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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.

flag	asked Mar 29 2010 at 18:00 Paolo Aceto 452●2●9

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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 2010 at 18:19

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I mean theoretical upper bounds depending on the number of crossing – Paolo Aceto Mar 29 2010 at 21:01

Steven Sivek · Accepted Answer · 2010-03-29 19:29:35Z UTC

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

2.68 ≤ lim inf (k_n)^1/n ≤ lim sup (k_n)^1/n ≤ 13.5.

The upper bound holds if you replace k_n by the much larger number l_n 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 a_n of prime alternating n-crossing links: lim (a_n)^1/n exists and is equal to (101+√21001)/40.

Sam Nead · Answer 2 · 2010-03-29 19:07:21Z UTC

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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!

answered Mar 29 2010 at 19:07

Sam Nead
5,302●12●26

The over-estimate is way too large! Once you get beyond 17 crossings the probability that the code will not be planar is way too large. – Scott Carter Mar 29 2010 at 22:47

@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 2010 at 1:18

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...I was wrong about the genus... – Paolo Aceto Mar 30 2010 at 1:58

Counting knots with fixed number of crossings

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