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Topic: Skein relation

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	Skein relation - Wikipedia, the free encyclopedia
		Informally, a skein relation gives a linear relation between the values of a knot polynomial on a collection of three links which differ from each other only in a small region.
		For some knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively.
		More formally, a skein relation can be thought of as defining the kernel of a quotient map from the planar algebra of tangles.
	en.wikipedia.org /wiki/Skein_relation (724 words)

	Skein relation - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-13)
		Skein relations occur in knot theory, where they are most often used to give a simple definition of a knot polynomial.
		Informally, a skein relation gives a linear relation between the values of a knot polynomial on a collection of links which differ from each other only in a small region.
		More formally, a skein relation should be thought of as defining the kernel of a quotient map from the planar algebra of tangles.
	www.godseye.com /stat/en/s/k/e/Skein_relation.html (561 words)

	Knots and their Polynomials-4 (Site not responding. Last check: 2007-10-13)
		So reversing the orientation in a diagram and then applying the skein relation to a crossing is the same as rotating the original diagram 180 degrees, applying the skein relation, rotating the products of the relation another 180 degrees, and then reversing the orientation.
		Suppose that it takes n applications of the skein relation to reduce all the diagrams to unknots, for which the Jones polynomial (symbolically J{ }) does not depend on orientation.
		Because to apply the skein relation correctly at a crossing, we need the two strands to be coherently oriented.
	www.math.sunysb.edu /~tony/whatsnew/column/knots-0499/knots4.html (267 words)

	Event Symmetric Space-Time
		It would be nice to think that the two are related, surely it is not a coincidence, but we must not become carried away.
		The commutation relations used to generate the closed string algebra will remind anyone who knows about knot polynomials of Skein relations.
		These could be subject to the familiar Skein relations which define the HOMFLY polynomial.
	www.weburbia.com /press/html/g09.htm (3600 words)

	Knot Theory Vocabulary: Skein Relation (Site not responding. Last check: 2007-10-13)
		A skein relation is a set of rules defining a knot polynomial invariant with givens and associations between crossings.
		For example, the skein relation for the Jones Polynomial is
		A crossing in a knot diagram is selected as the first step in using a skein relation.
	library.thinkquest.org /12295/data/Vocabulary/Skein_Relation.html (182 words)

	GTM Vol 4 (2002) Paper 21 (Abstract) (Site not responding. Last check: 2007-10-13)
		This paper is based on my talks (`Skein modules with a cubic skein relation: properties and speculations' and `Symplectic structure on colorings, Lagrangian tangles and its applications') given in Kyoto (RIMS), September 11 and September 18 respectively, 2001.
		The theory of skein modules is outlined in the problem section of these proceedings.
		In the third section we apply (2,2)-moves and a skein module deformation of a 3-move to approximate unknotting numbers of knots.
	www.msp.warwick.ac.uk /gt/GTMon4/paper21.abs.html (229 words)

	Differential topology
		Skein relation (explanation) for the Alexander and the Convey polynomial.
		Relation of w_2 to the quadratic form of 4-manifolds.
		Its relation to w_2, to the signature and the Euler characteristic.
	www.math.metu.edu.tr /~serge/courses/710-2005/710-2005.html (239 words)

	The HOMFLYPT Polynomial (Site not responding. Last check: 2007-10-13)
		When one applies a skein relation to a neighborhood that contains one crossing, one obtains one link with one crossing less, and one link with the same number of crossings but one crossing reversed.
		In 1985 however a number of mathematicians showed independently of each other, and almost simultaneously, that the three terms A(t), B(t) and C(t) did not have to be related to each other through some parameter t in order for the skein relations to define a knot invariant.
		Since the terms A and B in the skein relations are unrelated, one obtains the polynomial of the mirror symmetric knot by exchanging A with B and vice versa.
	www.inst.bnl.gov /~wei/homflypt.html (479 words)

	Georgia Tech Geometry-Topology Seminar (Site not responding. Last check: 2007-10-13)
		ABSTRACT: I will describe certain extensions and transversal refinements of constructions due to Chas and Sullivan, which describe the "interactions" and "self-interactions" of families of loops in an oriented 3-manifold M. This approach is naturally motivated by Vassiliev theory.
		The relation of the new structures with the Conway skein theory of M is discussed.
		In particular the interaction (intersection) theory of the free loop space of M is closely related with the Hoste-Przytycki homotopy skein module.
	www.math.gatech.edu /~stavros/gt.html (752 words)

