We introduce a quandle invariant of classical and virtual links, denoted by Qtc(L). This quandle has the property that Qtc(L)≅Qtc(L′) if and only if the components of L and L′ can be indexed in such a way that L = K1 ∪⋯ ∪ Kμ, L′ = K 1′∪⋯ ∪ K μ′ and for each index i, there is a multiplier 𝜖i ∈{−1, 1} that connects virtual linking numbers over Ki in L to virtual linking numbers over Ki′ in L′: ℓj/i(Ki,Kj) = 𝜖iℓj/i(Ki′,K j′) for all j≠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.

中文翻译：

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连接数字、quandles 和组

我们引入了经典链接和虚拟链接的 quadle 不变量，表示为问tc(大号). 这个 qudle 具有以下特性问tc(大号)≅问tc(大号')当且仅当大号和大号'可以这样索引大号 = ķ1 ∪⋯ ∪ ķμ,大号' = ķ 1'∪⋯ ∪ ķ μ'并且对于每个索引一世, 有一个乘数𝜖一世 ∈{-1, 1}将虚拟链接号码连接到ķ一世在大号到虚拟链接号码ķ一世'在大号'：ℓj/一世(ķ一世,ķj) = 𝜖一世ℓj/一世(ķ一世',ķ j')对所有人j≠一世. 我们还将陈的经典定理扩展到虚拟链接，它将链接数与幂零商联系起来G(大号)/G(大号)3.