Topological Representations of Free Numerical Semigroups via Iterated Torus Knots

2024/12/02 by Patricio Almirón, Almirón, Patricio, Adrián Olivares-Fernández +1
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2412.01613

openalex publication_date 2024/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

In this paper we will associate a family \K₁,…,K_l\⊂ \mathbbS³ of iterated torus knots to a given free numerical semigroup. We will describe the fundamental group of the knot complement of each knot of the family. Finally, we will show that all knots in the family have same Alexander polynomial and it coincides (up to a factor) with the Poincaré series of the free numerical semigroup. As a consequence, we will provide families of iterated torus knots with the same Alexander polynomial of an irreducible plane curve singularity but which are non-isotopic to its associated knot.

Related

• Numerical semigroups, Dedekind sums and Alexander polynomials of torus knots
• Semigroups of L-space knots and nonalgebraic iterated torus knots
• The Alexander polynomial of twisted torus knots
• Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations
• Hyperbolic L-space knots and their formal semigroups