Quantum Enhancements of Involutory Birack Counting Invariants

TL;DR

This paper introduces quantum weights to enhance the involutory birack counting invariant, providing a new approach to analyze unoriented tangles through quantum algebraic methods.

Contribution

It presents the first enhancement of involutory birack invariants using quantum weights, linking birack theory with quantum tangle functors.

Findings

Enhanced invariant with quantum weights

Framework for involutory birack-labeled tangle functors

Potential for new tangle classification tools

Abstract

The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series of several papers, we enhance the involutory birack counting invariant with quantum weights, which may be understood as tangle functors of involutory birack-labeled unoriented tangles.