Generalised Legendrian racks of Legendrian links

TL;DR

This paper introduces generalized Legendrian racks (GL-racks), algebraic structures that serve as invariants for Legendrian links, capable of distinguishing many Legendrian knots and providing a new algebraic perspective.

Contribution

It defines GL-racks as Legendrian link invariants, proves their invariance under Legendrian isotopy, and explores their algebraic properties and module categories.

Findings

GL-racks distinguish infinitely many Legendrian unknots and trefoils.

Every GL-rack admits a homogeneous representation.

Categories of GL-rack modules and Beck modules are equivalent.

Abstract

A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. Employing a purely rack theoretic approach, we associate a generalised Legendrian rack (or a GL-rack) to an oriented Legendrian link, and prove that it is an invariant under Legendrian isotopy. As immediate applications, we prove that this invariant distinguishes infinitely many oriented Legendrian unknots and oriented Legendrian trefoils. To comprehend their algebraic structure, we prove that every GL-rack admits a homogeneous representation. Further, using the idea of trunks, we define modules over GL-racks, and prove the equivalence of the category of GL-rack modules and the category of Beck modules over a fixed GL-rack.