arXiv subjects

David Cimasoni

Publications and source records attributed to David Cimasoni.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic concordance of links

Algebraic concordance of knots can be understood from the perspective of Seifert matrices, Blanchfield forms, and homology surgery. We initiate a systematic study of algebraic concordance for links from each of these viewpoints. The present article is concerned with algebraic concordance from the perspective of homology surgery and Blanchfield forms, whereas a companion article by the third named author focuses on C-complexes and generalised Seifert matrices. The outcome of the present work consists of two obstructions to $\mu$-component links being concordant. The first obstruction, called the homology surgery invariant, takes values in the Witt group of hermitian forms over the field of fractions $Q$ of $\mathbb{Z}[\mathbb{Z}^\mu]$. The second obtruction, called the Blanchfield invariant, takes values in a Witt group of $Q/\mathbb{Z}[\mathbb{Z}^\mu]$-valued hermitian linking forms. For $\mu\le 2$, we describe these invariants in terms of generalised Seifert matrices.

math.GT↗

arXiv2025

Extended signatures and link concordance

The Levine-Tristram signature admits an n-variable extension for n-component links: it was first defined as an integer valued function on $(S^1\setminus\{1\})^n$, and recently extended to the full torus $T^n$. The aim of the present article is to study and use this extended signature. First, we show that it is constant on the connected components of the complement of the zero-locus of some renormalized Alexander polynomial. Then, we prove that the extended signature is a concordance invariant on an explicit dense subset of $T^n$. Finally, as an application, we present an infinite family of 3-component links with the following property: these links are not concordant to their mirror image, a fact that can be detected neither by the non-extended signatures, nor by the multivariable Alexander polynomial, nor by the Milnor triple linking number.

math.GT↗

arXiv2024

A diagrammatic computation of abelian link invariants

We show how the multivariable signature and Alexander polynomial of a colored link can be computed from a single symmetric matrix naturally defined from a colored link diagram. In the case of a single variable, it coincides with the matrix introduced by Kashaev in [arXiv:1801.04632], which was recently proven to compute the Levine-Tristram signature and the Alexander polynomial of oriented links [arXiv:2311.01923, arXiv:2310.16729]. As a corollary, we obtain a multivariable extension of Kauffman's determinantal model of the Alexander polynomial, recovering a result of Zibrowius [arXiv:1601.04915v1].

math.GT↗

arXiv2023

On the Kashaev signature conjecture

In 2018, Kashaev introduced a square matrix indexed by the regions of a link diagram, and conjectured that it provides a novel way of computing the Levine-Tristram signature and Alexander polynomial of the corresponding oriented link. In this article, we show that for the classical signature (i.e. the Levine-Tristram signature at -1), this conjecture follows from the seminal work of Gordon-Litherland. We also relate Kashaev's matrix to Kauffman's "Formal Knot Theory" model of the Alexander polynomial. As a consequence, we establish the Alexander polynomial and classical signature parts of the conjecture for arbitrary links, as well as the full conjecture for definite knots.

math.GT↗

arXiv2023

On Arf invariants of colored links

Several classical knot invariants, such as the Alexander polynomial, the Levine-Tristram signature and the Blanchfield pairing, admit natural extensions from knots to links, and more generally, from oriented links to so-called colored links. In this note, we explore such extensions of the Arf invariant. Inspired by the three examples stated above, we use generalized Seifert forms to construct quadratic forms, and determine when the Arf invariant of such a form yields a well-defined invariant of colored links. However, apart from the known case of oriented links, these new Arf invariants turn out to be determined by the linking numbers.

math.GT↗

arXiv2023

Torres-type formulas for link signatures

We investigate the limits of the multivariable signature function $\sigma_L$ of a $\mu$-component link $L$ as some variable tends to $1$ via two different approaches: a three-dimensional and a four-dimensional one. The first uses the definition of $\sigma_L$ by generalized Seifert surfaces and forms. The second relies on a new extension of $\sigma_L$ from its usual domain $(S^1\setminus\{1\})^\mu$ to the full torus $\mathbb{T}^\mu$ together with a Torres-type formula for $\sigma_L$, results which are of independent interest. Among several consequences, we obtain new estimates on the value of the Levine-Tristram signature of a link close to $1$.

