Computational topology

Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory. A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving problems that arise naturally in fields such as computational geometry, graphics, robotics, structural biology and chemistry, using methods from computable topology. A large family of algorithms concerning 3-manifolds revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3-manifold theory into integer linear programming problems. Rubinstein and Thompson's 3-sphere recognition algorithm. This is an algorithm that takes as input a triangulated 3-manifold and determines whether or not the manifold is homeomorphic to the 3-sphere. It has exponential run-time in the number of tetrahedral simplexes in the initial 3-manifold, and also an exponential memory profile. Moreover, it is implemented in the software package Regina. Saul Schleimer went on to show the problem lies in the complexity class NP. Furthermore, Raphael Zentner showed that the problem lies in the complexity class coNP, provided that the generalized Riemann hypothesis holds. He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg on the complexity of knottedness detection. The connect-sum decomposition of 3-manifolds is also implemented in Regina, has exponential run-time and is based on a similar algorithm to the 3-sphere recognition algorithm. Determining that the Seifert-Weber 3-manifold contains no incompressible surface has been algorithmically implemented by Burton, Rubinstein and Tillmann and based on normal surface theory. The Manning algorithm is an algorithm to find hyperbolic structures on 3-manifolds whose fundamental group have a solution to the word problem. At present the JSJ decomposition has not been implemented algorithmically in computer software.