arXiv subjects
Publications and source records attributed to Peyton Phinehas Wood.
We study 4-Legendrian racks and their effectiveness at distinguishing Legendrian knots. We prove that permutation racks with 4-Legendrian rack structures cannot distinguish Legendrian knots that share the same knot type, Thurston-Bennequin number, and rotation number. However, they also recover these three classical invariants.
In this paper we introduce the notion of a spherical knot mosaic where a knot is represented by tiling the surface of a topological 2-sphere with 11 canonical knot mosaic tiles and show this gives rise to several novel knot (and link) invariants: the spherical mosaic number, spherical tiling number, minimal spherical mosaic tiling number, spherical face number, spherical n-mosaic face number, and minimal spherical mosaic face number. We show examples where this framework offers an improvement over classical knot mosaics. Furthermore, we explore several bounds involving classical knot invariants derived from these spherical mosaic invariants.
We prove that the displacement group of the dihedral quandle with n elements is isomorphic to the group generated by rotations of the n/2-gon when n is even and the n-gon when n is odd. We additionally show that any quandle with at least one trivial column has equivalent displacement and inner automorphism groups. Then, using a known enumeration of quandles which we confirm up to order 10, we verify the automorphism group and the inner automorphism group of all quandles (up to isomorphism) of orders less than or equal to 7, compute these for all 115,431 quandles orders 8, 9, and 10, and extend these results by computing the displacement group of all 115,837 quandles (up to isomorphism) of order less than or equal to 10.