Positive braids of maximal signature

TL;DR

This paper characterizes positive braid links with positive Seifert form using forbidden minors, establishing a correspondence with simply laced Dynkin diagrams and classifying alternating positive braid knots.

Contribution

It introduces a finite forbidden minor characterization and links positive braid links with Dynkin diagrams, providing a new classification framework.

Findings

Positive braid links with positive Seifert form are characterized by forbidden minors.

A one-to-one correspondence exists between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams.

A simple classification of alternating positive braid knots is provided.

Abstract

We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.