Comparing Bennequin-type inequalities

TL;DR

This paper compares various Bennequin-type inequalities, demonstrating that while some bounds are sharp, the difference between self-linking number and four-ball genus can grow arbitrarily large, highlighting limitations of certain inequalities.

Contribution

It provides examples showing the disparity between self-linking number and four-ball genus, and compares the sharpness of different Bennequin-type inequalities.

Findings

Examples where the difference between self-linking number and four-ball genus is arbitrarily large.

The $s$-Bennequin and $ au$-Bennequin inequalities are sharp in the provided examples.

The slice-Bennequin inequality can be far from sharp in certain cases.

Abstract

The slice-Bennequin inequality states an upper bound for the self-linking number of a knot in terms of its four-ball genus. The $s$-Bennequin and $\tau$-Bennequin inequalities provide upper bounds on the self-linking number of a knot in terms of the Rasmussen $s$ invariant and the Ozsv\'ath-Szab\'o $\tau$ invariant. We exhibit examples in which the difference between self-linking number and four-ball genus grows arbitrarily large, whereas the $s$-Bennequin inequality and the $\tau$-Bennequin inequality are both sharp.