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, social science, structural biology, and chemistry, using methods from computable topology.^[1][2][3]

Major algorithms by subject area

Algorithmic 3-manifold theory

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.^[4] Saul Schleimer went on to show the problem lies in the complexity class NP.^[5] Furthermore, Raphael Zentner showed that the problem lies in the complexity class coNP,^[6] provided that the generalized Riemann hypothesis holds. He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg ^[7] 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^[8] 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.^[9]

At present the JSJ decomposition has not been implemented algorithmically in computer software. Neither has the compression-body decomposition. There are some very popular and successful heuristics, such as SnapPea which has much success computing approximate hyperbolic structures on triangulated 3-manifolds. It is known that the full classification of 3-manifolds can be done algorithmically,^[10] in fact, it is known that deciding whether two closed, oriented 3-manifolds given by triangulations (simplicial complexes) are equivalent (homeomorphic) is elementary recursive.^[11] This generalizes the result on 3-sphere recognition.

Conversion algorithms

• SnapPea implements an algorithm to convert a planar knot or link diagram into a cusped triangulation. This algorithm has a roughly linear run-time in the number of crossings in the diagram, and low memory profile. The algorithm is similar to the Wirthinger algorithm for constructing presentations of the fundamental group of link complements given by planar diagrams. Similarly, SnapPea can convert surgery presentations of 3-manifolds into triangulations of the presented 3-manifold.
• D. Thurston and F. Costantino have a procedure to construct a triangulated 4-manifold from a triangulated 3-manifold. Similarly, it can be used to construct surgery presentations of triangulated 3-manifolds, although the procedure is not explicitly written as an algorithm in principle it should have polynomial run-time in the number of tetrahedra of the given 3-manifold triangulation.^[12]
• S. Schleimer has an algorithm which produces a triangulated 3-manifold, given input a word (in Dehn twist generators) for the mapping class group of a surface. The 3-manifold is the one that uses the word as the attaching map for a Heegaard splitting of the 3-manifold. The algorithm is based on the concept of a layered triangulation.

Algorithmic knot theory

• Determining whether or not a knot is trivial is known to be in the complexity classes NP^[13] as well as co-NP.^[14]
• The problem of determining the genus of a knot is known to have complexity class PSPACE.^[13]
• There are polynomial-time algorithms for the computation of the Alexander polynomial of a knot.^[15]
• The Jones polynomial, the HOMFLY polynomial and Reshetikhin–Turaev invariants are fixed parameter tracktable, while the Jones polynomial is also known to be #P-hard.^[16] Computing the first coefficient of the HOMFLYPT and Kauffman polynomial is in P.^[17]
• Additive approximation of the Jones polynomial is BQP-complete, i.e., equivalently hard as polynomial time quantum algorithms.^[18]
• Given two (tame) knots by link diagrams deciding whether they are equivalent is ER. This is established by reducing knot equivalence (isotopy) to equivalence (homeomorphy) of the associated knot complements which are 3-manifolds which in turn can be encoded by triangulations. Since these knot complements are Haken manifolds one then uses a result that the equivalence problem of these manifolds is in ER. There does not seem to be a good reference for this, these results are scattered across the literature in the field, but well-known by the community.

Computational homotopy

• Computational methods for homotopy groups of spheres.
• Computational methods for solving systems of polynomial equations.
• Brown has an algorithm to compute the homotopy groups of spaces that are finite Postnikov complexes,^[19] although it is not widely considered implementable.

Computational homology

Computation of homology groups of cell complexes reduces to bringing the boundary matrices into Smith normal form. Although this is a completely solved problem algorithmically, there are various technical obstacles to efficient computation for large complexes. There are two central obstacles. Firstly, the basic Smith form algorithm has cubic complexity in the size of the matrix involved since it uses row and column operations which makes it unsuitable for large cell complexes. Secondly, the intermediate matrices which result from the application of the Smith form algorithm get filled-in even if one starts and ends with sparse matrices.

• Efficient and probabilistic Smith normal form algorithms, as found in the LinBox library.
• Simple homotopic reductions for pre-processing homology computations, as in the Perseus software package.
• Algorithms to compute persistent homology of filtered complexes, as in the TDAstats R package.^[20]