Graduate Student Seminar
Fri, 7 October, 2016
7:00pm
Title: Torsion in rack and quandle homology and its applications to Knot Theory
Speaker: Seung Yeop Yang
Abstract: Rack homology theory was introduced between 1990 and 1995 by Fenn, Rourke, and Sanderson, and in 1999, Carter, Jelsovsky, Kamada, Langford, and Saito modified it to quandle homology theory in order to obtain knot invariants for classical knots and knotted surfaces in a state-sum form called cocycle knot invariants. In 1993, Fenn, Rourke, and Sanderson introduced rack spaces to define rack homotopy invariants and a modification to quandle spaces and quandle homotopy invariants of classical links was introduced by Nosaka in 2011.
In analogy to the well-known result in reduced group homology of finite groups that the order of a group annihilates its homology, we prove that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. It was an open conjecture for over 5 years. We also introduce an $m$-almost quasigroup quandle as a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups. Moreover, as a generalization of rack and quandle spaces, we define the Cayley-type graph and CW complex of a distributive structure and study their properties. Moreover, for a connected quandle we introduce the shadow homotopy invariant of a classical link.
Note: The Graduate Student Seminar is mandatory f