L

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. Orient G m so that G m with the vertex A Planar graph G black faces are to the faces colored black left of each edge. e delete contract Then, with this orientation of G m ,   ( ) ( ) = = + + k G ( ) q G ; , A x j G ; x x t G x ( ; 1, x 1) m m

Thistlethwaite’s Theorem • If L is an alternating link and G L is the blackface graph of the face two-colored knot universe, then − = − − + − − − − − − 1 1 w L ( ) (3 ( ) w L r G ( ) n G ( ))/4 k G ( ) 1 1 V L x ( ; ) ( 1) x ( x x ) T G ( ; x , x ) L L 2 2 L L G L

All these polynomials extend to embedded graphs Topological Transition Polynomial Transition Polynomial • Kauffman/Jones • Virtual Kauffman/Jones • Penrose • Topo Penrose • Martin • Martin • Classical Tutte • Bollobás-Riordan (via medial graph polynomial (via medial restricted to a graph restricted to a special line) special surface)

Insights from transition poly properties E-M & Moffatt The transition-Tutte relation lifts, now restricted to a surface instead of a line: Combine specialization to R and P for further insights: so… The Penrose polynomial is captured by the Tutte (Bollobás-Riordan) polynomial: The Penrose polynomial has a (twisted) deletion-contraction reduction

4/10/08 The generalized transition polynomial ( ) q G W x ; , , 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, + … abc + def = lmn x (New edge pairs have weight 1) Underlying Algebraic Structure This is a Hopf algebra map from the binomial bialgebra to a Hopf algebra of Eulerian graph (multiplication is disjoint union, comultiplication is summing over disjoint pairs of Eulerian subgraphs).

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

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?