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Topic: Jones polynomial

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	The Jones Polynomial (Site not responding. Last check: 2007-10-05)
		The Jones polynomial,V(t),emerged from a study of finte dimensional von Neumann algebras.
		Thus the Jones polynomial can sometimes distinguish a knot from it's mirror image and so is distinct from the Alexander polynomial.
		Jones, V. R., "A new knot polynomial and von Neumann algebras", Bull.
	www.indiana.edu /~knotinfo/descriptions/jones_polynomial.html (116 words)

	Knot polynomial
		A knot polynomial is a particular knot invariant.
		Technically, an Alexander polynomial is a generato r of a principal Alexander ideal related to the homology of the infinitely cyclic cover of a knot complement —where all the emphasised phrases have particular mathematical meanings.
		(this is an Alexander polynomial of the knot).
	www.sciencedaily.com /encyclopedia/knot_polynomial (1098 words)

	Knot invariant - Wikipedia, the free encyclopedia
		These are currently the most useful invariants for distinguishing knots from one another, though at the time of writing it is not known whether any of these distinguishes all knots from each other or even just the unknot from all other knots.
		This is a homology theory whose Euler characteristic is the Alexander polynomial of the knot.
		Along a different line of study, there is a cohomology theory of knots, called Khovanov homology, whose Euler characteristic is the Jones polynomial.
	en.wikipedia.org /wiki/Knot_invariant (362 words)

	Jones_Vaughan (Site not responding. Last check: 2007-10-05)
		In 1980 Jones moved to the United States spending the academic year 1980-81 as E R Hedrick Assistant Professor of Mathematics at the University of California at Los Angeles.
		Jones worked on the Index Theorem for von Neumann algebras, continuing work begun by Connes and others.
		Jones gave a lecture to the 1990 Congress dressed in a rather unusual way for a mathematics lecture.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Jones_Vaughan.html (719 words)

	Knot polynomial - Enpsychlopedia (Site not responding. Last check: 2007-10-05)
		It is a method of representing knots where the coefficients of the polynomial are used to encode the properties of the knot.
		A polynomial representation of a knot is a mapping of the mathematical properties of knots to those of polynomials.
		HOMFLYPT is a binary (two-variable) polynomial, with $P_{\rm unknot}(x,y)=1 as with the predecessors.$
	www.grohol.com /psypsych/Jones_polynomial (1328 words)

	[No title]
		The Kauffman bracket polynomial of LL2(n) is equal to that of the 2-component unlink for all n.
		The Jones polynomial of LLL(0) is equal to that of the 3-component unlink if one chooses orientations of the blue and green strands so that the writhe of the diagram is 2, whereas for n >= 1 the Jones polynomial of LLL(n) is equal to that of the 3-component unlink unconditionally.
		For m >= 1 the Jones polynomial of LL1(m, n) is precisely equal to that of the unlink of two components.
	www.math.utk.edu /~morwen/ekt.html (592 words)

	Vaughan Jones
		Vaughan Jones is a Professor of Mathematics at the University of California at Berkeley and Distinguished Alumni Professor of the University of Auckland, and a Co-Director (with Marston Conder) of the NZ Institute of Mathematics and its Applications (NZIMA).
		Jones was studying subfactors when he discovered that, rather than having continuous dimensions the only dimensions less than 4 were 4cos2g/n.
		The original polynomial for knots derived by Alexander in 1928 fails to separate a left-handed trefoil from a right-handed one, showing that much remained undiscovered, since some of the simplest knots could not be distinguished.
	www.math.auckland.ac.nz /Careers/vaughan/vaughan.htm (1249 words)

	Joint Mathematics Colloquium (Site not responding. Last check: 2007-10-05)
		Abstract: The Jones polynomial of a knot in 3-space is a fascinating polynomial invariant that encodes the noncommutativity of the over-under crossings of a knot.
		Despite the numerous definitions, the geometry and topology of the Jones polynomial is tighly hidden.
		It is conjectured (and verified for the simplest knots) that this complex curve coincides with the curve of deformations of SL(2, C) representations of the knot complement, viewed from the boundary.
	www.math.neu.edu /bhmn/garoufalidis.html (181 words)

	KNOTS (Site not responding. Last check: 2007-10-05)
		The Alexander polynomial is an algebraic modulus for the knot.
		What made the Jones polynomial such an exciting discovery for knot theorists was the fact that it could detect the difference between many knots and their mirror images.
		The tree reduces the calculation of the Jones polynomial of the trefoil diagram to the calculation of certain unknots and unlinks.
	www.math.uic.edu /~kauffman/Tots/Knots.htm (16163 words)

	Jones Polynomial (Site not responding. Last check: 2007-10-05)
		Combined with the link sum relationship, this allows Jones polynomials to be built up from simple knots and links to more complicated ones.
		polynomials is given by Jones (1985) for knots of up to eight crossings, and by Jones (1987) for knots of up to 10 crossings.
		Prime Knots with 10 or fewer crossings have distinct Jones polynomials.
	www.math.sdu.edu.cn /mathency/math/j/j065.htm (347 words)

