$N$-quandles of spatial graphs

TL;DR

This paper extends the concept of $N$-quandles from links to spatial graphs, investigates their finiteness, and proposes a classification conjecture, supported by partial verification and potential counterexamples.

Contribution

It generalizes $N$-quandles to spatial graphs and explores their finiteness, extending previous link-based classifications and conjectures.

Findings

Proved basic properties of $N$-quandles for spatial graphs.

Formulated a conjecture for classifying spatial graphs with finite $N$-quandles.

Presented partial verifications and a possible counterexample.

Abstract

The fundamental quandle is a powerful 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 quotients of the quandle, such as the $n$-quandle defined by Joyce \cite{JO}; in particular, Hoste and Shanahan \cite{HS2} classified the knots and links with finite $n$-quandles. Mellor and Smith \cite{MS} introduced the $N$-quandle of a link as a generalization of Joyce's $n$-quandle, and proposed a classification of the links with finite $N$-quandles. We generalize the $N$-quandle to spatial graphs, and investigate which spatial graphs have finite $N$-quandles. We prove basic results about $N$-quandles for spatial graphs, and conjecture a classification of spatial graphs with finite $N$-quandles, extending the conjecture for links in \cite{MS}. We verify the conjecture in several cases, and also present a possible counterexample.