physical knot theory Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...

, each realization of a link or

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

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 knotted curve

C

is defined as the ratio

L(C) = \operatorname(C)/\tau(C)

, where

\operatorname(C)

is the length of

C

and

\tau(C)

is the knot thickness of

C

. Ropelength can be turned into a

knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some i ...

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

quadrisecant In geometry, a quadrisecant or quadrisecant line of a space curve is a Line (geometry), line that passes through four points of the curve. This is the largest possible number of intersections that a Generic property, generic space curve can have ...

s shows that the ropelength of any nontrivial knot has to be at least

15.66

. 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 In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

C^1

. For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372.

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

\operatorname(K)^

, where

\operatorname(K)

denotes the crossing number. There exist knots and links, namely the

(k,k-1)

torus knot In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of cop ...

s and

k

Hopf link In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. Geometric realization A concrete model consists of ...

s, for which this lower bound is tight. That is, for these knots (in

big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...

L(K)=O(\operatorname(K)^).

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. This is nearly tight, as for every knot,

L(K)= O(\operatorname(K)\log^5(\operatorname(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. 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.

References

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Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...

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