Type of Document Dissertation Author Cohen, Moshe URN etd-07082010-142254 Title Dimer Models for Knot Polynomials Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee
Advisor Name Title Dasbach, Oliver Committee Chair Litherland, Richard Committee Member Morales, Jorge Committee Member Neubrander, Frank Committee Member Oporowski, Bogdan Committee Member Tohline, Joel Dean's Representative Keywords
- spanning trees
- knot theory
Date of Defense 2010-06-24 Availability unrestricted AbstractA 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.
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