Biquasiles and Dual Graph Diagrams

TL;DR

This paper introduces dual graph diagrams for knots and links, defines biquasiles as algebraic structures inspired by Reidemeister moves, and develops invariants for oriented knots and links based on these concepts.

Contribution

It presents a novel combinatorial and algebraic framework using dual graph diagrams and biquasiles to study knot invariants, extending previous algebraic approaches.

Findings

Defined dual graph diagrams for knots and links

Introduced biquasiles as new algebraic structures

Constructed invariants for oriented knots and links

Abstract

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 and provide examples.