The Kauffman Bracket of Virtual Links and the Bollobás–Riordan Polynomial

Chmutov, Sergei; Pak, Igor

doi:10.17323/1609-4514-2007-7-3-409-418

Cited by 47 publications

(92 citation statements)

References 14 publications

(13 reference statements)

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“…We see that these two theorems from [5] and [7] which relate the Bollobás-Riordan polynomial and the Kauffman bracket follow from two different interpretations of the same ribbon graph.…”

Section: The Connection With the Bollobás-riordan Polynomialmentioning

confidence: 87%

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Unsigned State Models for the Jones Polynomial

Moffatt

2011

Ann. Comb.

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It is a well-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 S 3 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 to the Jones and Bollobás-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.

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“…We see that these two theorems from [5] and [7] which relate the Bollobás-Riordan polynomial and the Kauffman bracket follow from two different interpretations of the same ribbon graph.…”

Section: The Connection With the Bollobás-riordan Polynomialmentioning

confidence: 87%

“…By taking each of these two perspectives in turn, we can recover and relate the results from [7] and [4] which relate the Bollobás-Riordan polynomial and the Kauffman bracket. The Bollobás-Riordan polynomial is given by…”

Section: The Connection With the Bollobás-riordan Polynomialmentioning

confidence: 99%

Unsigned State Models for the Jones Polynomial

Moffatt

2011

Ann. Comb.

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“…As mentioned previously, our main motivation for this work came from recent results connecting the Bollobás-Riordan polynomial and knot polynomials ( [6][7][8]17]) which generalize well-known relations between the Tutte polynomial and knot polynomials ( [12,20]). In particular, we were interested in generalizing connections between the behaviour of the Jones polynomial of an alternating link and the matroid properties of the Tutte polynomial discussed by the first author in [11].…”

Section: An Application To Knot Theorymentioning

confidence: 97%

“…The multivariate Bollobás-Riordan polynomial is the obvious extension of the multivariate Tutte polynomial. It has been used previously [6,16,17].…”

Section: Tutte and Bollobás-riordan Polynomialsmentioning

confidence: 99%

Expansions for the Bollobás-Riordan Polynomial of Separable Ribbon Graphs

Huggett

Moffatt

2011

Ann. Comb.

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We define 2-decompositions of ribbon graphs, which generalize 2-sums and tensor products of graphs. We give formulae for the Bollobás-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.

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“…Recently, there has been a lot of interest in connections between knots and ribbon graphs [3][4][5][6]15,16]. In particular, there are various constructions which realize the Jones polynomial of a link as an evaluation of Bollobás and Riordan's ribbon graph polynomial (defined in [1,2]) of an associated signed ribbon graph.…”

Section: Introductionmentioning

confidence: 99%

Partial duality and Bollobás and Riordan’s ribbon graph polynomial

Moffatt¹

2010

Discrete Mathematics

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a b s t r a c tRecently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan's ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation which used the homfly polynomial of a knot.

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The Kauffman Bracket of Virtual Links and the Bollobás–Riordan Polynomial

Cited by 47 publications

References 14 publications

Unsigned State Models for the Jones Polynomial

Unsigned State Models for the Jones Polynomial

Expansions for the Bollobás-Riordan Polynomial of Separable Ribbon Graphs

Partial duality and Bollobás and Riordan’s ribbon graph polynomial

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