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Topic: HOMFLY polynomial

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Arthur Guinness

	Knot Theory Invariants: The HOMFLY Polynomial
		The publication of the Jones Polynomial excited the mathematical community to the point that new polynomial invariants were being created at a stupendous rate.
		One of the objectives of the time was to find a polynomial that generalized both the Alexander and Jones polynomial.
		To calculate the polynomial, a crossing is selected and the skein relation is solved for the polynomial given that type of crossing.
	library.thinkquest.org /12295/data/Invariants/Articles/HOMFLY.html (284 words)

	The Dispatch - Serving the Lexington, NC - News
		The derivative of a polynomial is a polynomial
		The integral of a polynomial is a polynomial
		In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=polynomial (2721 words)

	Polynomial Summary
		For example, Z[x] is the subring of polynomials with coefficients in the ring Z of integers, Q[x] is the subring of polynomials with coefficients in the field Q of rational numbers, and ℜ[x] is the subring of polynomials with coefficients in the field ℜ of real numbers.
		Usually a polynomial function is simply called a polynomial, and it is clear from the context if the polynomial is to be regarded as a function or as an element of a ring of polynomials.
		In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power).
	www.bookrags.com /Polynomial (3406 words)

	Stephen Bigelow (Site not responding. Last check: 2007-10-12)
		This does for the HOMFLY polynomial what "A homological definition of the Jones polynomial" did for the Jones polynomial.
		Mostly a survey paper, including why the HOMFLY polynomial is the unique Markov trace of the Iwahori-Hecke algebra, and open problems.
		Obtains the Jones polynomial as an algebraic intersection pairing between certain elements of singular homology of a configuration space.
	www.math.ucsb.edu /~bigelow/papers.html (379 words)

	Abstracts of some of my papers
		We classify all knot diagrams of genus two and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic volume for weak genus two knots.
		As a consequence we show that the number of almost positive knots of given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, \em{inter alia} to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots.
		Using degree and coefficient growth properties of the Conway polynomial, further applications to Legendrian knots and the Casson-Walker invariant are given.
	www.kurims.kyoto-u.ac.jp /~stoimeno/abstracts.html (1858 words)

	Knot Polynomials: A state-space representation of HOMFLY (Site not responding. Last check: 2007-10-12)
		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)

	The Dispatch - Serving the Lexington, NC - News
		In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or 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.
		It generalizes both the Alexander polynomial and the Jones polynomial both of which can be obtained by appropriate substitutions from HOMFLY.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=HOMFLY_polynomial (163 words)

	Research Oriented Knot Theory Topology Math Science
		- Research paper on a state-space representation of the HOMFLY polynomial, by B. BollobÃ¡s, L. Pebody and D. Weinreich.
		The Jones polynomial is shown to give rise to physical states of quantum gravity.
		The number of different Alexander, Homfly and Jones polynomials for knots of up to 15 crossings is given.
	www.iaswww.com /ODP/Science/Math/Topology/Knot_Theory/Research_Oriented (166 words)

	A Dynamic Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links
		The HOMFLY polynomial invariant for directed knots and links is a powerful tool for distinguishing knots and links.
		The HOMFLY knot polynomial is a polynomial with variables M and L. The HOMFLY formula relates three links which differ by weaves containing a single crossing.
		Another apparent property (which the author has not been able to prove or cite) is that every link's HOMFLY polynomial evaluates to some power of -2 when M and L are assigned the value 1.
	burtleburtle.net /bob/knot/thesis.html (4206 words)

	Knot Table: Homfly polynomial (Site not responding. Last check: 2007-10-12)
		We follow some historical conventions: the Homfly polynomial is given in variables v and z (Hugh Morton), while the Kauffman polynomial is given in variables a and z (Kauffman).
		Any polynomial with two variables can be thought of as a polynomial in one variable with coefficients that are polynomial in the other variable.
		The Homfly and Kauffman polynomials are given as semicolon-delimited list of coefficients with respect to the variable z
	www.indiana.edu /~knotinfo/descriptions/homfly_polynomial.html (141 words)

	Knot data tables
		For the Jones and Alexander polynomial, if the absolute term occurs between its minimal and maximal degrees, then it is bracketed, else the minimal degree is recorded in braces before the coefficient list.
		Here is a list of all mutant groups (91 groups, consisting of 86 pairs and 5 triples) of prime knots through 12, and of 13 crossings (774 groups, consisting of 703 pairs, 38 triples, 32 groups of 4 and one group of 6).
		If the degree of the Alexander polynomial coincides with half the degree of the Alexander variable in the skein polynomial, then genus is equal to canonical genus; there are 49 non-alternating 11 and 12 crossing knots where this is not the case.
	www.kurims.kyoto-u.ac.jp /~stoimeno/ptab/index.html (1213 words)

