Related papers: Alexander polynomial of sextics

We compute Bernstein-Sato polynomials of some pairs of topologically equivalent plane curve singularities. Some pairs have the same Tjurina number but distinct Bernstein-Sato polynomials, which implies that they are not analytically…

Consider the $n$th degree polynomial equation, $X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than ${2n \choose 2}$ solutions, then it has infinitely many solutions. We show here that…

For the Borromean link, we determine its irreducible ${\rm SL}(2,\mathbb{C})$-character variety, and find a formula for the twisted Alexander polynomial as a function on the character variety.

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-bridge link,…

We give a complete classification of all Salem polynomials of length 5. For length 6 we show that all but finitely many Salem polynomials lie in one of 12 infinite families, and subject to Lehmer's Conjecture we give a complete list of the…

We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…

Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted…

We study the distribution of arithmetic invariants associated to Alexander polynomials for certain infinite families of links. The families of links we consider arise from braids on a fixed number of strings. We explore analogies with…

We give a short introduction to the theory of twisted Alexander polynomials of a 3--manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted…

We construct many irreducible polynomials within semigroups generated by sets of the form $S=\{x^2+c_1,\dots,x^2+c_s\}$ under composition.

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…

For a pair $(P,Q)$ of finite posets the generators of the ideal $L(P,Q)$ correspond bijectively to the isotone maps from $P$ to $Q$. In this note we determine all pairs $(P,Q)$ for which the Alexander dual of $L(P,Q)$ coincides with…

The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $\Delta_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied…

Multivariable Alexander invariants of algebraic links calculated in terms of algebro-geometric invariants (polytopes and ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity…

For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x^2), arithmetic properties of certain coefficients arising are…

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.

For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their…

We present explicit infinite families of twisted torus knots that are not fibered. Our approach relies on an explicit formula for the Alexander polynomial derived in our previous work. We show that the leading coefficients of the Alexander…

In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever…