Related papers: A bracket polynomial for 2-tangle shadows

An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence…

Let $\mathbb F_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over $\mathbb F_{q^2}$ with many rational points. The curves…

The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive'…

There is a digraph corresponding to every square matrix over $\mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence…

We compute an analogue of Pascal's triangle enriched in bilinear forms over a finite field. This gives an arithmetically meaningful count of the ways to choose $j$ ring homomorphisms into an algebraic closure from an \'etale extension of…

We construct an integer polynomial whose coefficients enumerate the Kauffman states of the two-bridge knot with Conway's notation C(n,r).

Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this…

We give a bracket polynomial expression for intermediate terms between discriminant and resultant for pair of binary forms. As an application of the bracket polynomial expression, we give an algebraic proof of the algebraic independence of…

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the…

The special shadow-complexity is an invariant of closed $4$-manifolds defined by Costantino using Turaev's shadows. We show that for any positive integer $k$, the special shadow-complexity of the connected sum of $k$ copies of $S^1\times…

A polynomial f (multivariate over a field) is decomposable if f = g(h) with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number…

We apply a symbolic approach of the general quadratic decomposition of polynomial sequences - presented in a previous article referenced herein - to polynomial sequences fulfilling specific orthogonal conditions towards two given…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve…

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study…

Let $A = (a_1,\dots,a_n)\in \mathbb{Z}^n$ be a sequence with sum $k(2g-2+n)$. The double ramification cycle $\mathsf{DR}_g(A) \in \mathsf{CH}^g(\bar{\mathcal{M}}_{g,n})$ is the virtual class of the locus of curves $(C,p_1,\dots,p_n)$ where…

Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic…

For any positive integer r, we exhibit a knot Kr with (20 $\times$ 2 r--1 + 1) crossings whose Jones polynomial V (Kr) is equal to 1 mod-ulo 2 r. Our construction rests on a certain 20-crossing tangle T 20 which is undetectable by the…

We give a monomial basis for the Kauffman bracket skein algebra of the $4$-holed disk, and find a presentation. This is based on an insight into the ${\rm SL}(2,\mathbb{C})$-character variety of the rank $4$ free group.

There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have…

We introduce a Kauffman-Jones type polynomial $\mathcal{L}_{\gamma}(A)$ for a curve $\gamma$ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial $\mathcal{L}_{\gamma}(A)$ is a Laurent polynomial in one…