A diagrammatic presentation and its characterization of non-split compact surfaces in the 3-sphere

We give a presentation for a non-split compact surface embedded in the 3-sphere [Formula: see text] by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces represented by the diagrams are ambient isotopic in [Formula: see text].

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.

Alexander- and Markov-type theorems for virtual trivalent braids

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on [Formula: see text]-moves.

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.

The Yang–Mills measure in the SU(3)-skein module

Let [Formula: see text] be a nonzero complex number which is not a root of unity. Let [Formula: see text] be a compact oriented surface, the [Formula: see text]-skein space of [Formula: see text], [Formula: see text], is the vector space over [Formula: see text] generated by framed oriented links (including framed oriented trivalent graphs in [Formula: see text]) quotient by the [Formula: see text]-skein relations due to Kuperberg [Spiders for rank [Formula: see text] Lie algebra, Comm. Math. Phys. 180(1) (1996) 109–151]. For closed [Formula: see text], with genus greater than [Formula: see text], we construct a local diffeomorphism invariant trace on [Formula: see text] when [Formula: see text] is a positive real number not equal to [Formula: see text].

The Markov theorems for spatial graphs and handlebody-knots with Y-orientations

We establish the Markov theorems for spatial graphs and handlebody-knots. We introduce an IH-labeled spatial trivalent graph and develop a theory on it, since both a spatial graph and a handlebody-knot can be realized as the IH-equivalence classes of IH-labeled spatial trivalent graphs. We show that any two orientations of a graph without sources and sinks are related by finite sequence of local orientation changes preserving the condition that the graph has no sources and no sinks. This leads us to define two kinds of orientations for IH-labeled spatial trivalent graphs, which fit a closed braid, and is used for the proof of the Markov theorem. We give an enhanced Alexander theorem for orientated tangles, which is also used for the proof.