We showed that the ropelength required to tie an N-crossing knot or link grows at a rate that varies at least between N to the three-fourths and N (that is, linear in N). We showed the three-fourths power is sharp.
| Ropelength | |||||
|
|||||
| Imagine a solid disc of radius R centred at x and normal to K at each point x along the parametrized smooth knot K. R(K) is the largest R so that the disks are disjointed. The rope length is L(K)=[arclength(K)]/[R(K)]. The crossing number C(K) of the knot-type K is the necessary number of crossings in the planar diagram of K.
We showed that the ropelength required to tie an N-crossing knot or link grows at a rate that varies at least between N to the three-fourths and N (that is, linear in N). We showed the three-fourths power is sharp. |
|||||
|
|||||
| Here is one paper on ropelength, and here is one with a broader view that also discusses ropelength. See also entanglement rates. | |||||