Narrow Results

Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e. we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We view braid groups as subgroups of braid-permutation groups. We construct a family of universal representations of braid-permutation groups, without using associators. All representations in the family are faithful, defined over ℚ by simple explicit formulae. We show that they give universal Vassiliev-type invariants for braid-permutation groups.

It had been known since old times (works of Murakami–Ohtsuki, Cheptea–Le and the second author) that there exists a universal finite type invariant ("an expansion") Z^old for knotted trivalent graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is "homomorphic"). Yet perhaps the most important operation on KTGs is the "edge unzip" operation, and while the behavior of Z^old under edge unzip is well understood, it is not plainly homomorphic as some "correction factors" appear. In this paper we present two equivalent ways of modifying Z^old into a new expansion Z, defined on "dotted knotted trivalent graphs" (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace "edge unzips" by "tree connected sums", and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of KTGs retains all the good qualities that KTGs have — it remains firmly connected with the Drinfel'd theory of associators and it is sufficiently rich to serve as a foundation for an "algebraic knot theory". As a further application, we present a simple proof of the good behavior of the LMO invariant under the Kirby II (band-slide) move, first proven by Le, Murakami, Murakami and Ohtsuki.

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that |Q| ≥ 32 if Q is not conjugacy closed. We also show that |Q| ≥ 64 if there exists x ∈ Q such that x² is not in the nucleus of Q. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that has both nilpotency class 3 and an abelian inner mapping group.

A gyrogroup is a non-associative structure that may be regarded as a suitable generalization of groups. In this paper, we introduce the notion of an associator in an arbitrary gyrogroup that, in some sense, measures the deviation from associativity of gyrogroup operations. We then study the normal closure of the set of associators and establish several relevant properties, including the universal property of the associativization of a gyrogroup. This leads to a few invariant properties of gyrogroups in terms of associator normal subgyrogroups and gives an effective method to study representations of gyrogroups.