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I am an undergraduate studying polynomial knots. Given a polynomial knot, it is common to generate a tube around the space curve for visualization purposes. I define the set T[c(t)] as the set of all polynomial space curves within the maximal tube generated about c(t) on the interval [a,b], where a maximal tube is a tube with the largest radius such that the tube is smooth and does not intersect itself. So, if p(t) is in T[c(t)], then max{||c(t)-p(t)||} < r, where r is the radius of the maximal tube about c(t). Also, since re-parametrization does not affect the curve, we assume that if p(t) is in T[c(t)] then p(a) is contained in the normal disc (of radius strictly less than r) at c(a) and p(b) is contained in the normal disc (of radius strictly less than r) at c(b). I want to define T[c(t)] as a metric space with some metric d(p,q). My question: Are there any interesting questions about this space? I can define the space and can think of a few things that I can say about the space, but does it matter? I want/NEED it to matter in some way, does anyone have any ideas?

Thank you

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While I am sympathetic to the reason behind this question, I do not think it is very focused. Please see mathoverflow.net/howtoask Also, please use LaTeX in your post. –  David Roberts Mar 3 '11 at 3:35
On a more mathematical point, it is not too tricky to define a metric on your space of knots. As far as interesting question, look up any resource on metric spaces, look the sort of properties they can have and see if your space has them. –  David Roberts Mar 3 '11 at 3:48
I like this question. What kind of stuff can you prove? The metric that you are ussing is the $l_\infty$ metric. Note that the tube around a knot might intersect another knot that is close to it in more than one point and so it's not so easy to reparametrize as you claimed right? –  Alfredo Hubard Nov 4 '11 at 23:56
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