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.
The ropelength of a knotted curve is defined as the ratio , where is the length of and is the knot thickness of .
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. For every knot , the ropelength of is at least proportional to , where denotes the crossing number. [4] There exist knots and links, namely the torus knots and - 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. [5] This is nearly tight, as for every knot,
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.
The ropelength of a knotted curve is defined as the ratio , where is the length of and is the knot thickness of .
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. For every knot , the ropelength of is at least proportional to , where denotes the crossing number. [4] There exist knots and links, namely the torus knots and - 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. [5] This is nearly tight, as for every knot,