Research ArticleNo Access

Linking numbers, quandles and groups

Lafayette College, Easton, PA 18042, USA

https://doi.org/10.1142/S0218216521500486Cited by:1 (Source: Crossref)

Abstract

We introduce a quandle invariant of classical and virtual links, denoted by $Q_{tc} (L)$ $Q_{tc} (L)$ . This quandle has the property that $Q_{tc} (L) ≅ Q_{tc} (L^{'})$ if and only if the components of $L$ and $L^{'}$ can be indexed in such a way that $L = K_{1} \cup \dots \cup K_{μ}$ , $L^{'} = K_{1}^{'} \cup \dots \cup K_{μ}^{'}$ and for each index $i$ , there is a multiplier $𝜖_{i} \in {- 1, 1}$ that connects virtual linking numbers over $K_{i}$ in $L$ to virtual linking numbers over $K_{i}^{'}$ in $L^{'}$ : $ℓ_{j / i} (K_{i}, K_{j}) = 𝜖_{i} ℓ_{j / i} (K_{i}^{'}, K_{j}^{'})$ for all $j \neq i$ . We also extend to virtual links a classical theorem of Chen, which relates linking numbers to the nilpotent quotient $G (L) / G {(L)}_{3}$ .

Keywords:

AMSC: 57K10

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