Editing Computational topology (section)

==Major algorithms by subject area==

===Algorithmic 3-manifold theory===

A large family of algorithms concerning [[3-manifold]]s 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 [[Triangulation (topology)|triangulated]] [[3-manifold]] and determines whether or not the manifold is [[homeomorphism|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.  Saul Schleimer went on to show the problem lies in the complexity class [[NP (complexity)|NP]].<ref>{{cite web |first=Saul |last=Schleimer |title=Sphere Recognition Lies in NP |url=http://www.warwick.ac.uk/~masgar/Maths/np.pdf |date=2011 |via=[[University of Warwick]]}}</ref>   Furthermore, Raphael Zentner showed that the problem lies in the complexity class coNP,<ref>{{cite journal |arxiv=1605.08530 |title=Integer homology 3-spheres admit irreducible representations in SL(2,C) |year=2018 |last1=Zentner |first1=Raphael |s2cid=119275434 |journal=[[Duke Mathematical Journal]] |volume=167 |issue=9 |pages=1643–1712 |doi=10.1215/00127094-2018-0004 }}</ref> provided that the generalized Riemann hypothesis holds. He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg <ref>{{cite journal |arxiv=1112.0845 |last1=Kuperberg |first1=Greg |s2cid=12634367 |title=Knottedness is in NP, modulo GRH |journal=[[Advances in Mathematics]] |year=2014 |volume=256 |pages=493–506 |doi=10.1016/j.aim.2014.01.007 |doi-access=free }}</ref> on the complexity of knottedness detection.
* The [[connected sum|connect-sum]] decomposition of 3-manifolds is also implemented in [[Regina (program)|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<ref>{{cite journal |arxiv=0909.4625 |last1=Burton |first1=Benjamin A. |last2=Hyam Rubinstein |first2=J. |last3=Tillmann |first3=Stephan |title=The Weber-Seifert dodecahedral space is non-Haken |year=2009 |journal=[[Transactions of the American Mathematical Society]] |volume=364 |number=2 |pages=911–932 |doi=10.1090/S0002-9947-2011-05419-X|s2cid=18435885 }}</ref> 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 for groups|word problem]].<ref>J.Manning, Algorithmic detection and description of hyperbolic structures on 3-manifolds with solvable word problem, Geometry and Topology 6 (2002) 1–26</ref>

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,<ref>S.Matveev, Algorithmic topology and the classification of 3-manifolds, Springer-Verlag 2003</ref> in fact, it is known that deciding whether two closed, oriented 3-manifolds given by [[Triangulation (topology)|triangulation]]s (simplicial complexes) are equivalent (homeomorphic) is [[elementary recursive]].<ref>{{cite journal | arxiv = 1508.06720| doi = 10.2140/pjm.2019.301.189| title = Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization| date = 2019| last1 = Kuperberg| first1 = Greg| journal = Pacific Journal of Mathematics| volume = 301| pages = 189–241| s2cid = 119298413}}</ref> This generalizes the result on 3-sphere recognition.

==== Conversion algorithms ====

* [[SnapPea]] implements an algorithm to convert a planar [[Knot (mathematics)|knot]] or [[Knot (mathematics)#knot diagram|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 theory|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.<ref>{{cite journal |doi=10.1112/jtopol/jtn017 |title=3-manifolds efficiently bound 4-manifolds |year=2008 |last1=Costantino |first1=Francesco |last2=Thurston |first2=Dylan |s2cid=15119190 |journal=[[Journal of Topology]] |volume=1 |issue=3 |pages=703–745 |arxiv=math/0506577 }}</ref>
* 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===

