In this paper we describe the algebraic relations satisfied by the harmonic and anti-harmonic moments of simply connected, but not necessarily convex planar polygons with a given number of vertices.
Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infinitely many variables. The ordinary chromatic polynomial is a specialization of Stanley's one.
To each orientable embedded graph with a single vertex, a simple graph is associated, which is called the intersection graph of the embedded graph. As a result, we can define Stanley's symmetrized chromatic polynomial for any orientable embedded graph with a single vertex. Our goal is to extend Stanley's chromatic polynomial to embedded graphs with arbitrary number of vertices, and not necessarily orientable. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs , our extension is based not on the structure of the underlying abstract graph and the additional information about the embedding. Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley chromatic polynomial as an invariant of binary delta-matroids. We show that, similarly to Stanley's symmetrized chromatic polynomial of graphs, which satisfies 4-term relations for simple graphs, the polynomial that we introduce satisfies the 4-term relations for binary delta-matroids . For graphs, Stanley's chromatic function produces a knot invariant by means of the correspondence between simple graphs and knots. Analogously we may interpret the suggested extension as an invariant of links, using the correspondence between binary delta-matroids and links. (C) 2021 Elsevier B.V. All rights reserved.
Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that the value of the weight system constructed from the Lie algebra sl2 on a chord diagram depends on the intersection graph of the diagram, rather than the diagram itself. This inspired a question due to Lando about whether it is possible to extend the weight system sl2 to a graph invariant satisfying the four term relations for graphs. We show that for all graphs with up to 8 vertices such an extension exists and is unique, thus answering in affirmative to Lando’s question for small graphs.
Let C_d be the space of non-singular, univariate polynomials of degree d. The Viète map V sends a polynomial to its unordered set of roots. It is a classical fact that the induced map V_∗ at the level of fundamental groups realises an isomorphism between π_1(C_d) and the Artin braid group B_d. For fewnomials, or equivalently for the intersection C of C_d with a collection of coordinate hyperplanes, the image of the map V_∗:π_1(C)→B_d is not known in general. In the present paper, we show that the map V_∗ is surjective provided that the support of the corresponding polynomials spans Z as an affine lattice. If the support spans a strict sublattice of index b, we show that the image of V_∗ is the expected wreath product of Z/bZ with B_d/b. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.
We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.
In the Hurwitz space of rational functions on CP^1 with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincaré dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.
We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle (Y−ΔY−Δ) transformation at the critical point n=2n=2. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to construct the recursion on the parameter nn. We provide two different proofs of the Zamolodchikov tetrahedron equation satisfied by the star-triangle transformation in the case of n=2n=2 multivariate Tutte polynomial, we extend the latter to the case of valency 2 points and show that the Biggs formula and the star-triangle transformation commute.
We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to AN, DN and BN singularities. In the case of AN Frobenius manifolds our hierarchy turns out to coincide with the KP hierarchy; for BN Frobenius manifolds it coincides with the BKP hierarchy; and for DN hierarchy it is a certain reduction of the 2-component BKP hierarchy. As a side product to these results we illustrate the enumerative meaning of certain coefficients of AN, DN and BN Frobenius potentials.
We present a study of real Hurwitz numbers enumerating a special kind of real meromorphic functions, which we call simple framed purely real functions. We deduce partial differential equations of cut-and-join type for generating functions for these numbers. We also construct a topological field theory for them.
We provide possible directions of generalizations of earlier found relations between the Chebyshev polynomials and the Catalan numbers arising in studying commuting difference operators. These generalizations are mostly related with ideas proposed by N.H. Abel in his publication in 1826, which then were reproduced by many authors in a modern language. As generalization of Chebyshev polynomials, we propose to consider polynomials with exactly two critical values well-studied in a so-called theory of dessins d'enfants. The Catalan numbers are located in the first column of the table of Harer-Zagier numbers related with the distribution by genus of orientable sewing of polygons with even number of sides. The commuting difference operators are implicitly contained in the Abel theory, who studied quasi-elliptic integrals, namely, the elliptic integrals of 3rd kind integrable in terms of logarithms. In the present work we formulate conjectures on relation between the main Abel theorem and commuting semi-infinite matrices. In the work we provide calculations supporting the conjectured relations.
The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations --- the Denef--Loeser conjecture on topological zeta functions --- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables.
A crucial difference from the case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this setting. We develop several new tools and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.
A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a generalization of this construction for the singularities of types A and D that gives a solution of the open WDVV equations. For the A-singularity, the resulting solution describes the intersection numbers on the moduli space of r-spin disks, introduced recently in a work of the 2nd author, E. Clader and R. Tessler. In the 2nd part of the paper, we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups.
A meromorphic differential on a Riemann surface is said to be real-normalized if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. Our goal is to develop tools to study this topology. We propose a combinatorial model for the real normalized differentials with a single order 2 pole and use it to analyze the corresponding absolute period foliation.
We derive a quadratic recursion relation for the linear Hodge integrals of the form $〈τ_2^nλ_k〉$. These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.
In this paper we study the so-called sigma form of the second Painleve hierarchy. To obtain this form, we use some properties of the Hamiltonian structure of the second Painleve hierarchy and of the Lenard operator.
To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4- regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for such graphs, called a transition polynomial. We specialize this polynomial to a multiplicative weight system, that is, a function on chord diagrams satisfying 4-term relations and determining thus a finite type knot invariant. We prove a similar statement for the transition polynomial of general ribbon graphs and binary delta-matroids defined by R. Brijder and H. J. Hoogeboom, which defines, as a consequence, a finite type invariant of links.
We construct a generalization of the Tutte polynomial for vertex-weighted graphs for which the coefficients of the “deletion–contraction” relation depend nontrivially on the vertex weights. We show that the corresponding relation on the coefficients coincides with the two-cocycle relation in the group cohomology. We obtain a representation of a new invariant by summing over subgraphs and establish its connection with four-invariants of graphs.
The notions of Newton polytope, toric variety, tropical geometry and Groebner basis established fundamental relations between the algebraic and convex geometries. This survey presents the state of the art of the interfaces between these notions.