	Computing the HOMFLY polynomial (Site not responding. Last check: 2007-10-12)
		My master's thesis was A Dynamic Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links.
		If you want just the Jones polynomial, and you want it for an alternating link, there is a very similar but faster algorithm (with no code available).
		"Computing the Tutte Polynomial of a Graph and Jones Polynomial of a Knot of Moderate Size" by K. Sekine, H. Imai, and S. Tani.
	www.burtleburtle.net /bob/knot/homfly.html (256 words)

	► » HOMFLY polynomial coefficients sum to power of -2?
		HOMFLY polynomial coefficients sum to power of -2?
		Setting both variables to 1, the polynomial is defined as x=-(y+z),
		raise or lower the polynomial by a power of -2, that would explain it.
	www.science-chat.org /HOMFLY-polynomial-coefficients-sum-to-power-of-2-6669949.html (113 words)

	Atlas: Coefficients of Homfly polynomial and Kauffman polynomial are not finite type invariants by Gyo Taek Jin (Site not responding. Last check: 2007-10-12)
		Atlas: Coefficients of Homfly polynomial and Kauffman polynomial are not finite type invariants by Gyo Taek Jin
		We show that the integer-valued knot invariants appearing as the coefficients of the HOMFLY polynomial and the Kauffman polynomial are not of finite type.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caea-31.
	atlas-conferences.com /c/a/e/a/31.htm (101 words)

	Northwestern University Mathematical Calendar (Site not responding. Last check: 2007-10-12)
		The starting point is a formula for a quantum polynomial, which presents it as an alternating sum over (non-invariant) polynomials with positive coefficients.
		A similar formula for the SU(N) HOMFLY polynomial was provided in a paper by Murakami, Ohtsuki and Yamada.
		However, that latter formula was based on `resolving' a link into a set of special graphs rather than simple circles, and this complicated the guessing of an appropriate differential for the categorifying comples.
	www.math.northwestern.edu /news/calendar/abstract.cgi?id=1111428569&dyear=2005 (204 words)

	Northwestern University Mathematical Calendar (Site not responding. Last check: 2007-10-12)
		The whole construction is based on elementary manipulations with polynomial algebra.
		The HOMFLY-PT polynomial has an important specialization: if we set t=q^N, then the resulting 1-variable polynomial is related to the representation theory of the quantum group SU_q(N).
		This polynomial admits a similar categorification, if we modify the polynomial algebra underlying the Soergel's construction, and replace the Koszul complexes with Koszul matrix factorizations.
	www.math.northwestern.edu /news/calendar/abstract.cgi?id=1141935750&dyear=2006 (203 words)

	Atlas: Approximating the coefficients of the HOMFLY polynomial by Vassiliev invariants by Laure Helme-Guizon (Site not responding. Last check: 2007-10-12)
		Atlas: Approximating the coefficients of the HOMFLY polynomial by Vassiliev invariants by Laure Helme-Guizon
		Approximating the coefficients of the HOMFLY polynomial by Vassiliev invariants
		For instance, Y Rong and I. Kofman found approximations by Vassiliev invariants for the coefficients of Jones polynomial.
	atlas-conferences.com /c/a/j/k/05.htm (149 words)

	Matt Fredrikson
		The HOMFLY polynomial for knots and links is a very powerful invariant that is practical for common use.
		However, software to calculate the polynomial is either very outdated (cannot compute on complex knots) or simply unreliable.
		I am currently working on a solution that can compute the HOMFLY polynomial for arbitrarily large/complex knots and links and does so quickly enough to remain a practical solution for such knots.
	www.mathcs.duq.edu /~fa03fredrikson (515 words)

	Class math.topol.Braid.polys
		In 1991, Hugh Morton and Hamish Short gave an algorithm for computing the so-called HOMFLY polynomial (from which it is easy to obtain the Jones Polynomial, the classical Alexander polynomial, and the ``2-variable'' polynomial) from a braid presentation of a link.
		The given method has worst-case running time proportional to the square of the number of crossings of a braid, once a fixed-cost (depending only on the braid-index) initialization is made.
		Since it initializes an structures which are factorially large in the size of the input, it may consume significant computational resources.
	sunsite.ubc.ca /Djun/thesis/java/doc/math.topol.Braid.polys.html (234 words)

	AT&T Worldnet Service - Directory
		A research paper describing an algorithm for calculating the HOMFLY polynomial of directed knots and links.
		Research paper showing that any knot with trivial Jones polynomial must have at least 18 crossings.
		Research paper on a state-space representation of the HOMFLY polynomial, by B. Bollobás, L. Pebody and D. Weinreich.
	dailynews.att.net /cgi-bin/webdrill?catkey=gwd/Top/Science/Math/Topology/Knot_Theory/Research_Oriented (314 words)

