## Cayley graphs for finite N-quandles

Below is a program to generate the Cayley graph of a finite *N*-quandle, based on an algorithm due to Winker. The program was used for the following papers:

*N*-quandles of links, by Blake Mellor and Riley Smith (*Topology and its Applications*, vol. 294, 2021)

**Abstract:** The fundamental quandle is a powerful invariant of knots and links, but it is difficult to describe in detail. It is often useful to look at quotients of the quandle, especially finite quotients. One natural quotient introduced by Joyce is the *n*-quandle. Hoste and Shanahan gave a complete list of the knots and links which have finite *n*-quandles. We introduce a generalization of *n*-quandles for links, denoted *N*-quandles (for a link with *k* components, *N* is a *k*-tuple of positive integers), and we conjecture a classification of the links with finite *N*-quandles.

N-quandles of spatial graphs, with Veronica Backer Peral, July 2022

**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.

*Mathematica* version: NQCayleyGraph.nb

*Python* version: NQCayleyGraph.py

Comments and questions may be sent to blake.mellor@lmu.edu. You are welcome to use and modify the code as you wish - but if you add any cool features, please share them with me!

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