# Generating Set for Nonzero Determinant Links Under Skein Relation

## TL;DR

This paper investigates the minimal initial set of link invariants needed to determine all nonzero determinant links via skein relations, showing that only the unknot suffices in unoriented cases, while the unknot and Hopf link are needed in oriented cases.

## Contribution

It establishes the minimal generating sets for nonzero determinant link invariants under skein relations, distinguishing between oriented and unoriented cases.

## Key findings

- Unknot alone generates all nonzero determinant links in unoriented skein relations.
- Unknot and Hopf link orientations generate all nonzero determinant links in oriented skein relations.
- Provides a recursive framework for understanding link invariants based on initial generators.

## Abstract

Traditionally introduced in terms of advanced topological constructions, many link invariants may also be defined in much simpler terms given their values on a few initial links and a recursive formula on a skein triangle. Then the crucial question to ask is how many initial values are necessary to completely determine such a link invariant. We focus on a specific class of invariants known as nonzero determinant link invariants, defined only for links which do not evaluate to zero on the link determinant. We restate our objective by considering a set $\mathcal{S}$ of links subject to the condition that if any three nonzero determinant links belong to a skein triangle, any two of these belonging to $\mathcal{S}$ implies that the third also belongs to $\mathcal{S}$. Then we aim to determine a minimal set of initial generators so that $\mathcal{S}$ is the set of all links with nonzero determinant. We show that only the unknot is required as a generator if the skein triangle is unoriented. For oriented skein triangles, we show that the unknot and Hopf link orientations form a set of generators.