	[No title]
		Of course it looks more like it if you call the variable z "hbar", but the real thing is to note that the two kinds of crossings in K and K' are analogous (somehow) to the different orderings in pq and qp.
		So now we see a close relation between quantum theory - to be precise, "statistics" in quantum theory - and the braid group.
		and the canonical commutation relations pq - qp = -i hbar.
	math.ucr.edu /home/baez/braids.ascii (9604 words)

	Knots and Jones polynomial
		The skein relation talks about three knots that look exactly the same except at one location, where one has downover, one has upover, and one has nocross.
		Notice that if in either of these two you were to replace the solitary crossing by a "nocross" then you would get two concentric circles or anyway two concentric unknots.
		Witten was awarded the Fields medal for discovering a relation between the Jones polynomial and Quantum Field Theory.
	www.physicsforums.com /showthread.php?p=47798 (709 words)

	Motivate : Stephen's talk
		In 1926 the topologist Kurt Reidemeister proved that two projections of the same knot can be related by a sequence of moves, which we now call the Reidemeister moves.
		After that, knot theory was relatively quiet until Vaughan Jones discovered a completely new polynomial invariant in 1985; this was the catalyst for a burst of ideas and results.
		One of the key new ideas is that of a skein relation, which gives us a recursive method of defining and calculating a knot polynomial.
	motivate.maths.org /conferences/conf28/c_28_talk.shtml (1174 words)

	Geometry and Topology, Volume 9 (2005)
		We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations.
		We calculate the knot invariant for two-bridge knots and relate it to double branched covers for general knots.
		In the appendix we show that the cord ring is determined by the fundamental group and peripheral structure of a knot and give applications.
	www.msp.warwick.ac.uk /gt/2005/09/p036.xhtml (105 words)

	publications GDR tresses
		In this paper we give several constructions which yield examples, both explicit and non-explicit, of pairs of transversal knots K, K' with the property: K and K' have the same topological knot type and the same Bennequin invariant, but are not transversally equivalent.
		In this paper we consider the cohomology of such groups with coefficients in the module $R$ (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement).
		We also define a notion of Garside submonoid (subgroup) of a Garside monoid (group), which is related to the notion of LCM-homomorphisms between Artin group, and prove that most of the properties extend.
	www.math.unicaen.fr /~picantin/GDRpub/GDRpub2003.html (3664 words)

	Springer Online Reference Works
		It has its root in the statistical mechanics model of the Jones–Conway polynomial by V.F.R. Jones.
		It has been applied to periodic links and to the building of a Hopf algebra structure on the Jones–Conway skein module of the product of a surface and an interval [a3], [a4], [a2].
		V.G. Turaev, "Skein quantization of Poisson algebras of loops on surfaces" Ann.
	eom.springer.de /j/j130010.htm (283 words)

	UC Davis Math: Profile of the Faculty
		On the algebra side, the Jones polynomial comes from a Lie group called SU(2), or rather, a modification which is called a quantum group.
		The Jones polynomial can also be defined directly by a simple skein relation.
		I found another set of skein relations, involving tangled graphs as well as knots and links, for the link invariant corresponding to the Lie group
	www.math.ucdavis.edu /research/profiles/greg (414 words)

	Alexander polynomial - Wikipedia, the free encyclopedia
		Let K be a knot in the 3-sphere.
		ISBN 0-8218-3678-1 (accessible introduction utilizing a skein relation approach)
		Joan Birman mentions in her paper New points of view in knot theory (Bull.
	en.wikipedia.org /wiki/Alexander_polynomial (813 words)

	AMCA: Introduction to skein modules by U. Kaiser, H. Morton, J. Przytycki (Site not responding. Last check: 2007-10-13)
		We discuss at length the Alexander polynomial (1928) and the skein relation described by Alexander.
		The next step in the history leading to skein modules was the normalization of the Alexander polynomial by J.H.Conway (c.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/l/g/76.htm (190 words)

