In physical knot theory, each realization of a link or knot has an associated 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.
Definition
The ropelength of a knot curve C is defined as the ratio, where Len is the length of C and τ is the thickness of the link defined by C.
Ropelength minimizers
One of the earliest knot theory questions was posed in the following terms:
Can I tie a knot on a foot-long rope that is one inch thick?
In our terms we are asking if there is a knot with ropelength 12. This question has been answered, and it was shown to be impossible: an argument using quadrisecants shows that the ropelength of any nontrivial knot has to be at least 15.66. However, the search for the answer has spurred a lot of research on both theoretical and computational ground. It has been shown that for each link type there is a ropelength minimizer although it is only of class C 1, 1. For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372.
An extensive search has been devoted to showing relations between ropelength and other knot invariants. As an example there are well known bounds on the asymptotic dependence of ropelength on the crossing number of a knot. It has been shown that and for a knot C with crossing number Cr and ropelength L, where the O and Ω are examples of big O notation and big Omega notation, respectively. The lower bound is shown with two families that realize this bound. A former upper bound of O3/2 has been shown using Hamiltonian cycles in graphs embedded in a cubic integer lattice. The current best near-linear upper bound was established with a divide-and-conquer argument to show that minimum projections of knots can be embedded as planar graphs in the cubic lattice. However, no one has yet observed a knot family with super-linear length dependence L > O and it is conjectured that the upper bound is in fact linear.
Ropelength can be turned into a knot invariant by defining the ropelength of a knot type to be the minimum ropelength over all realizations of that knot type. So far this invariant is impractical as we have not determined that minimum for the majority of knots.
