 
Summary: RESEARCH BLOG 7/12/04
Last week, Saul Schleimer posted a paper showing that recognition
of the 3sphere lies in NP. This generalizes work of Hass, Lagarias, and
Pippenger [3], who showed that recognition of the unknot lies in NP,
using Haken's algorithm. In both cases, the certificate can be chosen
to be a collection of coordinates for normal/almost normal surfaces.
Most likely, one can show that one need only specify the quadrilateral
types for the normal surfaces. In any case, once these coordinates are
given, the algorithm runs in time bounded by a polynomial function of
the number of tetrahedra. I've worked out an argument to show that
the unknot recognition problem is coNP (which means that there is a
certificate that a knot is nontrivial which can be checked in polynomial
time by an algorithm). It would be interesting to know if the sphere
recognition problem is in coNP as well, but this is likely much more dif
ficult (even using geometrization). The certificate in the case of Saul's
algorithm is coordinates for a collection of normal and almost normal
2spheres, given by the RubinsteinThompson algorithm [4]. One then
has to check that the complementary regions are balls and punctured
spheres, which Saul does by normalizing the almost normal 2spheres,
while keeping careful track of what happens during the normalization.
