Topological Representations of Free Numerical Semigroups via Iterated Torus Knots

TL;DR

This paper establishes a connection between free numerical semigroups and iterated torus knots, describing their fundamental groups and Alexander polynomials, revealing new topological invariants related to algebraic structures.

Contribution

It introduces a novel association between free numerical semigroups and families of iterated torus knots, analyzing their topological invariants and distinguishing non-isotopic knots with identical Alexander polynomials.

Findings

All knots in the family share the same Alexander polynomial.

The Alexander polynomial coincides with the Poincaré series of the semigroup.

Families of knots with the same Alexander polynomial as singularities but are non-isotopic.

Abstract

In this paper we will associate a family $\{K_1,\dots,K_l\}\subset \mathbb{S}^3$ 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 Poincar\'e 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.