Groups versus quandle-like invariants of 3-manifolds

Abstract

Risandles are nonassociative algebraic structures recently introduced to construct invariants of 3-manifolds. In this note, we show that the categories of groups and nonempty, faithful risandles are equivalent. In analogy to knot quandles, we also introduce fundamental risandles of 3-manifolds, which categorify the risandle coloring invariants of Ishii, Nakamura, and Saito. For infinitely many 3-manifolds, the equivalence of categories recovers the fundamental group from the fundamental risandle and vice versa.