Pachner moves, generic complexity, and randomising 3-manifold triangulations

We study the computational complexity of decision problems on triangulated 3-manifolds. In this setting there has been encouraging initial progress in recent years, but many important questions remain wide open. The “simple” problem of 3-sphere recognition and the related problem of unknot recognition are both known to be in NP, by work of Schleimer [18] and earlier work of Hass, Lagarias and Pippenger [9] respectively. In recent announcements by Kuperberg [13] and Hass and Kuperberg [8], these problems are also in co-NP if the generalised Riemann hypothesis holds. It remains a major open question as to whether either problem can be solved in polynomial time. There are very few hardness results for such problems. A notable example due to Agol, Hass and Thurston is knot genus: if we generalise unknot recognition to computing knot genus, and we generalise the ambient space from S to an arbitrary 3-manifold, then the problem becomes NP-complete [1]. The key construction in their result can also be adapted for problems relating to least-area surfaces [1, 7].