Minor theory for quasipositive surfaces

TL;DR

This paper investigates the order structure of quasipositive surfaces, showing it is almost a well-quasi-order and becomes one when restricted to surfaces with a fixed root of a full twist, advancing understanding in knot theory.

Contribution

It introduces a new order-theoretic framework for quasipositive surfaces and proves it is almost a well-quasi-order, becoming one under specific restrictions.

Findings

The set of quasipositive surfaces is closed under incompressible inclusion.

The induced order on fiber surfaces of positive braid links is almost a well-quasi-order.

Restricting to surfaces with a fixed root of a full twist yields an actual well-quasi-order.

Abstract

The set of quasipositive surfaces is closed under incompressible inclusion. We prove that the induced order on fibre surfaces of positive braid links is almost a well-quasi-order. When restricting to quasipositive surfaces containing a fixed root of a full twist, we get an actual well-quasi-order.