	AMCA: Mahler measure of the Jones polynomial, part I by Abhijit Champanerkar
		The Mahler measure of the Alexander polynomial (Silver, Williams) and A-polynomial (Boyd, Rodriguez-Villegas) has been related to the volume of the knot complement.
		For torus knots, we obtain the explicit limit from the HOMFLY polynomial.
		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/m/w/11.htm (174 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		The HOMFLY polynomial is an invariant of knots and links which generalizes both the classical Alexander polynonmial as well as the Jones polynonmial.
		Unfortunately, because of marshal law in Poland, word of the discovery by Przytycki and Traczyk was slow in reaching the United States.
		Thus the acronym, HOMFLY, (coined by David Yetter) does not include the initials P and T.
	pzacad.pitzer.edu /~jhoste/HosteWebPages/homfly.html (127 words)

	Proceedings of the American Mathematical Society
		Abstract: It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the least degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp.
		Furthermore, the relationships between these restrictions on the range of the Bennequin invariant are investigated, which leads to a new simple proof of the inequality involving the Kauffman polynomial.
		Chmutov, S., Goryunov, V., Murakami, H. Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves, Math.
	www.mathaware.org /proc/2002-130-04/S0002-9939-01-06153-6/home.html (436 words)

	Salvador Vera: Directorio - Topología
		Tables of the number of prime alternating knots and an upper bound for the number of prime knots up to 17 crossings; the number of different Homfly, Jones and Alexander polynomials for knots up to 15 crossings.
		A Dynamic Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links A Dynamic Programming Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links
		Research interests are in geometric topology and the theory of low-dimensional manifolds: particularly knot theory, polynomial invariants of links, and quantum invariants of 3-manifolds derived, via skein theory, from link invariants.
	www.satd.uma.es /matap/svera/links/matnet15.html (626 words)

	Math arXiv: Search results (Site not responding. Last check: 2007-10-12)
		math.QA/0601267 On the Hecke algebras and the colored HOMFLY polynomial.
		math.QA/0601266 On the Hecke algebras and the colored HOMFLY polynomial.
		math.GT/0403448 A volume-ish theorem for the Jones polynomial of alternating knots.
	front.math.ucdavis.edu /author/Lin-Xiao-Song (279 words)

	Kauffman: Combinatorics and topology - François Jaeger's work in knot theory
		JAEGER, Circuit partitions and the Homfly polynomial of closed braids, Trans.
		JAEGER, On the Kauffman polynomial of planar matroids, Proceedings of the Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity, Elsevier,
		SALEUR, The Conway polynomial in ℝ3 and in thickened surfaces: a new determinant formulation, J. Comb.
	www.numdam.org /numdam-bin/item?id=AIF_1999__49_3_927_0 (550 words)

	7.9 The Kauffman Polynomial (Site not responding. Last check: 2007-10-12)
		Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.
		Thus, for example, here's the Kauffman polynomial of the knot
		It is well known that the Jones polynomial is related to the Kauffman polynomial via
	www.math.toronto.edu /~drorbn/KAtlas/Manual/Kauffman.html (87 words)

	Mathematical Research Letters :: a mathematics journal
		This is a class of infinitely many knots closed under taking mirror images.
		Our proof relies on a non-standard parameterization of the HOMFLY polynomial.
		Another interesting corollary of this parameterization is that if all Vassiliev invariants up to degree $c$ vanish on a knot of crossing number $c$, then this knot has trivial HOMFLY polynomial.
	www.mrlonline.org /mrl/2001-008-005/2001-008-005-004.html (117 words)

	CADwire.net - Directory > Science > Math > Topology > Knot Theory
		A Dynamic Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links - A research paper describing an algorithm for calculating the HOMFLY polynomial of directed knots and links.
		Knot Polynomials - Research paper on a state-space representation of the HOMFLY polynomial, by B. BollobÃ¡s, L. Pebody and D. Weinreich.
		Knots in Braid Notation - Every knot of up to 11 crossings in braid notation and every link up to 10 crossings, compiled by Adam Chalcraft.
	www.cadwire.net /directory/dir.asp?/Science/Math/Topology/Knot_Theory/Research_Oriented (282 words)

	7.8 The HOMFLY-PT Polynomial (Site not responding. Last check: 2007-10-12)
		HOMFLYPT[K][a, z] computes the HOMFLY-PT (Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki and Traczyk) polynomial of a knot/link K, in the variables a and z.
		The HOMFLYPT program was written by Scott Morrison.
		It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at
	www.math.toronto.edu /~drorbn/KAtlas/Manual/HOMFLYPT.html (97 words)

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