math.GT↗

arXiv2022

Elliptic dimers on minimal graphs and genus 1 Harnack curves

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [arXiv:1503.00289]. Specific instances of such models were studied in [arXiv:052711, arXiv:1612.09082, arXiv1801.00207]; we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of [arXiv:math-ph/0202018, arXiv:math/0311062] on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the $\mathbb{Z}^2$-periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in [arXiv:math-ph/0311005] by Kenyon, Okounkov and Sheffield. We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay's trisecant identity.

math.PR↗

arXiv2022

The dimer and Ising models on Klein bottles

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph $Γ$ in the Klein bottle $K$. Let $Γ_{mn}$ denote the graph obtained by pasting $m$ rows and $n$ columns of copies of $Γ$, which embeds in $K$ for $n$ odd and in the torus $\mathbb{T}^2$ for $n$ even. We compute the dimer partition function $Z_{mn}$ of $Γ_{mn}$ for $n$ odd, in terms of the well-known characteristic polynomial $P$ of $Γ_{12}\subset\mathbb{T}^2$ together with a new characteristic polynomial $R$ of $Γ\subset K$. Using this result together with work of Kenyon, Sun and Wilson [arXiv:1310.2603], we show that in the bipartite case, this partition function has the asymptotic expansion $\log Z_{mn}=mn f_0/2 +\mathrm{fsc}+o(1)$, for $m, n$ tending to infinity and $m/n$ bounded below and above, where $f_0$ is the bulk free energy for $Γ_{12}\subset\mathbb{T}^2$ and $\mathrm{fsc}$ an explicit finite-size correction term. The remarkable feature of this later term is its universality: it does not depend on the graph $Γ$, but only on the zeros of $P$ on the unit torus and on an explicit (purely imaginary) shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of $P$. We then show that this asymptotic expansion holds for the Ising partition function as well, with $\mathrm{fsc}$ taking a particularly simple form: it vanishes in the subcritical regime, is equal to $\log(2)$ in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blöte, Cardy and Nightingale.

math-ph↗

arXiv2021

Minimal bipartite dimers and higher genus Harnack curves

This paper completes the comprehensive study of the dimer model on infinite minimal graphs with Fock's weights [arXiv:1503.00289] initiated in [arXiv:2007.14699]: the latter article dealt with the elliptic case, i.e., models whose associated spectral curve is of genus one, while the present work applies to models of arbitrary genus. This provides a far-reaching extension of the genus zero results of [arXiv:math-ph/0202018, arXiv:math/0311062], from isoradial graphs with critical weights to minimal graphs with weights defining an arbitrary spectral data. For any minimal graph with Fock's weights, we give an explicit local expression for a two-parameter family of inverses of the associated Kasteleyn operator. In the periodic case, this allows us to prove local formulas for all ergodic Gibbs measures, thus providing an alternative description of the measures constructed in [arXiv:math-ph/0311005]. We also compute the corresponding slopes, exhibit an explicit parametrization of the spectral curve, identify the divisor of a vertex, and build on [arXiv:math/0311062, arXiv:1107.5588] to establish a correspondence between Fock's models on periodic minimal graphs and Harnack curves endowed with a standard divisor.

math.PR↗

arXiv2021

Isoradial immersions

Isoradial embeddings of planar graphs play a crucial role in the study of several models of statistical mechanics, such as the Ising and dimer models. Kenyon and Schlenker give a combinatorial characterization of planar graphs admitting an isoradial embedding, and describe the space of such embeddings. In this paper we prove two results of the same type for generalizations of isoradial embeddings: isoradial immersions and minimal immersions. We show that a planar graph admits a flat isoradial immersion if and only if its train-tracks do not form closed loops, and that a bipartite graph has a minimal immersion if and only if it is minimal. In both cases we describe the space of such immersions. The techniques used are different in both settings, and distinct from those of Kenyon and Schlenker. We also give an application of our results to the dimer model defined on bipartite graphs admitting minimal immersions.

