\(N\)-quandles of spatial graphs (Q6585079)

Quandles are algebraic objects defined by a list of axioms that capture the properties of conjugation within a group. One of their most prominent applications is in providing an invariant of links in~\(\mathbb{S}^3\), the fundamental quandle, which \textit{M. Niebrzydowski} [J. Knot Theory Ramifications 19, No. 6, 829--841 (2010; Zbl 1221.05158)] generalized to spatial graphs, that is, embeddings of finite graphs into~\(\mathbb{S}^3\).\N\NThe fundamental quandle is a rather fine invariant, but this comes at the cost of quandles being generally difficult to compare. This problem becomes more manageable if one makes them smaller by adding relations, and one way of doing so is as follows: Every fundamental quandle \(Q\)~has finitely many orbits for its action on itself (called the components of~\(Q\)). Given a tuple~\(N\) with one positive integer~\(n_C\) for each component~\(C\), we impose the additional relation that the \(n_C\)-th power of every element of~\(C\) acts trivially. This produces the fundamental \(N\)-quandle~\(Q_N\) of the link or spatial graph, which is the central object of the article.\N\NThe second author and \textit{R. Smith} [Topology Appl. 294, Article ID 107662, 26 p. (2021; Zbl 1465.57062)] proved that for every link~\(L\) and tuple \(N\)~of suitable length, the fundamental \(N\)-quandle of~\(L\) is finite if~\(L\), with its components labeled by~\(N\), is the singular locus of a spherical \(3\)-orbifold with underlying space~\(\mathbb{S}^3\). Moreover, they conjectured that this is an equivalence. In the article under review, the authors extend this conjecture to spatial graphs. More precisely, they conjecture that the fundamental \(N\)-quandle of a spatial graph~\(G\) is finite if and only if \(G\)~is homeomorphic to a subgraph of the singular locus~\(H\) of a spherical \(3\)-orbifold with underlying space~\(\mathbb S^3\), and the entries in~\(N\) divide the corresponding labels of~\(H\).\N\NThe paper provides evidence in support of the ``if'' direction. By showing that finiteness of fundamental \(N\)-quandles is preserved under taking subgraphs, subdividing edges, and replacing~\(N\) with a tuple dividing it, the proof of this implication is reduced to verifying finiteness of the fundamental \(N\)-quandles of singular loci of spherical \(3\)-orbifolds with underlying space~\(\mathbb{S}^3\). In Dunbar's classification of geometric \(3\)-orbifolds [\textit{W. D. Dunbar}, Rev. Mat. Univ. Complutense Madr. 1, No. 1--3, 67--99 (1988; Zbl 0655.57008)], spherical orbifolds come in two types: one consisting of \(3\)-orbifolds that fiber over a \(2\)-orbifold (Type 2), and one consisting of \(18\)~``exceptional'' \(3\)-orbifolds (Type 4).\N\NThe authors verify the conjecture for all but one of the labeled spatial graphs in the ``Type~4'' list. To that end, they use a computer implementation of an algorithm due to \textit{S. Winker} [Quandles, knot invariants, and the \(n\)-fold branched cover, Ph.D. thesis, University of Illinois, Chicago, 1984, \url{http://homepages.math.uic.edu/~kauffman/Winker.pdf}] and \textit{J. Hoste} and \textit{P. D. Shanahan} [Math. Comput. 88, No. 317, 1427--1448 (2019; Zbl 1414.57008)], which enumerates the elements of a quandle given by a presentation, and halts precisely if the quandle is finite. Whether the one remaining fundamental \(N\)-quandle in this list is finite is left as an open question (though answered affirmatively in follow-up work).\N\NThen, the authors compute the sizes of fundamental quandles for an infinite family of spatial graphs that arise in Dunbar's ``Type 2''' orbifolds. The spatial graphs in this family are quite tame compared to the general case, yet this section of the paper is rather technical, suggesting that a different approach is necessary for proving the conjecture in general.\N\NThe article does a very good job of succintly acquainting the reader with some of the basic concepts, such as quandles, fundamental quandles of links and spatial graphs, their Wirtinger presentations, and Cayley graphs of quandles. Several helpful illustrations and examples support the exposition. The theory of geometric \(3\)-orbifolds is blackboxed in Dunbar's classification.\N\NIn follow-up work, the second author gives a proof of the aforementioned conjecture of Mellor-Smith, and establishes the direction under study of the main conjecture in this paper. Specifically, Mellor shows that for labeled spatial graphs \((G,N)\) that arise as the singular locus of a \(3\)-orbifold, the fundamental \(N\)-quandle is finite if and only if the orbifold is spherical and has underlying space~\(\mathbb{S}^3\) [\textit{B. Mellor}, ``Classifying links and spatial graphs with finite \(N\)-quandles'', Preprint, \url{arXiv:2304.05537}].