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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 .
These homology theories have contributed to further mainstreaming of knot theory. In the last several decades of the 20th century, scientists and mathematicians began finding applications of knot theory to problems in biology and chemistry. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or
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.
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. [ 1 ] [ 2 ] Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 / 2 {\displaystyle t^{1/2}} with integer coefficients .
A polygonal knot is a knot whose image in R 3 is the union of a finite set of line segments. [6] A tame knot is any knot equivalent to a polygonal knot. [6] [Note 2] Knots which are not tame are called wild, [7] and can have pathological behavior. [7] In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for ...
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
A (3,−7)-3D torus knot. EureleA Award showing a (2,3)-torus knot. (2,8) torus link. In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R 3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p ...
In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings. A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand un