Ropelength

Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot.

Knots and links that minimize ropelength are called ideal knots and ideal links respectively.

The ropelength of a knotted curve

Ropelength can be turned into a knot invariant by defining the ropelength of a knot

to be the minimum ropelength over all curves that realize

One of the earliest knot theory questions was posed in the following terms: In terms of ropelength, this asks if there is a knot with ropelength

The answer is no: an argument using quadrisecants shows that the ropelength of any nontrivial knot has to be at least

[1] However, the search for the answer has spurred research on both theoretical and computational ground.

It has been shown that for each link type there is a ropelength minimizer although it may only be of differentiability class

[2][3] For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372.

[1] An extensive search has been devoted to showing relations between ropelength and other knot invariants such as the crossing number of a knot.

denotes the crossing number.

[4] There exist knots and links, namely the

-Hopf links, for which this lower bound is tight.

That is, for these knots (in big O notation),[3]

On the other hand, there also exist knots whose ropelength is larger, proportional to the crossing number itself rather than to a smaller power of it.

The proof of this near-linear upper bound uses a divide-and-conquer argument to show that minimum projections of knots can be embedded as planar graphs in the cubic lattice.

[6] However, no one has yet observed a knot family with super-linear dependence of length on crossing number and it is conjectured that the tight upper bound should be linear.