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Generalized duality for graphs on surfaces and the signed BollobasRiordan polynomial
 J. Combin. Theory Ser. B
"... Abstract. We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed BollobásRiordan polynomials of dual graphs. This relation unifies various r ..."
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Abstract. We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed BollobásRiordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the BollobásRiordan polynomials.
RELATIVE TUTTE POLYNOMIALS FOR COLORED GRAPHS AND VIRTUAL KNOT THEORY
, 909
"... Abstract. We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial ..."
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Cited by 1 (1 self)
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Abstract. We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly. 1.
and
, 2005
"... The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific p ..."
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The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.
and
, 2005
"... The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific p ..."
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The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.
and
, 2005
"... The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific p ..."
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The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.
and
, 2005
"... The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific p ..."
Abstract
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The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.
New Invariants of Long Virtual Knots
, 705
"... This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from their classical counterparts with respect to their symmetries ..."
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This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from their classical counterparts with respect to their symmetries and the concatenation product. 1
Colored Tutte polynomials and composite knots
"... Abstract. Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski’s tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly complicated knots and, in the future, even virtual knots. Résumé ..."
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Abstract. Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski’s tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly complicated knots and, in the future, even virtual knots. Résumé. En faisant une revue de trois articles récents et de la recherche en cours, nous montrons comment une généralisation aux graphes colorés de la formule de Brylawski sur le produit tensoriel peut être utilisée à calculer le polynôme de Jones de quelques nœuds et, dans la future, même de quelques nœuds virtuels, bien compliqués.