On the genus defect of positive braid knots

TL;DR

This paper investigates the relationship between the Seifert genus and the topological 4-genus of prime positive braid knots, establishing bounds and characterizing knots with small genus differences via surface minors.

Contribution

It introduces a lower bound on the genus difference based on braid strand number and characterizes knots with small genus difference using finitely many forbidden minors.

Findings

The genus difference is bounded below by an affine function of the minimal braid strand number.

Prime positive braid knots with small genus difference are characterized by finitely many forbidden surface minors.

The property of having a small genus difference is linked to specific surface minor obstructions.

Abstract

We show that the difference between the Seifert genus and the topological 4-genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot. We deduce that among prime positive braid knots, the property of having such a genus difference less than any fixed constant is characterised by finitely many forbidden surface minors.