math.CO↗

arXiv2020

Graph coverings and twisted operators

Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator. The main result of this article is the fact that these operators behave in a controlled way under graph covering maps. When such an operator can be used to enumerate objects, or compute a partition function, this has concrete implications on the corresponding enumeration problem, or statistical mechanics model. For example, we show that if $\widetilde{\Gamma}$ is a finite connected covering graph of a graph $\Gamma$ endowed with edge-weights $x=\{x_e\}_e$, then the spanning tree partition function of $\Gamma$ divides the one of $\widetilde{\Gamma}$ in the ring $\mathbb{Z}[x]$. Several other consequences are obtained, some known, others new.

math.CO↗

arXiv2019

Topological complexity of photons' paths in biological tissues

In the present contribution three means of measuring the geometrical and topological complexity of photons' paths in random media are proposed. This is realized by investigating the behavior of the average crossing number, the mean writhe, and the minimal crossing number of photons' paths generated by Monte Carlo (MC) simulations, for different sets of optical parameters. It is observed that the complexity of the photons' paths increases for increasing light source/detector spacing, and that highly "knotted" paths are formed. Due to the particular rules utilized to generate the MC photons' paths, the present results may have an interest not only for the biomedical optics community, but also from a pure mathematical point of view.

physics.optics↗

arXiv2019

The topological hypothesis for discrete spin models

The topological hypothesis claims that phase transitions in a classical statistical mechanical system are related to changes in the topology of the level sets of the Hamiltonian. So far, the study of this hypothesis has been restricted to continuous systems. The purpose of this article is to explore discrete models from this point of view. More precisely, we show that some form of the topological hypothesis holds for a wide class of discrete models, and that its strongest version is valid for the Ising model on $\mathbb{Z}^d$ with the possible exception of dimensions $d=3,4$.

cond-mat.stat-mech↗

arXiv2017

Revisiting the combinatorics of the 2D Ising model

We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants, using the language of Kac-Ward matrices. We also give a brief account of the relations between various alternative formalisms which have been used in the combinatorial study of the planar Ising model: dimers and Grassmann variables, spin and disorder operators, and, more recently, s-holomorphic observables. In addition, we point out that these formulas can be extended to the double-Ising model, defined as a pointwise product of two Ising spin configurations on the same discrete domain, coupled along the boundary.

math.CO↗

arXiv2017

Colored tangles and signatures

Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considering the multivariable signature of the closure of a colored tangle. The corresponding defect is expressed in terms of the Maslov index and of the Lagrangian functor defined by Turaev and the first-named author.

math.GT↗

arXiv2016

A Burau-Alexander 2-functor on tangles

We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links.

math.GT↗

arXiv2016

Identities between dimer partition functions on different surfaces

Given a weighted graph $G$ embedded in a non-orientable surface $Σ$, one can consider the corresponding weighted graph $\widetilde{G}$ embedded in the so-called orientation cover $\widetildeΣ$ of $Σ$. We prove identities relating twisted partition functions of the dimer model on these two graphs. When $Σ$ is the Möbius strip or the Klein bottle, then $\widetildeΣ$ is the cylinder or the torus, respectively, and under some natural assumptions, these identities imply relations between the genuine dimer partition functions $Z(G)$ and $Z(\widetilde{G})$. For example, we show that if $G$ is a locally but not globally bipartite graph embedded in the Möbius strip, then $Z(\widetilde{G})$ is equal to the square of $Z(G)$. This extends results for the square lattice previously obtained by various authors.

math-ph↗

arXiv2016

Splitting numbers and signatures

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and nullity. Although very elementary and easy to compute, this bound turns out to be suprisingly efficient. In particular, it makes it a routine check to recover the splitting number of 129 out of the 130 prime links with at most 9 crossings. Also, we easily determine 16 of the 17 splitting numbers that were studied by Batson and Seed using Khovanov homology, and later computed by Cha, Friedl and Powell using a variety of techniques. Finally, we determine the splitting number of a large class of 2-bridge links which includes examples recently computed by Borodzik and Gorsky using a Heegaard Floer theoretical criterion.

math.GT↗