Topological Representations of Free Numerical Semigroups via Iterated Torus Knots

TL;DR

The paper addresses the problem of multiple fibered knots sharing a polynomial invariant by constructing a systematic family of iterated torus knots $\mathcal{K}_S$ from a free numerical semigroup $S$. It develops an explicit inductive cabling construction encoded by splice diagrams and provides detailed presentations of the knot groups, along with a unifying Alexander polynomial $\Delta_K(t)=P_S(t)(1-t)$ tied to the semigroup's generating series $P_S(t)$. This framework connects semigroup combinatorics to knot invariants, recovering algebraic knots and situating $L$-space knots within the same family, while showing the potential for non-isotopic knots to share invariants. The work also discusses topological realizability of free semigroups and how different generator orderings yield distinct knots in $\mathcal{K}_S$, enriching the interaction between singularity theory and knot theory.

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 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.

Figures (4)

• Figure 2.1: Splice diagram of a $(p,q)-$torus knot.
• Figure 2.2: Splice diagram of an iterated torus knot.
• Figure 2.3: Splice diagram of the links $\widetilde{L}_i$
• Figure 3.1: Splice diagram of iterated torus knot

Theorems & Definitions (17)

• Theorem
• Theorem
• Definition 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Example 2.4
• Proposition 2.5
• proof
• Proposition 3.1