Fundamental generalized Legendrian rack and classical invariants

TL;DR

This paper establishes a relationship between the fundamental generalized Legendrian rack (GL-rack) and classical Legendrian knot invariants, showing that isomorphic GL-racks imply specific relationships between Thurston-Bennequin and rotation numbers.

Contribution

It introduces the fundamental GL-rack as a new invariant and links it to classical invariants, revealing a symmetry property in Legendrian knot theory.

Findings

Isomorphic fundamental GL-racks imply equal or opposite Thurston-Bennequin numbers.

Isomorphic fundamental GL-racks imply equal or opposite rotation numbers.

The result provides a new perspective on Legendrian knot classification.

Abstract

In this paper, we prove that if two Legendrian knots have isomorphic fundamental GL-racks, then either they have the same Thurston-Bennequin number and the same rotation number, or they have the opposite Thurston-Bennequin numbers and opposite rotation numbers.