Commit 2020-09-29 07:37 22d034c1

feat(algebra/quandle): racks and quandles (#4247) This adds the algebraic structures of racks and quandles, defines a few examples, and provides the universal enveloping group of a rack. A rack is a set that acts on itself bijectively, and sort of the point is that the action act : α → (α ≃ α) satisfies

act (x ◃ y) = act x * act y * (act x)⁻¹

where x ◃ y is the usual rack/quandle notation for act x y. (Note: racks do not use has_scalar because it's convenient having x ◃⁻¹ y for the inverse action of x on y. Plus, associative racks have a trivial action.) In knot theory, the universal enveloping group of the fundamental quandle is isomorphic to the fundamental group of the knot complement. For oriented knots up to orientation-reversed mirror image, the fundamental quandle is a complete invariant, unlike the fundamental group, which fails to distinguish non-prime knots with chiral summands.