	Knots and their Polynomials-11 (Site not responding. Last check: 2007-10-05)
		Calculate the Jones polynomial of the Figure-Eight Knot.
		Show that when two knots are spliced together, the Jones polynomial of the splice is the product of the two Jones polynomials.
		Calculate the Jones polynomials of (Left-trefoil) splice (Left-trefoil) and (Right-trefoil) splice (Right-trefoil) = the two Granny knots.
	www.math.sunysb.edu /~tony/whatsnew/column/knots-0499/knots11.html (68 words)

	Energy for Knots
		But in the 1980s the subject was revolutionised by the discovery of the "Jones polynomial" by Vaughan Jones, a New Zealand mathematician.
		The Jones polynomial is superficially similar, but was reached by a very different route -via mathematical physics.
		In particular, the Jones polynomial of a reef knot is different from that of a granny knot, providing logically impeccable confirmation of what generations of boy scouts already know - that it does matter which one you use.
	www.fortunecity.com /emachines/e11/86/knotprob.html (1018 words)

	Math Trek: The Tangled Task of Distinguishing Knots, Science News Online, Feb. 22, 2003 (Site not responding. Last check: 2007-10-05)
		In the new scheme, the Jones polynomial for the unknot is 1.
		A systematic procedure (algorithm) allows the Jones polynomial to be computed for any knot, based on its pattern of crossings.
		Information on the Jones polynomial for knots is available at http://www.ams.org/new-in-math/cover/knots3.html and http://mathworld.wolfram.com/JonesPolynomial.html.
	www.sciencenews.org /20030222/mathtrek.asp (1127 words)

	The HOMFLYPT Polynomial (Site not responding. Last check: 2007-10-05)
		One may thus define a two-parameter invariant P(s,t), which is known as the homflypt polynomial from the initials of the mathematicians that discovered it.
		In addition to being more efficient than both the Alexander-Conway and the Jones polynomial in distinguishing inequivalent knots, the homflypt polynomial was the first known knot invariant to distinguish mirror symmetric knots.
		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 (481 words)

	Knotty Calculations: Science News Online, Feb. 22, 2003 (Site not responding. Last check: 2007-10-05)
		At the heart of the connection between computer science and quantum physics is a knot invariant called the Jones polynomial, which associates a given knot with an array of numbers.
		The Jones polynomial involves a complex mathematical formula, and although calculating it is easy for simple knots, it is enormously difficult for messy, tangled knots.
		Calculating the Jones polynomial for complicated knots is considered beyond the reach of even the fastest computers.
	www.sciencenews.org /articles/20030222/bob11.asp (2566 words)

	Members' Data (Site not responding. Last check: 2007-10-05)
		Vaughan Jones is elected to Honorary Membership of the Society in recognition of his profound achievements in the theory of von Neumann algebras and its applications.
		These include knot theory (which was completely revitalised by the introduction of the Jones polynomial), quantum field theory and statistical mechanics.
		The iterative construction of the Jones tower gives rise to a sequence of projections (the ‘Jones projections’) and an associated nested sequence of algebras whose generators satisfy the same relations as those of the braid group.
	www.lms.ac.uk /citation.html (414 words)

	Knots and Jones polynomial - Information Technology Services (Site not responding. Last check: 2007-10-05)
		The Jones polynomial is a topological invariant and key to a lot of knot theory.
		Witten was awarded the Fields medal for discovering a relation between the Jones polynomial and Quantum Field Theory.
		Jones studied oriented knots where there is an arrow along the rope.
	www.physicsforums.com /archive/forum/t-3751 (822 words)

	Citebase - Quantum Computing and the Jones Polynomial (Site not responding. Last check: 2007-10-05)
		This paper is an exploration of relationships between the Jones polynomial and quantum computing.
		We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group.
		We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0105255 (892 words)

	Paul Zinn-Justin's Alternating Virtual Link Database (Site not responding. Last check: 2007-10-05)
		An important modification is that the normalization of the Jones polynomials is now such that P(empty)=1.
		Jones polynomials for tangles have been similarly modified.
		Higher spin Jones polynomials are missing for tangles.
	ipnweb.in2p3.fr /~lptms/membres/pzinn/virtlinks (612 words)

	1
		Since these two polynomials are different we know their associated knots are different.
		The Alexander polynomial was remarkable for how successful it was in distinguishing the knots in Tait's orginal table and it gave witness to how thorough a researcher Tait was.
		(One of these polynomial was simultaneously discovered by six different mathematicians and its name is an acronym of their last names---HOMFLY.) Moreover, Jones' polynomial quickly led to the proofs that established all of Tait's original conjectures on knot projections.
	www.math.buffalo.edu /~menasco/Knottheory.html (1177 words)

	Linka Bet Top > Science > Math > Topology > Knot Theory > Research Oriented (Site not responding. Last check: 2007-10-05)
		Research paper on a state-space representation of the HOMFLY polynomial, by B. BollobÃ¡s, L. Pebody and D. Weinreich.
		The number of different Alexander, Homfly and Jones polynomials for knots of up to 15 crossings is given.
		The Jones polynomial is shown to give rise to physical states of quantum gravity.
	www.linkabet.com /odp/world_directory.php?CatID=5843421 (212 words)

