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David Cimasoni

Publications and source records attributed to David Cimasoni.

At least 37 records · Page 2Linked to original sources
arXiv2015

The geometry of dimer models

This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV 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 geometrical 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 contain any new results, but give a (relatively original) account of the works of Kasteleyn, Cimasoni-Reshetikhin and Kenyon-Okounkov-Sheffield.

math-ph↗
arXiv2014

Kac-Ward operators, Kasteleyn operators, and s-holomorphicity on arbitrary surface graphs

The conformal invariance and universality results of Chelkak-Smirnov on the two-dimensional Ising model hold for isoradial planar graphs with critical weights. Motivated by the problem of extending these results to a wider class of graphs, we define a generalized notion of s-holomorphicity for functions on arbitrary weighted surface graphs. We then give three criteria for s-holomorphicity involving the Kac-Ward, Kasteleyn, and discrete Dirac operators, respectively. Also, we show that some crucial results known to hold in the planar isoradial case extend to this general setting: in particular, spin-Ising fermionic observables are s-holomorphic, and it is possible to define a discrete version of the integral of the square of an s-holomorphic function. Along the way, we obtain a duality result for Kac-Ward determinants on arbitrary weighted surface graphs.

math-ph↗
arXiv2014

Link Floer homology categorifies the Conway function

Given an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multivariate Alexander polynomial, an invariant only defined up to a sign and powers of the variables. In this paper, we get rid of this ambiguity by proving that this Euler characteristic is equal to the so-called Conway function, the representative of the multivariate Alexander polynomial introduced by Conway in 1970 and explicitly constructed by Hartley in 1983. This is achieved by creating a model of the Conway function adapted to rectangular diagrams, which is then compared to the Euler characteristic of the combinatorial version of link Floer homology.

math.GT↗
arXiv2012

The critical temperature for the Ising model on planar doubly periodic graphs

We provide a simple characterization of the critical temperature for the Ising model on an arbitrary planar doubly periodic weighted graph. More precisely, the critical inverse temperature βfor a graph G with coupling constants (J_e)_{e\in E(G)} is obtained as the unique solution of a linear equation in the variables (\tanh(βJ_e))_{e\in E(G)}. This is achieved by studying the high-temperature expansion of the model using Kac-Ward matrices.

math-ph↗
arXiv2012

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.

math-ph↗
arXiv2010

A generalized Kac-Ward formula

The Kac-Ward formula allows to compute the Ising partition function on a planar graph G with straight edges from the determinant of a matrix of size 2N, where N denotes the number of edges of G. In this paper, we extend this formula to any finite graph: the partition function can be written as an alternating sum of the determinants of 2^{2g} matrices of size 2N, where g is the genus of an orientable surface in which G embeds. We give two proofs of this generalized formula. The first one is purely combinatorial, while the second relies on the Fisher-Kasteleyn reduction of the Ising model to the dimer model, and on geometric techniques. As a consequence of this second proof, we also obtain the following fact: the Kac-Ward and the Fisher-Kasteleyn methods to solve the Ising model are one and the same.

math-ph↗
arXiv2010

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.

math-ph↗
arXiv2009

Slicing Bing doubles

Bing doubling is an operation which produces a 2-component boundary link B(K) from a knot K. If K is slice, then B(K) is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if B(K) is boundary slice, then K is algebraically slice. We also show that the Rasmussen invariant can tell that certain Bing doubles are not smoothly slice.

math.GT↗
arXiv2009

Dimers on graphs in non-orientable surfaces

The main result of this paper is a Pfaffian formula for the partition function of the dimer model on a graph G embedded in a closed, possibly non-orientable surface S. This formula is suitable for computational purposes, and it is obtained using purely geometrical methods. The key step in the proof consists of a correspondence between some orientations on G and the set of pin^- structures on S. This generalizes (and simplifies) the results of a previous paper [2].

math-ph↗
arXiv2007

Dimers on surface graphs and spin structures. II

In a previous paper, we showed how certain orientations of the edges of a graph G embedded in a closed oriented surface S can be understood as discrete spin structures on S. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on G. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as a quantum field theory on surface graphs.

math-ph↗
arXiv2006

Dimers on surface graphs and spin structures. I

Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.

math-ph↗
arXiv2006

A generalization of several classical invariants of links

We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the Alexander-Conway polynomial and the Murasugi-Tristram-Levine signatures.

math.GT↗
arXiv2005

Generalized Seifert surfaces and signatures of colored links

In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this m-variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and the Atiyah-Singer G-signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.

math.GT↗
arXiv2004

The Conway function of a splice

We give a closed formula for the Conway function of a splice in terms of the Conway function of its splice components. As corollaries, we refine and generalize results of Seifert, Torres, and Sumners-Woods.

math.GT↗
arXiv2004

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.

math.GT↗
arXiv2004

Long line knots

We study continuous embeddings of the long line L into L^n (n>1) up to ambient isotopy of L^n. 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 n>3 for differentiable ones. Finally, we prove that the classification of smooth embeddings L \to L^3 is equivalent to the classification of classical oriented knots.

math.GN↗
arXiv2004

L'homologie de Novikov des entrelacs de Waldhausen

A graph multilink is a link with multiplicities in a homology 3-sphere whose exterior is a graph manifold. In this Note, we compute the Novikov homology of graph multilinks. As a corollary, we give a majoration for the number of Novikov modules on a given graph link.

math.GT↗