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