	Not Even Wrong » Blog Archive » Khovanov Homology (Site not responding. Last check: 2007-10-13)
		The only known definitions of it are kind of like the pre-Witten skein relation definitions of Jones polynomials.
		Gukov is also the co-author of a paper that just appeared on the arXiv entitled “Topological M-theory as Unification of Form Theories of Gravity”.
		Like M-theory itself, it appears that no one knows what “topological M-theory” is, but it is supposed to be some sort of seven-dimensional theory that is related to topological strings on 6d Calabi-Yaus in much the same way M-theory is a conjectural 11d theory related to 10d superstrings.
	www.math.columbia.edu /~woit/wordpress/archives/000104.html (542 words)

	Preprints and Publications
		In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).
		Abstract: We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems.
		We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram.
	myweb.lmu.edu /bmellor/papers.html (1371 words)

	Braids
		In other words - and this has a lot of physical/philosophical significance - the braid group may be thought of as the obvious generalization of the group of permutations to a situation in which "switching" two things' places twice does NOT get you back to the original situation.
		Actually I'm going to be sneaky and use a variant of the Heisenberg algebra where we stick in a square root of hbar, which I'll call h, just to confuse the heck out of the physicists.
		Anyway, now suppose that the magnetic field strength is a constant B. Let U denote the unitary operator corresponding to translation by a unit distance in the x direction, and let V be the unitary operator corresponding to a unit translation in the y direction.
	math.ucr.edu /home/baez/braids.html (10505 words)

	A Conway Polynomial for Virtual Links (Site not responding. Last check: 2007-10-13)
		A polynomial invariant is defined for virtual link diagrams which arises from the Conway polynomial for links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur.
		The polynomial fulfills a Conway-type skein relation and has some interesting properties.
		Examples are given that the invariant can detect chirality and even non-invertibility of virtual knots and links.
	www.mathematik.uni-dortmund.de /~sawollek/klm2001.html (97 words)

	References beginning with B
		The Århus integral of rational homology 3-spheres III: The relation with the Le-Murakami-Ohtsuki invariant, Selecta Mathematica, New Series 10 (2004) 305-324,
		Skein relations for link invariants coming from exceptional Lie algebras, Bern University preprint, September 1998.
		Feynman diagrams as a weight system: four-loop test of a four-term relation, hep-th/9612011, Open University UK (OUT-4102-66), and Mainz University (MZ-TH/96-37) preprint, November 1996.
	www.math.toronto.edu /drorbn/VasBib/References_beginning_with_B.html (671 words)

	Amazon.com: The Geometry and Physics of Knots (Lezioni Lincee): Books: Michael Atiyah (Site not responding. Last check: 2007-10-13)
		Deals with an area of research that lies at the crossroads of mathematics and physics.
		The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions.
		projective flatness, symplectic quotient, skein relation, topological quantum field theory, holomorphic bundles, moduli space, conformal field theory, symplectic structure, abelian case, marked points
	www.amazon.com /Geometry-Physics-Knots-Lezioni-Lincee/dp/0521395542 (1214 words)

	Atlas: KNOTS in Poland 2003: The mini-semester on Knot Theory and its Ramifications - List of Speakers (Site not responding. Last check: 2007-10-13)
		Uwe Kaiser Skein theory and topology of mapping spaces
		Thang Le Quantum invariants: recursion relations and cyclotomic expansion
		Maciej Mroczkowski Homflypt and Kauffman Skein modules of L(2, 1)
	atlas-conferences.com /cgi-bin/abstract/calg-01 (702 words)

	Link Invariants (Site not responding. Last check: 2007-10-13)
		The values of the unknots, Hopf link, Trefoil knot, Figure-eight knot, Connection with the Jones polynomial, Bracket connection, Reconstruction theorems
		Twist variable, Recovered field theory, Links in a solid torus, Satellites, Skein relation, Projectors, Borromean rings, Connected sums, Mutations
		Fundamental skein relation, Casimir operator, Composite states, Pattern links, Higer dimensional representations, Polynomial structure, SU(3) examples
	www.df.unipi.it /~guada/KNOT.html (194 words)

	AV #81380 - Video Cassette - Gague Theories and the Jones Polynomial (Site not responding. Last check: 2007-10-13)
		Edward Witten describes a three-dimensional quantum gage theory in this video.
		Knot invariants, the skein relation, Lagrangian quantum gauge theory, Hilbert spaces, the Chern Simons Invariant, Teichmuller space and Heegard splittings are covered.
		Last modified on August 17, 2006 by av@sfsu.edu
	www.sfsu.edu /~avitv/avcatalog/81380.htm (62 words)

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