ropelength

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

One of the earliest knot theory questions was posed in the following terms:

{{block indent|left=1.6|Can I tie a knot on a foot-long rope that is one inch thick?}}

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 $15.66$.{{r|DDS}} 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 $C^1$.{{r|GMSV|CKS}} For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372.{{r|DDS}}

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 $\backslash operatorname\{Cr\}(K)^\{3/4\}$, where $\backslash operatorname\{Cr\}(K)$ denotes the crossing number.{{r|BS}} 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),{{r|CKS}}

$$L(K)=O(\backslash operatorname\{Cr\}(K)^\{3/4\}).$$

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.{{r|DET}} This is nearly tight, as for every knot,

$$L(K)=\; O(\backslash operatorname\{Cr\}(K)\backslash log^5(\backslash operatorname\{Cr\}(K))).$$

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.{{r|DEPZ}} 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.{{r|DE}}

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Category:Knot invariants

Category:Geometric topology