Bikei, Involutory Biracks and Unoriented Link Invariants

Abstract

We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N = 1 are biquandles, which we call bikei or 双圭. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, and we give an example of a non-involutory birack whose counting invariant detects the non-invertibility of a virtual knot.

Symmetric enhancements of involutory virtual birack counting invariants

“…involutory virtual biracks with π = Id : X → X, first introduced in [10]. The definitions and results in this section are straightforward generalizations, with new notation, of results from [1,13] etc. See [4,10,11] for more about oriented biracks and virtual biracks.…”

“…Virtual biracks are algebraic structures generalizing virtual biquandles (see [10]) with axioms motivated by the framed virtual Reidemeister moves. Involutory biquandles, useful for defining invariants of unoriented knots and links, were considered in [1]. Each finite example of these algebraic structures defines a computable integer-valued invariant of the related type of knots and links.…”

Bikei homology

“…Example 4. (See [1] for more) Let D be an unoriented knot or link diagram representing an unoriented knot or link K and let G be a set of generators corresponding to semiarcs in D. The set W of bikei words in G is defined recursively by the rules…”

Quantum enhancements of involutory birack counting invariants

“…It is easy to verify that X is involutory (see [2]), and it is straightfoward (if somewhat tedious) to verify that the following quadruple defines a quantum weight of X where V = Q 2 :…”