On Rack Invariants Of Legendrian Knots

Abstract

In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invariant, but detect some of the geometric properties such as cusps in a Legendrian knot. In the case of topologically trivial Legendrian knots, we test this family of invariants for its strengths and limitations. We further prove that these invariants form a natural generalization of the quandle invariant, by which we mean that any rack invariant under certain restrictions is equivalent to a Legendrian rack. The axioms of these racks are expressible in first order logic, and were discovered through a series of experiments using an automated theorem prover for first order logic. We also present the results from the experiments on Legendrian unknots involving auto-mated theorem provers, and describe how they led to our current formulation.