Editing Computational topology (section)

===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 />