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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 .
Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3.
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 .
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 ...
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.
Trefoil knot without 3-fold symmetry being unknotted by one crossing switch. Whitehead link being unknotted by undoing one crossing. In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it.
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.
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant , and hence can be used to distinguish between two different (non- isotopic ) knots.