Heegaard genus, degree-one maps, and amalgamation of 3-manifolds

TL;DR

This paper investigates how the Heegaard genus behaves under certain degree-one maps between 3-manifolds formed by amalgamation, establishing inequalities that relate the genus of the original and modified manifolds.

Contribution

It proves that the Heegaard genus does not decrease under a specific degree-one map involving amalgamation along a torus, providing new insights into 3-manifold topology.

Findings

Heegaard genus of M is at least that of N

Tunnel number of satellite knots is at least that of pattern knots

Degree-one maps preserve or increase Heegaard genus

Abstract

Let $M=W\cup_T V$ be an amalgamation of two compact 3-manifolds along a torus, where $W$ is the exterior of a knot in a homology sphere. Let $N$ be the manifold obtained by replacing $W$ with a solid torus such that the boundary of a Seifert surface in $W$ is a meridian of the solid torus. This means that there is a degree-one map $f\colon M\to N$, pinching $W$ into a solid torus while fixing $V$. We prove that $g(M)\ge g(N)$, where $g(M)$ denotes the Heegaard genus. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.