Rulings of Legendrian knots as spanning surfaces

TL;DR

This paper explores the relationship between rulings of Legendrian links and their associated surfaces, establishing bounds on the genus of these surfaces in relation to the knot's genus, especially for 2-graded rulings of certain knots.

Contribution

It proves that for 2-graded rulings of homogeneous knots, the surface genus does not exceed the knot's genus, and shows the canonical genus bounds the genus of 2-graded rulings in general.

Findings

Genus of surfaces from rulings is at most the knot genus for certain knots.

2-graded rulings correspond to orientable surfaces.

Canonical genus bounds the genus of 2-graded rulings.

Abstract

Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2-graded if and only if the surface is orientable. For 2-graded rulings of homogeneous (in particular, alternating) knots, we prove that the genus of this surface is at most the genus of the knot. While this is not true in general, we do prove that the canonical genus (a.k.a. diagram genus) of any knot is an upper bound for the genera of its 2-graded rulings.