[2207.08893] $N$-quandle spatial graph

Download the PDF of the paper entitled $N$-quandles of spatial graphs by Veronica Backer Peral and Blake Mellor

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overview: The fundamental quandle is a strong invariant of knots, links, and spatial graphs, but it is often difficult to determine whether two quandles are isomorphic. One approach is to look at the quotient of quandles, such as the $n$-quandle defined by Joyce \cite{JO}. In particular, Hoste and Shanahan \cite{HS2} classified knots and links with finite $n$ quandles. Mellor and Smith \cite{MS} introduced the link’s $N$-quandle as a generalization of Joyce’s $n$-quaddle and proposed a classification of links by finite $N$-quandles. We generalize the $N$-quandles to spatial graphs and investigate which spatial graphs have finite $N$-quandles. We prove basic results on $N$-quandles of spatial graphs and extend the link conjecture of \cite{MS} to infer the classification of spatial graphs with finite $N$-quandles. We test our conjecture in several cases and present possible counterexamples.