Fundamentals of cubic skein modules

Abstract: Over the past thirty-seven years, the study of linear and quadratic skein modules has produced a rich and far-reaching skein theory, intricately connected to diverse areas of mathematics and physics, including algebraic geometry, hyperbolic geometry, topological quantum field theories, and statistical mechanics. However, despite these advances, skein modules of higher degree-those depending on more parameters than the linear and quadratic cases-have received comparatively little attention, with only a few isolated explorations appearing in the literature. In this article, we undertake a systematic study of the cubic skein module, the first representative of this broader class. We begin by investigating its structure and properties in the $3$-sphere, and then extend the analysis to arbitrary $3$-manifolds. The results presented here aim to establish a foundational framework for the study of higher skein modules, thereby extending the scope of skein theory beyond its classical domains. Furthermore, studying the structure of cubic skein modules may lead to new polynomial invariants of knots.