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. 



Counting knots with fixed number of crossings
There are some known exponential bounds on the number. For example, if kn is the number of prime knots with n crossings, then Welsh proved in "On the number of knots and links" (MR1218230) that 



> 
> 2.68 ≤ lim inf (kn)1/n ≤ lim sup (kn)1/n ≤ 13.5.
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> 
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The upper bound holds if you replace kn by the much larger number ln of prime n-crossing 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 an of prime alternating n-crossing links: lim (an)1/n exists and is equal to (101+√21001)/40.


