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Jul 17, 2013 at 16:19 comment added Dave Futer Ilya's comment is very much on point. In general, part of what makes this question difficult is that the answer is likely to depend in a huge way on your model of "random knots". Choosing uniformly at random from a list of all $n$-crossing diagrams and choosing uniformly at random from a list of $n$-crossing knots are only two of many different models. Some others: stick knots, knot complements built out of $n$ tetrahedra (Champanerkar-Kofman-Patterson started an enumeration), etc. I don't believe the answer to your question is known for any of these...
Jul 17, 2013 at 14:24 comment added Ilya Kofman Your second paragraph is about knots & links, not diagrams. The answer could be different for a generic knot (chosen randomly from any ordered list of knot types) because a semi-adequate knot generally has many non-semi-adequate diagrams.
Jul 15, 2013 at 11:12 history edited Daniel Moskovich CC BY-SA 3.0
minor edit.
Jul 15, 2013 at 8:48 history asked Daniel Moskovich CC BY-SA 3.0