The Surgery Unknotting Number of Legendrian Links

TL;DR

This paper introduces the concept of the surgery unknotting number for Legendrian links, providing bounds and explicit calculations for various classes, linking it to classical invariants and the smooth 4-ball genus.

Contribution

It defines the surgery unknotting number for Legendrian links and computes it for several classes, establishing a formula relating it to classical invariants and the smooth 4-ball genus.

Findings

Calculated for twist knots, torus links, and positive Legendrian rational links.

Established lower bounds based on classical invariants.

Found that for non-slice links, the surgery unknotting number equals (j-1) plus twice the smooth 4-ball genus.

Abstract

The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting number is calculated for every Legendrian link that is topologically a twist knot or a torus link and for every positive, Legendrian rational link. In addition, the surgery unknotting number is calculated for every Legendrian knot in the Legendrian knot atlas of Chongchitmate and Ng whose underlying smooth knot has crossing number 7 or less. In all these calculations, as long as the Legendrian link of $j$ components is not topologically a slice knot, its surgery unknotting number is equal to the sum of $(j-1)$ and twice the smooth 4-ball genus of the underlying smooth link.