An infinite family of Legendrian torus knots distinguished by cube number

TL;DR

This paper introduces the Legendrian cube number as a new invariant that distinguishes certain Legendrian torus knots based on their Thurston-Bennequin and rotation numbers, expanding understanding of knot invariants.

Contribution

It defines the Legendrian cube number and demonstrates its ability to differentiate Legendrian torus knots with specific Thurston-Bennequin and rotation number configurations.

Findings

Legendrian cube number distinguishes knots with different rotation numbers.

The invariant applies to Legendrian left hand torus knots.

It provides a new tool for classifying Legendrian knots.

Abstract

For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number.