A few years ago I tried to solve for the tightest (smallest constant slope) helical rope about the z-axis in R^3. I.e., the lowest-slope helix about the z-axis such that if each point P of the helix has a flat, closed, perpendicular disk of radius = 1 centered at P that touches the z-axis, the interiors of all such disks are disjoint. This results in a transcendental equation that can only be solved numerically, but it was a fun endeavor and is in some sense the simplest "tightest-knot" type question I can think of. --Dan On 2013-12-31, at 12:07 AM, Bill Gosper wrote:
Are there still no explicitly solved tightenings of it (or any other true knot)? I expect the trefoil to get sorted fairly soon. A crude program of mine came up with 17 and 13 as the minimal "lengths" of the trefoil, and the subarc of the trefoil that could not spontaneously "untie itself". Evidently, I used tube diameter where the convention seems to be radius.