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Register a new userFrom ribbon categories to generalized Yang-Baxter operators and link invariants (after Kitaev and Wang)
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Authors
Seung-moon Hong
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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 $8times 8$ matrices with the parameter $N$ and the link invariants are specializations of the two-variable Kauffman polynomial invariant $F$.
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Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators. At present two examples of generalized Yang-Baxter operators are known and recently three types of variations for one of these were discovered. We present the definition of enhanced generalized YB-operators and show that all known examples of generalized YB-operators can be enhanced to give corresponding invariants of oriented links. Most of these invariants are specializations of the polynomial invariant $P$. Invariants from generalized YB-operators are multiplicative after a normalization.
A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
A heap is a set with a certain ternary operation that is self-distributive (TSD) and exemplified by a group with the operation $(x,y,z)mapsto xy^{-1}z$. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus and pretzel links. We show that for these families of links there exist epimorphisms from fundamental heaps to Vinberg and Coxeter groups, implying that corresponding groups are infinite. A relation to the Wirtinger presentation is also described. The cocycle invariant is defined using ternary self-distributive (TSD) cohomology, by means of a state sum that uses ternary heap $2$-cocycles as weights. It is shown that the cohomology splits into two types, called degenerate and nondegenerate, and that the degenerate part is one dimensional. Subcomplexes are constructed based on group cosets, that allow computations of the nondegenerate part. We apply colorings inferred from fundamental heaps to compute cocycle invariants, and prove that the invariant values can be used to derive algebraic properties of the cohomology.
Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Gra~na. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases.
We construct solutions to the set-theoretic Yang-Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.
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