Recognizing trivial links in polynomial time
  Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure progress toward a trivial embedding. In conjunction with a modified form of arc presentations of links, we obtain a strictly monotonic, deterministic algorithm that recognizes triviality in links within polynomial time and space. 
Chad Musick
math.GT