	Citebase - Categorifications of the colored Jones polynomial (Site not responding. Last check: 2007-10-05)
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
		The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1.
		Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2...
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0302060 (799 words)

	Ilya Kofman (Site not responding. Last check: 2007-10-05)
		The ``volume conjecture'' and its variants propose that colored Jones polynomials, which are weighted sums of Jones polynomials of cablings, determine the volume of hyperbolic knots.
		Yet the Jones polynomial is still not understood in terms of the knot complement.
		My aim is to establish the relationship between the Jones polynomial of a knot and the geometry of the knot complement.
	www.math.columbia.edu /~ikofman (686 words)

	UC Berkeley Mathematics
		One of the ket questions is the relation between the Jones polynomial and the topology and geometry of the knot complement.
		The Hyperbolic Volume Conjecture states that a growth rate of the Jones polynomial of a knot and its parallels is proportional to the Volume of a hyperbolic knot complement.
		The characteristic variety is obtained from the Jones polynomial (and its parallels) via a q-diﬀerence equation, and its specialization at q = 1.
	math.berkeley.edu /calendar-event125.html (220 words)

	Knot Polynomials: A state-space representation of HOMFLY (Site not responding. Last check: 2007-10-05)
		Kauffman relaxed the conditions which are desirable for a link invariant to create the Kauffman polynomial, a polynomial defined explicitly in terms of the states of the crossings of the link.
		He then showed that this polynomial could be adapted to create a state-space definition of the Jones polynomial, a vast improvement over the original formulation.
		The form of the polynomial is sufficiently nice to enable a slightly streamlined and tight proof of the existence of the HOMFLY polynomial, avoiding the misstatements made in some of the other proofs.
	www.gettysburg.edu /~dweinrei/research/homfly.html (242 words)

	[No title] (Site not responding. Last check: 2007-10-05)
		Jones polynomial and Geometry of Knots Complements -- Abhijit Champanerkar, July 13, 2004
		Discovered 20 years ago, the Jones polynomial caused a revolution in knot theory.
		Although easily computable, it is still not understood in terms of the topology of the knot complement.
	www.math.columbia.edu /~ums/abstracts/Su04Champanerkar.html (71 words)

	A volume-ish theorem for the Jones polynomial of alternating knots (Site not responding. Last check: 2007-10-05)
		A volume-ish theorem for the Jones polynomial of alternating knots
		polynomial: The ratio of the Volume and certain sums of coefficients of the
		Jones polynomial is bounded from above and from below by constants.
	www.math.lsu.edu /~kasten/Papers/abs_VolumeIsh.html (102 words)

	NZMS Newsletter 37 Centrefold - Vaughan Jones (Site not responding. Last check: 2007-10-05)
		The trace, which in matrix notation corresponds to the sum of the diagonal entries, is all important in the development of the Jones polynomial.
		The announcement of the Jones polynomial led to another astounding piece of mathematical serendipity when eight mathematicians in five different groups independently and simultaneously produced a two variable generalization of both the Jones and Alexander polynomials which could even more sensitively distinguish links up to the two shown below.
		Because the Q polynomial relationship involved two uncrossed types, it is possible to use it to reduce the projection of a link to unknotted components.
	ifs.massey.ac.nz /mathnews/centrefolds/37/Aug1986.shtml (1441 words)

	Louis H. Kauffman (Site not responding. Last check: 2007-10-05)
		We shall describe the state summation model for the bracket polynomial and its relation to the original Jones polynomial.
		With this adjustment the bracket polynomial is invariant under the basically flat Reidemeister II and III moves and multiplies by -A^3 or -A^(-3) under a type I Reidemeister move.
		The KT and the C are both 11 crossing knots, with non-trivial Jones polynomial.
	www.math.uic.edu /~kauffman (1558 words)

	[No title]
		The smoothing that respects the orientation of the link $o$ is the smoothing that is used in the construction of the Jones polynomial.
		Next the Jones polynomial will be used as a tool to find the $N_i$, to do this we need to understand the links associated with $N-d$ and $N/d$.
		The polynomial $\Gamma _G$ is defined through examining the spanning trees of G and assigning the edges each a state (either part of a cycle or a cut, and as active or inactive), and to each state assigning a value~\cite{This}.
	www.math.ucla.edu /~daodonno/Danielle.txt (2120 words)

	Mailgate: sci.fractals: Re: New tiling form Jones Polynomial generalization (Site not responding. Last check: 2007-10-05)
		Your phrasing here is ambiguous; I don't know whether you're referring to the minimal Pisot number, or to the minimal polynomial of an arbitrary Pisot number.
		Anyway, the figure generated by the program isn't related to the minimal Pisot number, but is one of 4 2-element tiles with an areal contraction defined by the polynomial c + c^3 = 1.
		These are related to a different number, (~1.465571232 rather than ~1.324717957) which I wouldn't be surprised to find is a Pisot number.
	mailgate.supereva.it /sci/sci.fractals/msg02857.html (210 words)

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