A standard form of incompressible surfaces in 3-dimensional handlebodies

TL;DR

This paper introduces a practical algorithm, based on Morse theory, to standardize and determine the incompressibility of surfaces in 3D handlebodies, including all properly embedded incompressible surfaces.

Contribution

It provides a new standard form for these surfaces and a complete criterion for their incompressibility, along with an effective algorithm and illustrative examples.

Findings

Algorithm effectively tests incompressibility

Standard form simplifies analysis of surfaces

Practical method demonstrated with examples

Abstract

Applying Morse theory, we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies. We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form. Our algorithm is practical. Several examples are given to test the algorithm.