Defined in Joyce, 1982, and Matveev, 1984, the fundamental quandle is a complete invariant of oriented classical knots up to ambient homeomorphism. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce a generalization of the Alexander quandle of a knot known as the fundamental Latin Alexander quandle and consider its Gröbner basis-valued invariants, which generalize the Alexander polynomial. We show via example that the invariant is not determined by the generalized Alexander polynomial for virtual knots.

The first author was partially supported by Simons Foundation collaboration grant 316709