Notes for AKT-170110-1/0:48:55
Roland van der Veen wrote Dror:
Dear Dror,
I enjoyed watching your first hour of AKT and had a minor gripe about your sentence at around 48:55 Your statement "Relative to difficult to compute things, the Jones polynomial is in fact surprisingly easy to compute" is in some sense very wrong as it ignores the fact that computing Jones is #P-hard, shown by Jaeger. Toda's theorem implies that using an oracle computing Jones one can obtain a polynomial time algorithm to solve all NP problems and even much more difficult problems https://en.wikipedia.org/wiki/Toda's_theorem
I suppose what you mean is you can write a short program to do the job.
Best, Roland
Dror: Thanks for the comment, Roland! Do you have a link to the Jaeger result? Roland: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100068936
No, I mean that I can write a short program that does the job well, in practice even if not in theory. So even though the Jones polynomial is in-theory complicated, in practice it can be computed for knots with about 200 crossings.
Aside. Jones once claimed to me (in person) that the Jones polynomial can be computed with knots with up to 200 crossings, but I've never verified it with my directly. Yet I've seen Khovanov homology computed (using my algorithm!) on knots with about 70 crossings, and in the light of that, 200 for Jones seems reasonable.
