Related papers: Poincar\'e duality and degrees of twisted Alexande…
The Tutte polynomial is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G, $T_G(X,Y)\; =\; {T}_{G^*}(Y,X)$ where $G^*$ denotes the…
In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely…
A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal…
We show how the Turaev--Viro invariant can be understood within the framework of Chern--Simons theory with gauge group SU(2). We also describe a new invariant for certain class of graphs by interpreting the triangulation of a manifold as a…
We introduce a version of the Alexander polynomial for singular knots and tangles and show how it can be strengthened considerably by introducing a perturbation. For singular long knots, we also prove that our Alexander polynomial agrees…
We prove a duality theorem for certain graded algebras and show by various examples different kinds of failure of tameness of local cohomology.
We define a generalization of the idempotented Schur algebra which gives a unified setting for a quantum group presentation of the Homfly-Pt polynomial, together with its specializations to the Alexander polynomial and the slm…
This paper presents two kinds of division polynomials for twisted Edwards curves. Their chief property is that they characterise the $n$-torsion points of a given twisted Edwards curve. We also present results concerning the coefficients of…
In this sequel to arxiv:arXiv:1012.0835 we develop Bezout type theorems for semidegrees (including an explicit formula for {\em iterated semidegrees}) and an inequality for subdegrees. In addition we prove (in case of surfaces) a Bernstein…
We connect Dedekind sums and Alexander polynomials of torus knots.
We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker…
We compute the equations and multidegrees of the biprojective variety that parametrizes pairs of symmetric matrices that are inverse to each other. As a consequence of our work, we provide an alternative proof for a result of Manivel,…
Given an oriented closed manifold $M$ of odd dimension and a unitary representation $\rho : \pi_1(M) \ra \GL_n(\F)$, we define a Reidemeister torsion, even if the cohomology associated with $\rho$ is not acyclic. As corollaries, we…
We calculate the tropical Dolbeault cohomology for the analytifications of the projective line and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincar\'e duality and behaves analogously to the…
We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative…
In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant…
We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the…
The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for…
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…
We give a brief survey of abelian torsions of 3-manifolds.