Rectangle condition for compression body and 2-fold branched covering

TL;DR

This paper introduces a rectangle condition for analyzing the irreducibility of Heegaard splittings in 3-manifolds with boundary, with applications to branched covers and knot theory.

Contribution

It provides a new rectangle condition for strong irreducibility and applies it to branched covers and knot width additivity, advancing understanding of 3-manifold topology.

Findings

Any thin meridional level surface in the link complement is incompressible.

The additivity of knot width holds for composite knots satisfying the condition.

The rectangle condition ensures strong irreducibility of certain Heegaard splittings.

Abstract

We give the rectangle condition for strong irreducibility of Heegaard splittings of $3$-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a $2$-fold covering of $S^3$ branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of knot width holds for a composite knot satisfying the condition.