Hyperbolic L-space knots and their formal semigroups

TL;DR

This paper constructs an infinite family of hyperbolic L-space knots with formal semigroups generated by five elements, expanding the known examples beyond those generated by three.

Contribution

It introduces the first infinite family of hyperbolic L-space knots with formal semigroups generated by five elements, advancing understanding of their algebraic structures.

Findings

Existence of hyperbolic L-space knots with formal semigroups generated by five elements

Extension of previous examples with semigroups generated by three elements

First infinite family demonstrating this property

Abstract

For an L-space knot, the formal semigroup is defined from its Alexander polynomial. It is not necessarily a semigroup. That is, it may not be closed under addition. There exists an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by three elements. In this paper, we give the first infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by five elements.