Ropelength

Template:Short description 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.

A numeric approximation of an ideal trefoil.

A numeric approximation of an ideal trefoil.

Definition

The ropelength of a knotted curve $C$ is defined as the ratio $L(C)=\mathrm{Len}(C)/\tau (C)$, where $\mathrm{Len}(C)$ is the length of $C$ and $\tau (C)$ is the knot thickness of $C$.

Ropelength can be turned into a knot invariant by defining the ropelength of a knot $K$ to be the minimum ropelength over all curves that realize $K$.

Ropelength minimizers

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 $12$. The answer is no: an argument using quadrisecants shows that the ropelength of any nontrivial knot has to be at least $15.66$.Template:R/ref 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 $C^1$.Template:R/refTemplate:R/ref For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372.Template:R/ref

Dependence on crossing number

An extensive search has been devoted to showing relations between ropelength and other knot invariants such as the crossing number of a knot. For every knot $K$, the ropelength of $K$ is at least proportional to $\mathrm{Cr}(K{)}^{3/4}$, where $\mathrm{Cr}(K)$ denotes the crossing number.Template:R/ref There exist knots and links, namely the $(k,k-1)$ torus knots and $k$-Hopf links, for which this lower bound is tight. That is, for these knots (in big O notation),Template:R/ref $$L(K)=O(\mathrm{Cr}(K{)}^{3/4}).$$

The ropelength of any knot or link must be greater than a universal constant times the three-quarter power of the crossing number, but this constant is not known exactly. This constant in proven to be above 1.1 Template:R/ref, and torus knots have been tightened with computer simulations that show that this constant must not exceed 10.76 Template:R/ref.

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.Template:R/ref This is nearly tight, as for every knot, $$L(K)=O(\mathrm{Cr}(K){\log }^{}(\mathrm{Cr}(K))).$$ 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.Template:R/ref 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.Template:R/ref