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This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. These topologists in the early part of the 20th century— Max Dehn , J. W. Alexander , and others—studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial .
A few major discoveries in the late 20th century greatly rejuvenated knot theory and brought it further into the mainstream. In the late 1970s William Thurston's hyperbolization theorem introduced the theory of hyperbolic 3-manifolds into knot theory and made it of prime importance. In 1982, Thurston received a Fields Medal, the highest honor ...
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum ... 6 2 knot: 6 2: 6a2 4 8 10 12 2 6 [312] 123456:234165 1231 ...
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Three-twist knot is the twist knot with three-half twists, also known as the 5 2 knot. Trefoil knot A knot with crossing number 3; Unknot; Knot complement, a compact 3 manifold obtained by removing an open neighborhood of a proper embedding of a tame knot from the 3-sphere. Notation used in knot theory: Conway notation
A reduced diagram is one in which all the isthmi are removed. Tait came up with his conjectures after his attempt to tabulate all knots in the late 19th century. As a founder of the field of knot theory, his work lacks a mathematically rigorous framework, and it is unclear whether he intended the conjectures to apply to all knots, or just to alternating knots.
The full set of fundamental transformations and operations on 2-tangles, alongside the elementary tangles 0, ∞, ±1 and ±2. The trefoil knot has Conway notation [3].. In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear.
Knot theory is a branch of topology that concerns itself with abstract properties of mathematical knots — the spatial arrangements that in principle could be assumed by a closed loop of string. The main article for this category is Knot theory .