Bilinear Enhancements of Quandle Invariants

Abstract

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

arXiv comment9 pages. Version 2 includes typo corrections