Bilinear enhancements of quandle invariants

Abstract

We enhance the quandle counting invariants of oriented classical and virtual links using a construction similar to quandle modules but inspired by symplectic quandle operations rather than Alexander quandle operations. Given a finite quandle [Formula: see text] and a vector space [Formula: see text] over a field, sets of bilinear forms on [Formula: see text] indexed by pairs of elements of [Formula: see text] satisfying certain conditions yield new enhanced multiset- and polynomial-valued invariants of oriented classical and virtual links. We provide examples to illustrate the computation of the invariants and to show that the enhancement is proper.