Virtual Biquandle Cocycle Quiver Representations
Alexander Bishop, Jose Ceniceros, Sam Nelson
TL;DR
The paper addresses enriching virtual knot invariants by generalizing cocycle quiver representations to virtual biquandles. It introduces virtual biquandle Boltzmann quiver representations parameterized by $((X,v),A,C,k,S)$ and proves isotopy invariance, enabling four polynomial invariants $\Phi_{\chi}^E$, $\Phi_{\chi}^P$, $\Phi_{p_M}^E$, and $\Phi_{p_M}^P$ via decategorification. Through concrete computations, it demonstrates that these invariants are computable and can distinguish virtual knots that agree under some other invariants, while also highlighting their complementary nature. The framework links colorings, Boltzmann weights, and quiver representations to yield infinite families of invariants with potential for scale and application to broader classes of classical and virtual knots.
Abstract
We introduce quiver representation-valued invariants of oriented virtual knots and links associated to a choice of finite virtual biquandle, abelian group, set of virtual Boltzmann weights, commutative unital ring and set of virtual biquandle endomorphisms. As an application we define new infinite families of polynomial virtual knot and link invariants via decategorification.