Topology Seminar

In his final paper in 1954, Alan Turing wrote `No systematic method is yet known by which one can tell whether two knots are the same.' Within the next 20 years, Wolfgang Haken and Geoffrey Hemion had discovered such a method. However, the computational complexity of this problem remains unknown. In my talk, I will give a survey on this area, that draws on the work of many low-dimensional topologists and geometers. Unfortunately, the current upper bounds on the computational complexity of the knot equivalence problem remain quite poor. However, there are some recent results indicating that, perhaps, knots are more tractable than they first seem. Specifically, I will explain a theorem that provides, for each knot type K, a polynomial p_K with the property that any two diagrams of K with n_1 and n_2 crossings differ by at most p_K(n_1) + p_K(n_2) Reidemeister moves.

In one of the foundational articles of PL topology in 1930, James Alexander laid the groundwork for a field that would shape topology for decades to come. One question from his original manuscript famously open: After he proved that every PL homeomorphism could be factorized into certain elementary moves, called stellar and inverse stellar moves. He asked whether these moves could be reordered, so that stellar moves preceded their inverses.

We prove that this is correct. Moreover, we prove a related conjecture in birational geometry due to Oda: Two birational toric varieties have a common blowup.