A characteristics for a surface sum of two handlebodies along an annulus or a once-punctured torus to be a handlebody

TL;DR

This paper characterizes when the sum of two handlebodies along an annulus or a once-punctured torus results in a handlebody, based on core curves and primitive collections of curves.

Contribution

It provides necessary and sufficient conditions for such surface sums of handlebodies to be handlebodies, extending understanding of handlebody decompositions.

Findings

Annulus sum is a handlebody iff core curve is a longitude in one handlebody.

Sum along a once-punctured torus is a handlebody if certain primitive curve conditions are met.

Conditions involve primitive collections of simple closed curves on the torus.

Abstract

The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum $H=H_1\cup_A H_2$ of two handlebodies $H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a longitude for either $H_1$ or $H_2$. 2. Let $H=H_1\cup_T H_2$ be a surface sum of two handlebodies $H_1$ and $H_2$ along a once-punctured torus $T$. Suppose that $T$ is incompressible in both $H_1$ and $H_2$. Then $H$ is a handlebody if and only if the there exists a collection $\{\delta, \sigma\}$ of simple closed curves on $T$ such that either $\{\delta, \sigma\}$ is primitive in $H_1$ or $H_2$, or $\{\delta\}$ is primitive in $H_1$ and $\{\sigma\}$ is primitive in $H_2$.