Biquasiles and Dual Graph Diagrams

Deanna Needell, Sam Nelson

Published 2016-10-21Version 1

We introduce \textit{dual graph diagrams} representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call \textit{biquasiles} whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links. In particular, we identify an example of a finite biquasile whose counting invariant distinguishes the chiral knot $9_{32}$ from its mirror image, demonstrating that biquasile counting invariants are distinct from biquandle counting invariants.