Virtual tangles and fiber functors

We define a category [Formula: see text] of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to [Formula: see text] which induces an equivalence of categories. On the other hand, we show that [Formula: see text] is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

Graphs, foams, tensors, polytopes, and homology

In the study of knotted trivalent graphs and their higher dimensional analogue, knotted foams, some of the moves have alternative interpretations. Here three interpretations are given. (1) As the boundaries of chains in a homology theory, (2) as a system of abstract tensor relations, and (3) as a collection of polyhedra that include the permutohedron. The homological interpretation will allow for a solution to the abstract tensor system.

HOMOMORPHIC EXPANSIONS FOR KNOTTED TRIVALENT GRAPHS

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.