[[Unknotting problem|Determining whether or not a knot is trivial]] is known to be in the complexity classes [[NP (complexity)|NP]]<ref name=HLP>{{citation | last1 = Hass | first1 = Joel | author1-link = Joel Hass
 | last2 = Lagarias | first2 = Jeffrey C. | author2-link = Jeffrey Lagarias | last3 = Pippenger | first3 = Nicholas | s2cid = 125854 | author3-link = Nick Pippenger | doi = 10.1145/301970.301971 | issue = 2
 | journal = [[Journal of the ACM]] | pages = 185–211 | title = The computational complexity of knot and link problems | volume = 46 | year = 1999 | arxiv = math/9807016}}</ref> as well as [[co-NP]].<ref>{{citation
 | last = Lackenby | first = Marc | author-link = Marc Lackenby | arxiv = 1604.00290 | title = The efficient certification of Knottedness and Thurston norm | journal = [[Advances in Mathematics]] | volume = 387 | date = 2021 | pages = 107796 | doi = 10.1016/j.aim.2021.107796| s2cid = 119307517 }}</ref> The problem of determining the [[knot genus|genus of a knot in a 3-manifold]] is [[NP-complete]];<ref name=AHT>{{citation | last1=Agol | first1 = Ian | last2 = Hass | first2 = Joel | author2-link = Joel Hass | last3 = Thurston | first3 = William | doi = 10.1090/S0002-9947-05-03919-X | volume = 358 | journal = Trans. Amer. Math. Soc. | pages = 3821–3850 | title = The computational complexity of knot genus and spanning area | number = 9 | year = 2006 | arxiv = math/0205057}}</ref> however, while [[NP_(complexity) | NP]] remains an upper bound on the complexity of determining the genus of a knot in R<sup>3</sup> or S<sup>3</sup>, as of 2006 it was unknown whether the algorithmic problem of determining the genus of a knot in those particular 3-manifolds was still [[NP-hard]].<ref name=AHT />

===Computational homotopy===
* Computational methods for [[Homotopy groups of spheres#Computational methods|homotopy groups of spheres]].
* Computational methods for solving [[Systems of polynomial equations#Homotopy continuation method|systems of polynomial equations]].
* Brown has an algorithm to compute the homotopy groups of spaces that are finite [[Postnikov system|Postnikov complexes]],<ref>{{citation
 | last1 = Brown | first1 = Edgar H. | authorlink1 = Edgar H. Brown
 | journal = Annals of Mathematics (2)
 | pages = 1–20
 | title = Finite Computability of Postnikov Complexes
 | volume = 65
 | year = 1957
 | issue = 1 |doi=10.2307/1969664| jstor = 1969664 }}</ref> although it is not widely considered implementable.

===Computational homology===

Computation of [[homology group]]s of [[CW complex|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 [http://www.linalg.org LinBox] library.
* Simple homotopic reductions for pre-processing homology computations, as in the [http://www.sas.upenn.edu/~vnanda/perseus/index.html Perseus] software package.
* Algorithms to compute [[persistent homology]] of [[Filtration (mathematics)|filtered]] complexes, as in the [https://CRAN.R-project.org/package=TDAstats TDAstats] R package.<ref>{{Cite journal|title = TDAstats: R pipeline for computing persistent homology in topological data analysis|journal = Journal of Open Source Software|date = 2018|pages=860|volume = 3|issue = 28| pmid=33381678|  doi = 10.21105/joss.00860|first1 = Raoul|last1 = Wadhwa|first2 = Drew|last2 = Williamson|first3 = Andrew|last3 = Dhawan|first4 = Jacob|last4 = Scott| pmc=7771879 |bibcode = 2018JOSS....3..860R|doi-access = free}}</ref>
*In some applications, such as in TDA, it is useful to have representatives of (co)homology classes that are as "small" as possible. This is known as the problem of (co)homology localization. On triangulated manifolds, given a chain representing a homology class, it is in general NP-hard to approximate the minimum-support homologous chain.<ref>{{Cite journal|title = Hardness results for homology localization|journal = Discrete & Computational Geometry|date = 2011|pages=425–448|volume = 45|issue = 3| mr=2770545|  doi = 10.1007/s00454-010-9322-8|first1 = Chao|last1 = Chen|first2 = Daniel|last2 = Freedman}} Preliminary version appeared at SODA 2010.</ref> However, the particular setting of approximating 1-cohomology localization on triangulated 2-manifolds is one of only three known problems whose hardness is equivalent to the [[Unique Games Conjecture]].<ref>{{cite conference
 | last1 = Grochow | first1 = Joshua
 | last2 = Tucker-Foltz | first2 = Jamie
 | doi = 10.4230/LIPIcs.SoCG.2018.43
 | conference = 34th Internat. Symp. Comput. Geom. (SoCG) '18
 | mr = 3824287
 | page = 43:1-43:16
 | title = Computational Topology and the Unique Games Conjecture
 | year = 2018 
 | doi-access = free
 | eprint = 1803.06800}}.
</ref>