Enver Karadayi recently wrote a dissertation <http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=4707&context=etd&sei-redir=1&referer=http%3A%2F%2Fwww.google.ca%2Furl%3Fsa%3Dt%26rct%3Dj%26q%3Drandom%2Bknots%2Bkaradayi%2Bsaito%26source%3Dweb%26cd%3D1%26ved%3D0CDEQFjAA%26url%3Dhttp%253A%252F%252Fscholarcommons.usf.edu%252Fcgi%252Fviewcontent.cgi%253Farticle%253D4707%2526context%253Detd%26ei%3DRg7ZUMGoF5Ka8gTtq4HQDg%26usg%3DAFQjCNEajRC1Yot3Co5DHg86QDoVjoYwAA#search=%22random%20knots%20karadayi%20saito%22>in which he generated successively n random points in the unit cube. He connected successive points with straight line segments and then connected the first point to the last point. His simulations (if I understand correctly) show that after n = 40, one almost always get a non-trivial knot. The difficulty is determining when you have a non-trivial knot. One runs up against the unknotting problem<http://en.wikipedia.org/wiki/Unknotting_problem> . Karadayi's method seems to be to compute the determinant of the knot and if it is divisible by a prime p > 2 then it is "p-colorable" which implies that it is not the unknot. If the determinant of the knot is 1 then it may be knotted, but presumably the probability is low. (KnotInfo <http://www.indiana.edu/~knotinfo/> shows 17 prime knots with crossing numbers 10 to 12 which have determinant = 1). It should be easy (well, at least possible) to do similar simulations for random walks. --Edwin On Mon, Dec 24, 2012 at 8:56 PM, <rcs@xmission.com> wrote:
Or just draw a straight line from one end of the walk to the other. Maybe exclude cases where this line intersects the path-so-far.
Rich
--- Quoting meekerdb <meekerdb@verizon.net>:
On 12/24/2012 4:47 PM, Fred lunnon wrote:
On 12/25/12, Bill Gosper<billgosper@gmail.com> wrote:
Awful thought: What is the expected time at which a 3D random walk first becomes "knotted"? (W.r.t. pulling on the ends.)
Bearing in mind that the ends may in general lie deep within the convex hull of the walk, perhaps this question requires rather more careful definition ... WFL
You could constrain the question to those random walks that return to the starting point (although of measure zero) or you could try defining a different kind of 'random walk' that starts with a unit loop and randomly expands and rearranges it, with crossovers allowed.
Brent
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