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.

There are some known exponential bounds on the number. For example, if k_{n} is the number of prime knots with n crossings, then Welsh proved in "On the number of knots and links" (MR1218230) that
The upper bound holds if you replace k_{n} by the much larger number l_{n} of prime ncrossing links. Sundberg and Thistlethwaite ("The rate of growth of the number of prime alternating links and tangles," MR1609591) also found asymptotic bounds on the number a_{n} of prime alternating ncrossing links: lim (a_{n})^{1/n} exists and is equal to (101+√21001)/40. 


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! 

