Generating Set for Nonzero Determinant Links Under Skein Relation

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.