From ribbon categories to generalized Yang-Baxter operators and link invariants (after Kitaev and Wang)

TL;DR

This paper establishes a connection between link invariants derived from ribbon categories and generalized Yang-Baxter operators, showing they produce the same invariants when appropriately normalized, and explores new operators from specific fusion categories.

Contribution

It proves the equivalence of link invariants from ribbon categories and generalized Yang-Baxter operators and introduces new operators from $SO(N)_2$ categories.

Findings

The two approaches yield identical link invariants after normalization.

New generalized Yang-Baxter operators are constructed from $SO(N)_2$ categories.

Link invariants are specializations of the two-variable Kauffman polynomial.

Abstract

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang-Baxter operators with appropriate enhancements. The generalized Yang-Baxter operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these generalized Yang-Baxter operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a generalized Yang-Baxter operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of generalized Yang-Baxter operators which is obtained from the ribbon fusion categories $SO(N)_2$, where $N$ is an odd integer. These operators are given by $8\times 8$ matrices with the parameter $N$ and the link invariants are specializations of the two-variable Kauffman polynomial invariant $F$.