Hard Legendrian unknots

TL;DR

This paper explores the complexity of Legendrian unknot diagrams, using normal rulings to identify and bound the minimal writhe of hard unknot front projections, with extensive computational analysis.

Contribution

It introduces methods to obstruct and analyze the Reidemeister hardness of Legendrian unknot diagrams, including infinite families and bounds on writhe, supported by computational data.

Findings

Identified infinite families of hard Legendrian unknot diagrams.

Bound the minimum possible writhe of these diagrams.

Analyzed 2.6 million diagrams, including 1.7 million identified as hard.

Abstract

We initiate the study of Reidemeister hardness of Legendrian unknot front projections. Using normal rulings, we obstruct several infinite families of hard unknot diagrams from being drawn with max-tb unknot fronts, along with 1.7 million of the 2.6 million hard unknot diagrams studied in \cite{applebaum2024unknottingnumberhardunknot}. We construct infinitely many smoothly hard max-tb unknot diagrams, and bound their minimum possible writhe. With respect to these bounds, our constructions are conjecturally sharp.