61 research outputs found
Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices
Let S be a flat surface of genus g with cone type singularities. Given a
bipartite graph G isoradially embedded in S, we define discrete analogs of the
2^{2g} Dirac operators on S. 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 (S,G) for these discrete Dirac operators
to be Kasteleyn matrices of the graph G. As a consequence, if these conditions
are met, the partition function of the dimer model on G can be explicitly
written as an alternating sum of the determinants of these 2^{2g} discrete
Dirac operators.Comment: 39 pages, minor change
The Alexander module of links at infinity
Walter Neumann showed that the topology of a ``regular'' algebraic curve V in
C^2 is determined up to proper isotopy by some link in S^3 called the link at
infinity of V. In this note, we compute the Alexander module over C[t^{\pm 1}]
of any such link at infinity.Comment: 14 pages, 2 figure
Studying the multivariable Alexander polynomial by means of Seifert surfaces
We show how Seifert surfaces, so useful for the understanding of the
Alexander polynomial \Delta_L(t), can be generalized in order to study the
multivariable Alexander polynomial \Delta_L(t_1,...,t_\mu). In particular, we
give an elementary and geometric proof of the Torres formula.Comment: 10 pages, 2 figure
A geometric construction of the Conway potential function
We give a geometric construction of the multivariable Conway potential
function for colored links. In the case of a single color, it is Kauffman's
definition of the Conway polynomial in terms of a Seifert matrix.Comment: 21 pages, 27 figure
A Lagrangian representation of tangles
We construct a functor from the category of oriented tangles in R^3 to the
category of Hermitian modules and Lagrangian relations over Z[t,t^{-1}]. This
functor extends the Burau representations of the braid groups and its
generalization to string links due to Le Dimet.Comment: 36 pages, 8 figure
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