Invariants of Links From the Generalized Yang–baxter Equation

Abstract

Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators (gYB-operators). At present two examples of gYB-operators are known and recently three types of variations for one of these were discovered. We present the definition of enhanced generalized Yang-Baxter operators and show that all known examples of gYB-operators can be enhanced to give corresponding invariants of oriented links. Most of these invariants are specializations of the polynomial invariant P . Invariants from gYB-operators are multiplicative after a normalization.We define enhanced generalized Yang-Baxter operators (briefly, EgYBoperators). These operators are new, but very few nontrivial examples of gYB-operators are known. In fact, so far only unitary solutions have been considered. We discuss a (2, 3, 2)-type gYB-operator given in [9] as one of the main examples. Another example of gYB-operator, (2, 3, 1)-type, appeared in [3] and three families of its variations were discussed in [2]. In this paper, we enhance these gYB-operators and discuss corresponding isotopy invariants of oriented links.Note that from any ribbon category we have a link invariant for each object. If there is a (generalized) localization in the sense of [3,8], then some enhancement should exist and it is reasonable to expect that the corresponding invariant recovers the one defined directly from the category. In [3], it is shown that SU(3) 3 theory has no 4 × 4 ordinary localization which corresponds to the usual YB-operator, but does have an 8 × 8 generalized localization which corresponds to gYB-operator. No 4 × 4 unitary YB-operator can yield the invariant discussed in [7] (and identified in [6]), and it seems unlikely that any ordinary YB-operator can. This is one of the motivations for this work. On the other hand, it is possible to find a 4 × 4 operator that together with associativities can produce this invariant. So it is possible that the gYB-operator can somehow absorb the nontrivial associativities.After online version of this work appeared, a method to obtain gYB-operators from certain objects in ribbon fusion categories was suggested [5]. Following this line a family of (2, 3, 1) gYB-operators were constructed from the unitary modular categories SO(N ) 2 and corresponding invariants of links were studied [4]. Furthermore, it is proved that the invariants are the same as those obtained directly from the categories.Here are the contents of this paper. In Sec. 2, we recall the notion of gYBoperator and introduce the EgYB-operators. In the definition of EgYB-operator, the orthogonality condition is weaker than the corresponding condition in the definition of EYB-operator. In Sec. 3, we define an invariant of oriented links associated with each EgYB-operator in a similar way as in [10]. The invariant is projectively multiplicative on a disjoint union of links, that is, multiplicative up to a (uniform) constant factor. In Sec. 4, some examples of EgYB-operators and corresponding lin...

Parameter-dependent Gaussian $(z,N)$-generalized Yang-Baxter operators

“…Moreover, these give rise to representations of the braid group, which plays a central role in the topological model for quantum computation ( [6]). A cascade of papers [7,2,14,15,19] followed these definitions, yielding new solutions and exploring new applications.…”