Ropelength

In 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.

Definition

The ropelength of a knot curve "C" is defined as the ratio $\{scriptstyle\; L,\; =,\; operatorname\{Len\}(C)/\; au(C)\}$, where τ("C") 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. 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}. Cantarella et al 2002.

Dependence of ropelength on other knot invariants

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

:$O(operatorname\{Cr\}(L)^\{3/4\})\; leq\; L(L)\; leq\; O(operatorname\{Cr\}(L)^\{3/2\}).,$

The first has been shown to be a lower bound with two families (("k", "k"−1) torus knots and "k"-Hopf links) that realize this bound. The upper bound has been shown using Hamiltonian cycles in graphs embedded in a cubic integer lattice. However, no one has yet observed a knot family with super-linear length dependence "L" > O(Cr("K")) and it is conjectured that the upper bound is in fact linear.

Ropelength as a knot invariant

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.

External links

* [http://www.math.uncc.edu/preprint/2003/2003_04.pdf "Realizable Powers of Ropelengths by Nontrivial Knot Families"]
*Jason Cantarella, Robert B. Kusner and John M. Sullivan, On the minimum ropelength of knots and links. Inventiones Mathematicae. Volume 150, Number 2 / November, 2002 [http://www.springerlink.com/content/jjkq7d3bdw1qqkqd/]