Distinguishing Power of 4-Legendrian Permutation Racks

TL;DR

This paper investigates the capabilities of 4-Legendrian permutation racks in differentiating Legendrian knots, showing their limitations and their ability to recover classical invariants.

Contribution

It demonstrates that 4-Legendrian permutation racks cannot distinguish knots with identical classical invariants but can recover these invariants.

Findings

Permutation racks with 4-Legendrian structures cannot distinguish certain Legendrian knots.

They can recover classical invariants like knot type, Thurston-Bennequin number, and rotation number.

The study clarifies the limitations and capabilities of these racks in knot theory.

Abstract

We study 4-Legendrian racks and their effectiveness at distinguishing Legendrian knots. We prove that permutation racks with 4-Legendrian rack structures cannot distinguish Legendrian knots that share the same knot type, Thurston-Bennequin number, and rotation number. However, they also recover these three classical invariants.