GL-racks and coloring invariants of Legendrian knots

TL;DR

This paper studies the algebraic structure of GL-racks and their application to Legendrian knot invariants, showing how finite GL-racks decompose and relate to knot colorings.

Contribution

It introduces the decomposition of finite GL-racks into permutation and block types and links these to Legendrian knot invariants.

Findings

Finite GL-racks decompose into permutation and block GL-racks.

Legendrian knots with identical classical invariants have equivalent coloring invariants.

The algebraic structure of GL-racks informs knot invariant analysis.

Abstract

In this paper, we explore the algebraic structure of GL-racks, and demonstrate that finite GL-racks decompose canonically into permutation GL-racks and block GL-racks. As a corollary, we verify that two Legendrian knots with the same classical invariants share equivalent coloring invariants with respect to any given finite GL-rack.