A standard form of incompressible surfaces in 3-dimensional handlebodies

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.

19 pages, 13 figures. Accepted for publication in SCIENCE CHINA Mathematics