A note on Fox colorings of virtual tangles

TL;DR

This paper investigates Fox colorings of virtual tangles, revealing how boundary color vectors are characterized in classical cases and how these conditions change in virtual settings, especially over different rings.

Contribution

It provides a characterization of boundary color vectors for classical tangles and explores their realizability in virtual tangles over various rings.

Findings

Classical tangle boundary vectors satisfy an alternating sum condition.

In the virtual setting over integers, realizability depends on a divisibility condition.

Over rac{p}{}Z, all boundary vectors are realizable in virtual tangles.

Abstract

We study Fox colorings of tangle diagrams by $R=\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}$, where $p\geq3$ is an odd integer. For an $R$-colored $m$-string tangle diagram, the colors at the $2m$ boundary points form a vector $v\in R^{2m}$. We show that for classical tangle diagrams, such vectors are completely characterized by the alternating sum condition $\Delta(v)=0$. We then investigate how this restriction changes in the virtual setting. For $R=\mathbb{Z}$, the realizability of $v$ is determined by a divisibility condition on $\Delta(v)$. For $R=\mathbb{Z}/p\mathbb{Z}$, every vector is realizable by a virtual tangle diagram.