Positive braid knots of maximal topological 4-genus

TL;DR

This paper establishes a precise condition linking the maximal topological 4-genus of positive braid knots to their signature invariant, and applies this to classify and compute the 4-genus for knots with up to 12 crossings.

Contribution

It provides a characterization of positive braid knots with maximal topological 4-genus via their signature invariant and enumerates these knots up to 12 crossings.

Findings

Positive braid knots have maximal topological 4-genus if and only if their signature is maximal.

All positive braid knots with up to 12 crossings with maximal 4-genus are classified.

The topological 4-genus is computed explicitly for all positive braid knots up to 12 crossings.

Abstract

We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.