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A dimer model for the Jones polynomial of pretzel knots

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Moshe Cohen (Bar-Ilan University)

Abstract: The Tutte polynomial gives remarkable insight into the structure of a graph, especially in terms of its bulky and computationally expensive definition using "activity". By narrowing our focus to a specific class of graphs coming from pretzel knots, we can recover these activity words easily as the determinant of a matrix.
This determinant encodes information about the spanning tree expansion of the graph, something that has proved useful in this context. When viewed a different way, however, this determinant encodes information about perfect matchings of a different bipartite graph. This "dimer model" arises from statistical mechanics and has been studied extensively in different contexts.
The Jones polynomial of a knot is a specialization of the Tutte polynomial of a graph obtained from a knot. No background in Knot Theory will be expected. This talk is accessible to anyone with a basic understanding of graph theory and matrix determinants.