(Disk, Essential surface) pairs of Heegaard splittings that intersect in one point

TL;DR

This paper explores special pairs of Heegaard splittings in 3-manifolds where an essential disk and surface intersect at exactly one point, leading to new constructions of manifolds with multiple, distinct genus splittings.

Contribution

It introduces a method to generate new Heegaard splittings from existing ones using essential disks and surfaces intersecting at one point, demonstrating examples with distinct genera.

Findings

Constructs new Heegaard splittings by cutting and attaching along essential surfaces.

Provides examples of 3-manifolds with multiple Heegaard splittings of different genera.

Shows existence of strongly irreducible non-minimal genus splittings derived from other splittings.

Abstract

We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty "minus" boundary.) Let F be a genus g surface with n boundary components. From S, we obtain a genus g(S)+2g+n-2 Heegaard splitting M=H'_1 \cup_S' H'_2 by cutting H_2 along F and attaching F \times [0,1] to H_1. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds having two Heegaard splittings of distinct genera where one of the two Heegaard splittings is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction.