Examples of different knots including the trivial knot (top left) and the trefoil knot (below it)A knot diagram of the trefoil knot, the simplest non-trivial knot
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 so it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.
A complete algorithmic solution to this problem exists, which has unknown complexity.[1] In practice, knots are often distinguished using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.
The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). For example, a higher-dimensional knot is an n-dimensional sphere embedded in (n+2)-dimensional Euclidean space.
^As first sketched using the theory of Haken manifolds by Haken (1962). For a more recent survey, see Hass (1998)
In topology, knottheory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope,...
mathematics known as knottheory. Knots and knotting have been used and studied throughout history. For example, Chinese knotting is a decorative handicraft...
In knottheory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two...
of knottheory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. The...
In knottheory, 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...
Physical knottheory is the study of mathematical models of knotting phenomena, often motivated by considerations from biology, chemistry, and physics...
mathematical field of knottheory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence...
knot is one of the most fundamental knots, and it forms the basis of many others, including the simple noose, overhand loop, angler's loop, reef knot...
significant stimulus in knottheory would arrive later with Sir William Thomson (Lord Kelvin) and his vortex theory of the atom. Different knots are better at different...
In the mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus...
of knottheory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or...
the "Ballantine rings". The first work of knottheory to include the Borromean rings was a catalog of knots and links compiled in 1876 by Peter Tait....
In knottheory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed...
Knottheory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs...
duality. Knottheory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs...
knots. Applying chemical topology and knottheory to molecular knots allows biologists to better understand the structures and synthesis of knotted organic...
as knottheory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is...
mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either...
field of knottheory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent...
structure. Knottheory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs...
related to, among other things, knottheory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry...
Homology theory Homotopy theory Ideal theory Index theory Information theory Invariant theory Iwasawa theory K-theoryKnottheory L-theory Lattice theory Lie...
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