My apologies if this is too closely related to this closed post.
I have been collaborating with a physicist looking at long polymer chains. These chains form knots with 2D projections having on the order of 100s of crossings, but which may in some/most cases all cancel out to give knots of relatively few crossings. I realize that there is no absolute knot invariant to definitively classify the knots that arise. But what is the best or best combination of knot invariants that would be 1) straightforward to compute on static diagrams and 2) distinguish most of the knots?
(By "straightforward to compute" I mean something like skein relations and not something like the fundamental group of the complement space. I realize that this requirement might be somewhat ill-defined, so feel free to lead me in the direction of computational references.)
For instance, I seem to recall somewhere that the trio of the Alexander Polynomial along with the 2nd and 3rd Vassiliev invariants was pretty powerful. Can anyone substantiate that and/or provide a reference?
Best Answer
Probably a better answer will come along soon, but in the meantime...
I remember listening to a talk by biologists in the late 1980's about this very problem, but with knotted DNA. They used the Jones polynomial (or perhaps the Kauffman bracket polynomial), but perhaps because that invariant was trendy in those days.
I think professional knot enumerators first compute the Kauffman bracket (or some similar polynomial), and then count representations of the fundamental group into small symmetric groups. I've heard that in practice this almost always distinguishes distinct prime knots, but of course these are knots with relatively few crossings.
If you have hundreds of crossings, I think the most efficient thing to do would be to first find all Reidemeister 1 and 2 moves which reduce the number of crossings. Once that's done hopefully there are many fewer crossings and you can compute the Kauffman bracket using a straightforward algorithm.