In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations
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History
The first knot polynomial, the Alexander polynomial, was introduced by J. W. Alexander in 1923, but other knot polynomials were not found until almost 60 years later. In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of
In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander-Conway polynomial. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups A central question in the mathematical theory of knots is whether two Knot diagrams represent the same knot In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial. Vaughan Frederick Randal Jones DCNZM (born 31 December 1952) is a New Zealand Mathematician, known for his work on In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983 This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial. In the mathematical field of Knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial
Soon after Jones' discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a state-sum model, which involved the bracket polynomial, an invariant of framed knots. Louis H Kauffman ( 3 February, 1945) is an American Mathematician, topologist, and professor of Mathematics in the Department of Mathematics In the mathematical field of Knot theory, the bracket polynomial (also known as the Kauffman bracket) is a Polynomial invariant of Framed In the mathematical theory of knots, a framed knot is the extension of a tame knot to an embedding of the Solid torus D 2 × This opened up avenues of research linking knot theory and statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics
In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern-Simons theory. Edward Witten (born August 26, 1951) is an American Theoretical physicist and Professor at the Institute for Advanced Study The Chern-Simons theory is a 3-dimensional Topological quantum field theory of Schwarz type, developed by Shiing-Shen Chern and James Harris Simons Viktor Vassiliev and Mikhail Goussarov started the theory of finite type invariants of knots. Mikhail Goussarov (March 8 1958 Leningrad – June 25 1999 Tel Aviv) was a Russian mathematician who worked in low-dimensional topology In the mathematical theory of knots, a finite type invariant is a knot invariant that can be extended (in a precise manner to be described to an invariant of certain singular The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable "change of variables").
In recent years, the Alexander polynomial has been shown to be related to Floer homology. Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology. The graded Euler characteristic of the knot Floer homology of Oszvath and Szabo is the Alexander polynomial. Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology.
References
- Colin Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9
- W. B. R. Lickorish, An introduction to knot theory. William Bernard Raymond Lickorish is a Mathematician. He is Emeritus professor of Geometric topology in DPMMS, University of Cambridge Graduate Texts in Mathematics, 175. Springer-Verlag, New York, 1997. ISBN 0-387-98254-X
See also
Specific knot polynomials
Related topics
- skein relationship for a formal definition of the Alexander polynomial, with a worked-out example. In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type In the mathematical field of Knot theory, the bracket polynomial (also known as the Kauffman bracket) is a Polynomial invariant of Framed In the mathematical field of Knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983 The Kauffman polynomial is a 2-variable Knot polynomial due to Louis Kauffman. A central question in the mathematical theory of knots is whether two Knot diagrams represent the same knot
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