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The geometry of dimer models

David Cimasoni — 2014

Winter Braids Lecture Notes

This is an expanded version of a three-hour minicourse given at the winterschool held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometric tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture notes do not...

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

David Cimasoni — 2012

Journal of the European Mathematical Society

Let $\sum$ be a flat surface of genus $g$ with cone type singularities. Given a bipartite graph $Γ$ isoradially embedded in $\sum$ , we define discrete analogs of the $2^{2 g}$ Dirac operators on $\sum$ . These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair $Γ \subset \sum$ for these discrete Dirac operators to be Kasteleyn matrices of the graph $Γ$ . As a consequence, if these conditions are met, the partition function of the dimer...

A Lagrangian representation of tangles II

David Cimasoni; Vladimir Turaev — 2006

Fundamenta Mathematicae

The present paper is a continuation of our previous paper [Topology 44 (2005), 747-767], where we extended the Burau representation to oriented tangles. We now study further properties of this construction.

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Currently displaying 1 – 3 of 3

The geometry of dimer models

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

A Lagrangian representation of tangles II