Skein theories for virtual tangles

TL;DR

This paper classifies virtual knot polynomials and trivalent graph invariants using skein-theoretic methods, revealing new categories and invariants, including those from $ ext{Rep}(O(2))$ and Deligne's $S_t$, under smallness conditions.

Contribution

It provides a comprehensive classification of virtual knot polynomials and trivalent graph invariants with smallness constraints, introducing new skein theories and braided categories.

Findings

Classified all virtual knot polynomials below Higman-Sims spin model

Identified skein theories from $ ext{Rep}(O(2))$ with braiding

Classified invariants of non-planar trivalent graphs from symmetric categories

Abstract

In this paper, we use skein-theoretic techniques to classify all virtual knot polynomials and trivalent graph invariants with certain smallness conditions. The first half of the paper classifies all virtual knot polynomials giving non-trivial invariants strictly smaller than the one given by the Higman-Sims spin model. In particular, we exhibit a family of skein theories coming from $\text{Rep}(O(2))$ with an interesting braiding. In addition, all skein theories of oriented virtual tangles with some smallness conditions are classified. In the second half of the paper, we classify all non-trivial invariants of (perhaps non-planar) trivalent graphs coming from symmetric trivalent categories. For each of these categories, we also classify when the sub-category generated by only the trivalent vertex is braided. An interesting example of this arise from the tensor category Deligne's $S_t$.