A note on Fox colorings of 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 $Δ(v)=0$. 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 $Δ(v)$. For $R=\mathbb{Z}/p\mathbb{Z}$, every vector is realizable by a virtual tangle diagram.

Figures (8)

• Figure 1: ${\mathbb{Z}}$-colored classical and virtual tangle diagrams
• Figure 2: Two moves (a) and (b)
• Figure 3: Adding a circle to $T$
• Figure 4: Closing $(\beta,C')$ with $m$ arcs
• Figure 5: Connecting $(\beta,C')$ to $(T,C)$

Theorems & Definitions (23)

• Proposition 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2: NNSW
• Lemma 2.3: NNSW
• Theorem 2.4: NNSW
• Lemma 3.1: cf. Prz
• proof