Biquandle Module Quiver Representations
Yewon Joung, Sam Nelson
TL;DR
This work develops a new infinite family of quiver representation-valued invariants for classical, virtual, and surface-knots by marrying finite biquandles with module coefficients and endomorphism-induced quivers. It extends biquandle bead-coloring invariants to full quiver representations via a data triple $(X,M,k,S)$ and defines a decategorified invariant, the natural path polynomial $\Phi_{\vec{D}}^{MP}(L)$, as a sum over identity-length paths. The approach yields a two-variable polynomial invariant that can distinguish knots sharing the biquandle counting invariant and applies to virtual knots and oriented surface-links, with concrete Python-based computations. Overall, the framework integrates coloring, module enhancements, and quiver representations to produce computable invariants with potential broad applicability to knot theory and its generalizations.
Abstract
We introduce an infinite family of quiver representation-valued invariants of classical, virtual and surface-knots and links associated to a choice of finite biquandle, commutative unital ring, biquandle module and set of biquandle endomorphisms. As an application, we use this quiver to define a new infinite family of two-variable polynomial invariants.