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 knot curve C is defined as the ratio , where Len(C) is the length of C and τ(C) is the thickness of the link defined by C.
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: an argument using quadrisecants shows that the ropelength of any nontrivial knot has to be at least 15.66. [1] 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. [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. 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
and
for a knot C with crossing number Cr(C) and ropelength L(C), where the O and Ω are examples of big O notation and big Omega notation, respectively.
The lower bound (big Omega) is shown with two families ((k, k−1) torus knots and k-Hopf links) that realize this bound. A former upper bound of O(Cr(C))3/2 has been shown using Hamiltonian cycles in graphs embedded in a cubic integer lattice. [4] The current best near-linear upper bound was established with a divide-and-conquer argument to show that minimum projections of knots can be embedded as planar graphs in the cubic lattice. [5] However, no one has yet observed a knot family with super-linear length dependence L(C) > O(Cr(CK)) and it is conjectured that the upper bound is in fact linear. [6]
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.
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
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In geometry, a quadrisecant or quadrisecant line of a curve is a line that passes through four points of the curve. Every knotted curve in three-dimensional Euclidean space has a quadrisecant. The number of quadrisecants of an algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley. Quadrisecants of skew lines are also associated with ruled surfaces and the Schläfli double six configuration.
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In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links. Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery construction. These invariants were discovered by Nicolai Reshetikhin and Vladimir Turaev in 1991, and were meant to be a mathematical realization of Witten's proposed invariants of links and 3-manifolds using quantum field theory.
