Legendrian knots and multi-crossings

TL;DR

This paper extends the concept of "ubercrossing projections to Legendrian knots, showing their isotopy properties and providing methods to compute classical invariants like tb and rotation numbers in different projections.

Contribution

It introduces the extension of "ubercrossing and petal projections to Legendrian knots and develops techniques to compute key invariants in these projections.

Findings

Legendrian knots with "ubercrossing projections are isotopic to the unknot.

Methods to compute tb and rotation numbers for petal projections.

Any Legendrian knot can be represented with an "ubercrossing projection in the front projection.

Abstract

It was shown in arXiv:1208.5742 that any smooth knot can be represented by an \"ubercrossing projection, i.e. a knot projection with no crossings aside from a single multi-crossing. We extend this idea to Legendrian knots and investigate \"ubercrossing and petal projections in the front and Lagrangian projections. We show that any Legendrian knot with an \"ubercrossing projection in the front projection is smoothly isotopic to the unknot and we demonstrate how to compute the $tb$ and rotation numbers for petal projections in the Lagrangian projection.