120 research outputs found
Alexander representation of tangles
A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on
the two disks in the boundary of the cylinder. Using an algebraic tool
developed by Lescop, we extend the Burau representation of braids to a functor
from the category of oriented tangles to the category of Z[t,t^{-1}]-modules.
For (1,1)-tangles (i.e., tangles with one endpoint on each disk) this invariant
coincides with the Alexander polynomial of the link obtained by taking the
closure of the tangle. We use the notion of plat position of a tangle to give a
constructive proof of invariance in this case.Comment: 13 pages, 5 figure
Long line knots
Abstract.: We study continuous embeddings of the long line into up to ambient isotopy of . We define the direction of an embedding and show that it is (almost) a complete invariant in the case n = 2 for continuous embeddings, and in the case for differentiable ones. Finally, we prove that the classification of smooth embeddings is equivalent to the classification of classical oriented knot
The critical Ising model via Kac-Ward matrices
The Kac-Ward formula allows to compute the Ising partition function on any
finite graph G from the determinant of 2^{2g} matrices, where g is the genus of
a surface in which G embeds. We show that in the case of isoradially embedded
graphs with critical weights, these determinants have quite remarkable
properties. First of all, they satisfy some generalized Kramers-Wannier
duality: there is an explicit equality relating the determinants associated to
a graph and to its dual graph. Also, they are proportional to the determinants
of the discrete critical Laplacians on the graph G, exactly when the genus g is
zero or one. Finally, they share several formal properties with the Ray-Singer
\bar\partial-torsions of the Riemann surface in which G embeds.Comment: 30 pages, 10 figures; added section 4.4 in version
The critical Z-invariant Ising model via dimers: locality property
We study a large class of critical two-dimensional Ising models, namely
critical Z-invariant Ising models. Fisher [Fis66] introduced a correspondence
between the Ising model and the dimer model on a decorated graph, thus setting
dimer techniques as a powerful tool for understanding the Ising model. In this
paper, we give a full description of the dimer model corresponding to the
critical Z-invariant Ising model, consisting of explicit expressions which only
depend on the local geometry of the underlying isoradial graph. Our main result
is an explicit local formula for the inverse Kasteleyn matrix, in the spirit of
[Ken02], as a contour integral of the discrete exponential function of
[Mer01a,Ken02] multiplied by a local function. Using results of [BdT08] and
techniques of [dT07b,Ken02], this yields an explicit local formula for a
natural Gibbs measure, and a local formula for the free energy. As a corollary,
we recover Baxter's formula for the free energy of the critical Z-invariant
Ising model [Bax89], and thus a new proof of it. The latter is equal, up to a
constant, to the logarithm of the normalized determinant of the Laplacian
obtained in [Ken02].Comment: 55 pages, 29 figure
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