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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 ...
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 .
The Bing double of a knot K is defined by placing the Bing double of the unknot in the solid torus surrounding it, as shown in the figure, and then twisting that solid torus into the shape of K. [2] This definition is similar to that for Whitehead doubles. The Bing double of the unknot is also called the Bing link. [3]
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 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. A numeric approximation of an ideal trefoil.
Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by considerations from biology, chemistry, and physics (Kauffman 1991). ). Physical knot theory is used to study how geometric and topological characteristics of filamentary structures, such as magnetic flux tubes, vortex filaments, polymers, DNAs, influence their physical properties and