Linking numbers, quandles and groups

Abstract

We introduce a quandle invariant of classical and virtual links, denoted by [Formula: see text]. This quandle has the property that [Formula: see text] if and only if the components of [Formula: see text] and [Formula: see text] can be indexed in such a way that [Formula: see text], [Formula: see text] and for each index [Formula: see text], there is a multiplier [Formula: see text] that connects virtual linking numbers over [Formula: see text] in [Formula: see text] to virtual linking numbers over [Formula: see text] in [Formula: see text]: [Formula: see text] for all [Formula: see text]. We also extend to virtual links a classical theorem of Chen, which relates linking numbers to the nilpotent quotient [Formula: see text].