Related papers: Mock Alexander Polynomials
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…
We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.
Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a…
Fox's conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remains open in general, a number of special cases have been…
In this paper we study the theory of knotoids and braidoids and the theory of pseudo knotoids and pseudo braidoids on the torus T. In particular, we introduce the notion of {\it mixed knotoids} in $S^2$, that generalize the notion of mixed…
A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By…
In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball $B^{3}$ with suitable identification of boundary points, then we can project the links on the equatorial…
We consider a braid $\beta$ which acts on a punctured plane. Then we construct a local system on this plane and find a homology cicle $D$ in its symmetric power such that $D\cdot \beta(D)$ coincides with the Alexander polynomial of the…
Let s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial…
For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot…
For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the…
We introduce a framework to analyze knots and links in an unmarked solid torus. We discuss invariants that detect when such links are equivalent under an ambient homeomorphism, and show that the multivariable Alexander polynomial is such in…
We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation…
We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids…
The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two…
For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…
In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface,…
This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…
The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincar\'e sphere: the intersection of the surface $S=\{(z_1,z_2,z_3)\in {\mathbb C}^3: z_1^5+z_2^3+z_3^2=0\}$…
We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps…