Normal Surfaces in 3-Manifolds: Algorithms, Experiments and Questions
Stephan Tillmann Sydney University
The theory of normal surfaces, introduced by Kneser in the 1920s and further developed by Haken in the 1960s plays a crucial role in 3-manifold topology. Normal surfaces allow topological problems to be translated into algebraic problems or linear programs, and they are the key to many important advances over the last 50 years, including the solution of the unknot recognition problem by Haken, the 3-sphere recognition problem by Rubinstein and Thompson and the homeomorphism problem by Haken, Hemion and Matveev. In this talk, I will summarise Haken 's blueprint for algorithmic 3-manifold topology, discuss the "difficulty" of the computational problems from a theoretical and experimental perspective and state some open questions and challenges.