Dimer Models for Knot Polynomials

A dimer model consists of all perfect matchings on a (bipartite) weighted signed graph, where the product of the signed weights of each perfect matching is summed to obtain an invariant. In this paper, the construction of such a graph from a knot diagram is given to obtain the Alexander polynomial....

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Main Author:	Cohen, Moshe
Other Authors:	Oporowski, Bogdan
Format:	Others
Language:	en
Published:	LSU 2010
Subjects:	Mathematics
Online Access:	http://etd.lsu.edu/docs/available/etd-07082010-142254/

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spelling	ndltd-LSU-oai-etd.lsu.edu-etd-07082010-1422542013-01-07T22:52:57Z Dimer Models for Knot Polynomials Cohen, Moshe Mathematics A dimer model consists of all perfect matchings on a (bipartite) weighted signed graph, where the product of the signed weights of each perfect matching is summed to obtain an invariant. In this paper, the construction of such a graph from a knot diagram is given to obtain the Alexander polynomial. This is further extended to a more complicated graph to obtain the twisted Alexander polynomial, which involved "twisting" by a representation. The space of all representations of a given knot complement into the general linear group of a fixed size can be described by the same graph. This work also produces a bipartite weighted signed graph to obtain the Jones polynomial for the infinite class of pretzel knots as well as for some other constructions. This is a corollary to a stronger result that calculates the activity words for the spanning trees of the Tait graph associated to a pretzel knot diagram, and this has several other applications, as well, including the Tutte polynomial and the spanning tree model of reduced Khovanov homology. Oporowski, Bogdan Neubrander, Frank Litherland, Richard Dasbach, Oliver Tohline, Joel Morales, Jorge LSU 2010-07-08 text application/pdf http://etd.lsu.edu/docs/available/etd-07082010-142254/ http://etd.lsu.edu/docs/available/etd-07082010-142254/ en unrestricted I hereby certify that, if appropriate, I have obtained and attached herein a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to LSU or its agents the non-exclusive license to archive and make accessible, under the conditions specified below and in appropriate University policies, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.
collection	NDLTD
language	en
format	Others
sources	NDLTD
topic	Mathematics
spellingShingle	Mathematics Cohen, Moshe Dimer Models for Knot Polynomials
description	A dimer model consists of all perfect matchings on a (bipartite) weighted signed graph, where the product of the signed weights of each perfect matching is summed to obtain an invariant. In this paper, the construction of such a graph from a knot diagram is given to obtain the Alexander polynomial. This is further extended to a more complicated graph to obtain the twisted Alexander polynomial, which involved "twisting" by a representation. The space of all representations of a given knot complement into the general linear group of a fixed size can be described by the same graph. This work also produces a bipartite weighted signed graph to obtain the Jones polynomial for the infinite class of pretzel knots as well as for some other constructions. This is a corollary to a stronger result that calculates the activity words for the spanning trees of the Tait graph associated to a pretzel knot diagram, and this has several other applications, as well, including the Tutte polynomial and the spanning tree model of reduced Khovanov homology.
author2	Oporowski, Bogdan
author_facet	Oporowski, Bogdan Cohen, Moshe
author	Cohen, Moshe
author_sort	Cohen, Moshe
title	Dimer Models for Knot Polynomials
title_short	Dimer Models for Knot Polynomials
title_full	Dimer Models for Knot Polynomials
title_fullStr	Dimer Models for Knot Polynomials
title_full_unstemmed	Dimer Models for Knot Polynomials
title_sort	dimer models for knot polynomials
publisher	LSU
publishDate	2010
url	http://etd.lsu.edu/docs/available/etd-07082010-142254/
work_keys_str_mv	AT cohenmoshe dimermodelsforknotpolynomials
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Dimer Models for Knot Polynomials

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