Racks, quandles and virtual knots

Abstract

We begin with a brief survey of the theory of virtual knots, which was announced in 1996 by Kauffman. This leads naturally to the subject of quandles and quandle homology, which we also briefly introduce. Chapter 2 contains a proof in terms of Gauss diagrams that the forbidden moves unknot virtual knots. This chapter includes material which has appeared in the Journal of Knot Theory and its Ramifications and is reprinted here by permission of World Scientific Publishing. In chapter 3 (cowritten with my advisor R.A.Litherland) we confirm a conjecture of J.S.Carter et.al. that the long exact sequence in rack homology is split. We go on to show that for racks with homogeneous orbits, the lower bounds for the Betti numbers are exact. We end chapter three with some explicit isomorphisms between Alexander quandles of certain forms and we describe some calculations of the second and third homology groups for a selection of quandles. Chapter 4 contains a classification result for the category of finite Alexander quandles. This result give us easy conditions for comparing finite Alexander quandles as well as a general procedure for listing all Alexander quandles with a given number of elements. As an application we list the number of distinct Alexander quandles (and how many of these are connected) with up to 15 elements.