Linking numbers, quandles and groups

TL;DR

The paper introduces a new quandle invariant for classical and virtual links that captures linking number information and extends classical linking number theorems to virtual links.

Contribution

It defines a novel quandle invariant $Q_{tc}(L)$ that characterizes virtual link components up to linking number sign changes and extends Chen's linking number theorem to virtual links.

Findings

$Q_{tc}(L)$ distinguishes links based on linking number patterns.

Extension of Chen's theorem to virtual links.

Provides a new algebraic tool for virtual link classification.

Abstract

We introduce a quandle invariant of classical and virtual links, denoted $Q_{tc} (L)$. This quandle has the property that $Q_{tc} (L) \cong Q_{tc} (L')$ if and only if the components of $L$ and $L'$ can be indexed in such a way that $L=K_1 \cup \dots \cup K_{\mu}$, $L'=K'_1 \cup \dots \cup K'_{\mu}$ and for each index $i$, there is a multiplier $\epsilon_i \in \{-1,1\}$ that connects virtual linking numbers over $K_i$ in $L$ to virtual linking numbers over $K'_i$ in $L'$: $\ell_{j/i}(K_i,K_j)= \epsilon_i \ell_{j/i}(K'_i,K'_j)$ for all $j \neq i$. We also extend to virtual links a classical theorem of Chen, which relates linking numbers to the nilpotent quotient $G(L)/G(L)_3$.