Way to determine the type of a knot given by a diagram Is there a general way to determine the type of a knot given by a diagram? I am using KLO (Kirby calculator) and I encounter some nontrivial knots while doing this. For example, can we determine the type of the following diagram? 

The best currently known algorithm for unknot recognition is due to Marc Lackenby, it runs in quasi-polynomial time $c^{\log c}$, where $c$ is the crossing number of the knot diagram. There is not (yet) a paper out, but Lackenby gave several talks on this in the last year or so (see Ian Agol's answer here for references). The oldest known algorithm for knot recognition (which has exponential run time) is due to Haken. It is implemented in SnapPy and Regina programs.  Lackenby's algorithm is a very clever (and major!) improvement of Haken's idea. However, afaik, it is not yet implemented.
