# Counting knots with fixed number of crossings

How to obtain an upperbound for knots up to k crossings? I think I've found something which involves the genus but I'm not sure.

• Paolo- This is a really vague and unhelpful question. The reader is left completely unclear on what you are looking for. Are you looking for references? For theoretical upper bounds? For actual data? You'll get much better answers if you're more specific. – Ben Webster Mar 29 '10 at 18:19
• I mean theoretical upper bounds depending on the number of crossing – Paolo Aceto Mar 29 '10 at 21:01

Dowker codes can be used to get an (over)estimate for the number of knots with $k$ crossings. Hoste has written a few, extremely clear, papers on using Dowker codes for enumeration of knot tables. I don't see how genus could be used - crossing number is an invariant defined in terms of diagrams while genus is much more topological... Very curious!
• @Scott - I did say it was an (over)estimate. :) My bound is of the order $n^n$ while the reality of the situation is more like an exponential (which I wasn't aware of). I've voted up Steven's answer. Hmmm. I will point out that my answer is elementary... – Sam Nead Mar 30 '10 at 1:18