Ribbon Graph Minors and Low-Genus Partial Duals

We give an excluded minor characterisation of the class of ribbon graphs that admit partial duals of Euler genus at most one

Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric

Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs

We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.Comment: Version 2 has minor changes. To appear in Annals of Combinatoric

A conjecture of Kauffman on amphicheiral alternating knots

Also published as: SFB-303--94831SIGLEAvailable from TIB Hannover: RN 4052(94831) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Does the Jones polynomial detect unknottedness?

There were many attempts to settle the question whether there exist non trivial knots with trivial Jones polynomial. In this paper we show that such a knot must have crossing-number at least 17. Furthermore we give an upper bound for the number of prime knots that have 14 or 15 crossings and compute the number of different Homfly, Jones and Alexander polynomials for knots up to 15 crossingsSIGLEAvailable from TIB Hannover: RN 4052(93798) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman