Louis H. Kauffman | |
---|---|

Born | February 3, 1945 |

Nationality | American |

Alma mater | Princeton University Massachusetts Institute of Technology |

Scientific career | |

Fields | Mathematics |

Institutions | University of Illinois at Chicago |

Doctoral advisor | William Browder |

**Louis Hirsch Kauffman** (born February 3, 1945) is an American mathematician, topologist, and professor of Mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the introduction and development of the bracket polynomial and the Kauffman polynomial.

Kauffman was valedictorian of his graduating class at Norwood Norfolk Central High School in 1962. He received his B.S. at the Massachusetts Institute of Technology in 1966 and his Ph.D. in mathematics from Princeton University in 1972 (with William Browder as thesis advisor).

Kauffman has worked at many places as a visiting professor and researcher, including the University of Zaragoza in Spain, the University of Iowa in Iowa City, the Institut des Hautes Études Scientifiques in Bures Sur Yevette, France, the Institut Henri Poincaré in Paris, France, the University of Bologna, Italy, the Federal University of Pernambuco in Recife, Brazil, and the Newton Institute in Cambridge England.^{ [1] }

He is the founding editor and one of the managing editors of the * Journal of Knot Theory and Its Ramifications *, and editor of the *World Scientific Book Series On Knots and Everything*. He writes a column entitled Virtual Logic for the journal *Cybernetics and Human Knowing*

From 2005 to 2008 he was president of the American Society for Cybernetics. He plays clarinet in the ChickenFat Klezmer Orchestra in Chicago.

Kauffman's research interests are in the fields of cybernetics, topology and foundations of mathematics and physics. His work is primarily in the topics of knot theory and connections with statistical mechanics, quantum theory, algebra, combinatorics and foundations.^{ [2] } In topology he introduced and developed the bracket polynomial and Kauffman polynomial.

In the mathematical field of knot theory, the bracket polynomial, also known as the *Kauffman bracket*, is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister moves), a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to state sum invariants of 3-manifolds. Recently the bracket polynomial formed the basis for Mikhail Khovanov's construction of a homology for knots and links, creating a stronger invariant than the Jones polynomial and such that the graded Euler characteristic of the Khovanov homology is equal to the original Jones polynomial. The generators for the chain complex of the Khovanov homology are states of the bracket polynomial decorated with elements of a Frobenius algebra.

The Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is defined as

where is the writhe and is a regular isotopy invariant which generalizes the bracket polynomial.

In 1994, Kauffman and Tom Etter wrote a draft proposal for a non-commutative *discrete ordered calculus* (DOC), which they presented in revised form in 1996.^{ [3] } In the meantime, the theory was presented in a modified form by Kauffman and H. Pierre Noyes together with a presentation of a derivation of free space Maxwell's equations on this basis.^{ [4] }

He won a Lester R. Ford Award (with Thomas Banchoff) in 1978.^{ [5] } Kauffman is the 1993 recipient of the Warren McCulloch award of the American Society for Cybernetics and the 1996 award of the Alternative Natural Philosophy Association for his work in discrete physics. He is the 2014 recipient of the Norbert Wiener award of the American Society for Cybernetics.^{ [6] }

In 2012 he became a fellow of the American Mathematical Society.^{ [7] }

Louis H. Kauffman is author of several monographs on knot theory and mathematical physics. His publication list numbers over 170.^{ [1] } Books:

- 1987,
*On Knots*, Princeton University Press 498 pp. - 1993,
*Quantum Topology (Series on Knots & Everything)*, with Randy A. Baadhio, World Scientific Pub Co Inc, 394 pp. - 1994,
*Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds*, with Sostenes Lins, Princeton University Press, 312 pp. - 1995,
*Knots and Applications (Series on Knots and Everything, Vol 6)* - 1995,
*The Interface of Knots and Physics: American Mathematical Society Short Course January 2–3, 1995 San Francisco, California (Proceedings of Symposia in Applied Mathematics)*, with the American Mathematical Society. - 1998,
*Knots at Hellas 98: Proceedings of the International Conference on Knot Theory and Its Ramifications*, with Cameron McA. Gordon, Vaughan F. R. Jones and Sofia Lambropoulou, - 1999,
*Ideal Knots*, with Andrzej Stasiak and Vsevolod Katritch, World Scientific Publishing Company, 414 pp. - 2002,
*Hypercomplex Iterations: Distance Estimation and Higher Dimensional Fractals (Series on Knots and Everything , Vol 17)*, with Yumei Dang and Daniel Sandin. - 2006,
*Formal Knot Theory*, Dover Publications, 272 pp. - 2007,
*Intelligence of Low Dimensional Topology 2006*, with J. Scott Carter and Seiichi Kamada. - 2012,
*Knots and Physics*(4th ed.), World Scientific Publishing Company, ISBN 978-981-4383-00-4

Articles and papers, a selection:

- 2001, The Mathematics of Charles Sanders Peirce, in:
*Cybernetics & Human Knowing*, Vol.8, no.1–2, 2001, pp. 79–110

**Edward Witten** is an American mathematical and theoretical physicist. He is currently the Charles Simonyi Professor in the School of Natural Sciences at the Institute for Advanced Study. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. In addition to his contributions to physics, Witten's work has significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, awarded for his 1981 proof of the positive energy theorem in general relativity. He is considered to be the practical founder of M-theory.

In topology, **knot theory** is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

In the mathematical field of knot theory, a **knot invariant** is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory. Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics.

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.

In the mathematical field of knot theory, the **HOMFLY polynomial** or **HOMFLYPT polynomial**, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables *m* and *l*.

In the mathematical field of knot theory, the **Jones polynomial** is a knot polynomial discovered by Vaughan Jones in 1984. 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 with integer coefficients.

In mathematics, the **Alexander polynomial** is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the **Alexander–Conway polynomial**, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.

In mathematics, **Khovanov homology** is an oriented link invariant that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial.

The **Tait conjectures** are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.

In the mathematical field of knot theory, the **bracket polynomial** is a polynomial invariant of framed links. Although it is not an invariant of knots or links, a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of 3-manifolds.

In mathematics, the **unknotting problem** is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether the problem lies in the complexity class P.

In knot theory, a **virtual knot** is a generalization of knots in 3-dimensional Euclidean space, **R**^{3}, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. Here is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).

In the mathematical field of knot theory, the **Arf invariant** of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface. If *F* is a Seifert surface of a knot, then the homology group H_{1}(*F*, **Z**/2**Z**) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an embedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot.

In knot theory, the **Kauffman polynomial** is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as

**Mikhail Khovanov** is a Russian-American professor of mathematics at Columbia University who works on representation theory, knot theory, and algebraic topology. He is known for introducing Khovanov homology for links, which was one of the first examples of categorification.

Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early, significant stimulus in knot theory would arrive later with Sir William Thomson and his vortex theory of the atom.

In the branch of mathematics called knot theory, the **volume conjecture** is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements.

In theoretical physics, the **six-dimensional (2,0)-superconformal field theory** is a quantum field theory whose existence is predicted by arguments in string theory. It is still poorly understood because there is no known description of the theory in terms of an action functional. Despite the inherent difficulty in studying this theory, it is considered to be an interesting object for a variety of reasons, both physical and mathematical.

**Hernando Burgos Soto** is a Canadian writer and mathematician, professor of mathematics at George Brown College. He is the author of several math papers in which he introduced some mathematics concepts and extended to tangles some celebrated results of knot theory about the Khovanov homology and the Jones polynomial. During his career as a mathematician, his interests have included Mathematical Statistics, Knot Theory, Algebraic Topology and more recently Mathematical Finance. He is comfortable writing in English and Spanish. When writing in Spanish, he works in the area of prose fiction writing short stories. Some of his short stories were published at the website Cuentos y Cuentos.

- 1 2 http://www.math.uic.edu/~kauffman/569.html
- ↑ "Presentation". Archived from the original on 2008-09-17. Retrieved 2007-09-26.
- ↑ T. Etter, L.H. Kauffman, ANPA West Journal, vol. 6, no. 1, pp. 3–5
- ↑ Louis H. Kauffman, H. Pierre Noyes, Discrete physics and the derivation of electromagnetism from the formalism of quantum mechanics, Proceedings of the Royal Society London A (1996), vol. 452, pp. 81–95
- ↑ Kauffman, Louis; Banchoff, Thomas (1977). "Immersions and Mod-2 quadratic forms".
*Amer. Math. Monthly*.**84**: 168–185. doi:10.2307/2319486. JSTOR 2319486. - ↑ About SSC: Awards, retrieved 2014-11-02.
- ↑ List of Fellows of the American Mathematical Society, retrieved 2013-01-27.

Wikiquote has quotations related to: Louis Kauffman |

- Louis Kauffman homepage at uic.edu
- Hypercomplex Fractals
- Arxiv Papers
- Louis Kauffman at the Mathematics Genealogy Project
- ChickenFat Klezmer Orchestra

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