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Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

Published online by Cambridge University Press:  18 November 2009

Y. DIAO  and
G. HETYEI
Y. DIAO
Affiliation:
Department of Mathematics and Statistics, UNC Charlotte, Charlotte, NC 28223, USA (e-mail: ydiao@uncc.edu, ghetyei@uncc.edu)
G. HETYEI
Affiliation:
Department of Mathematics and Statistics, UNC Charlotte, Charlotte, NC 28223, USA (e-mail: ydiao@uncc.edu, ghetyei@uncc.edu)
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Abstract

We introduce the concept of a relative Tutte polynomial of coloured 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 (and 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.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 3 , May 2010 , pp. 343 - 369
Copyright
Copyright © Cambridge University Press 2009

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