On Distortion and Thickness of Knots*

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What length of rope (of given diameter) is required to tie a particular knot ? Or, to turn the problem around, given an embedded curve, how thick a regular neighborhood of the curve also is embedded ? Intuitively, the diameter of the possible rope is bounded by the distance between strands at the closest crossing in the knot. But of course the distance between two points along a curve goes to zero as the points approach each other, so to make the notion precise, we need to exclude some neighborhood of the diagonal.

The first author was funded in part by NSF grant DMS 94-04278, and both authors enjoyed the hospitality of the IMA during the Spring 1996 term.