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	Knot polynomial - Wikipedia, the free encyclopedia
		In knot theory, a knot polynomial is a polynomial whose coefficients encode some of the properties of a given knot.
		Generally, such a polynomial is not meant to be evaluated as a function, but instead is used as a means of differentiating between knots that are not equivalent.
		Technically, an Alexander polynomial is a generator of a principal Alexander ideal related to the homology of the infinitely cyclic cover of a knot complement.
	en.wikipedia.org /wiki/Knot_polynomial (1202 words)

	Knot invariant - Wikipedia, the free encyclopedia
		Some knot invariants are worked out from a knot diagram, in which case they must be unchanged (that is to say, invariant) under the Reidemeister moves; knot polynomials are examples of this.
		The complement of a knot itself (as a topological space) is known to be a complete invariant of the knot, meaning that it distinguishes the given knot from all other knots up to isotopy.
		This is a homology theory whose Euler characteristic is the Alexander polynomial of the knot.
	en.wikipedia.org /wiki/Knot_invariant (387 words)

	Knot polynomial -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Technically, an Alexander polynomial is a generator 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).
		HOMFLYPT is a binary (two-variable) polynomial, with as with the predecessors.
	www.absoluteastronomy.com /encyclopedia/k/kn/knot_polynomial.htm (1366 words)

	Skein relation - Wikipedia, the free encyclopedia
		Skein relations occur in knot theory, where they are most often used to give a simple definition of a knot polynomial.
		For some knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively.
		Depending on the knot polynomial in question, the links (or tangles) appearing in a skein relation may be oriented or unoriented.
	en.wikipedia.org /wiki/Skein_relation (538 words)

	Authors and papers citing work by Lee Rudolph
		Transversal torus knots, Geometry and Topology 3 (1998), 253-268.
		On a conjecture by Kidwell and Stoimenow, preprint (2005)
		Knot Floer homology and the four-ball genus, Geometry and Topology 7 (2003), 615-639.
	black.clarku.edu /~lrudolph/research/citation.html (2336 words)

	Knot Theory
		Knots whose ends were glued together and their classification form the subject of a branch of Topology known as the Knot Theory.
		However, it must be noted that the two knots are topologically equivalent in the sense that there exists a topological transformation that maps one into another.
		In the Knot Theory until 1984 the main tool to tell the knots apart was the Alexander polynomials so named after the American mathematician J.W.Alexander.
	www.cut-the-knot.org /do_you_know/knots.shtml (709 words)

	really bad knot theory puns (Site not responding. Last check: 2007-10-21)
		We must define a knot, because if we do not, then we do not know what is a knot and what is not a knot.
		Knots are fun, and you would think so too if you had Emily and sarah-marie for this mini.
		Of course we were left with the evil task of showing that not all knots are the not knots.
	cerebro.xu.edu /~smbelcas/knotpuns.html (460 words)

	[No title]
		"A planar classification of pretzel knots and Montesinos knots", Pre'publications Orsay, 1980.
		"On topological knots and knot cobordism", Topology, 12 (1973), 33-40.
		Durfee, A. "The characteristic polynomial of the mono- dromy", Pacific J. Math., 59 (1975), 21-26.
	www.maths.gla.ac.uk /~ajb/btop/knotsbib.txt (9944 words)

	The Knot Table (Site not responding. Last check: 2007-10-21)
		For each knot, the first line denotes its referrence number, starting from 0 for the trivial knot, and ending to 2977 for the last 12 crossing knot.
		Once all knots have thus been presented, we continue by showing how knots with identical polynomials are distinguished.
		Knots here are presented through their referrence numbers, which were given before.
	www.inst.bnl.gov /~wei/knottable.html (250 words)

	NEW KNOT TABLES (Site not responding. Last check: 2007-10-21)
		After Redmeister [4], all knot tables that can be found in knot theory books are simple copies of the first: sometimes, some projection is slightly changed, or turned upside down, and that's all.
		A prime knot or link with singular digons, expressed by a Conway symbol, is called generating, and a knot or link without digons is called a basic polyhedron [14,16,17].
		Because the complete concept of new knot tables is based on the notion of generating knots and links and families originating from them, one of the possible future aims can be a search for new knot and link invariants that will be the invariants of families.
	www.mi.sanu.ac.yu /~jablans/knotab (1879 words)

	Knotted Word Worms
		For my first attempt at creating knotted text, I chose to employ a different method, which was to draw a closed and knotted path that I wanted to trace out, and then try and piece together words whose worm follows the path as closely as possible.
		It is known that the "stick number" of the trefoil knot (minimum number of sticks required to construct it) is six, but it is not clear if one can argue from this fact and achieve the desired conclusion.
		This suffices to make any knot using a series of words; with a bit more effort it should always be possible to make sentences or poems as well, utilizing the fact that a number of words are available for each direction.
	users.aol.com /s6sj7gt/knotted.htm (1914 words)

	California State University Chico - NREUP Programs
		Investigating which features of the polynomial come from the topology of the knot and which are algebraic is an important open question that the students could attack effectively.
		Mattman discovered a family of polynomials that, in all likelihood, are the trace polynomials for the (-2, 3, n) pretzel knots.
		However, for a knot of n crossings, this involves beginning with a system of n polynomial equations in n+1 indeterminates and using elimination theory to reduce to a single equation in two unknowns.
	www.maa.org /nreup/csuc.html (384 words)

	NDG2. Polynomial Invariants in Knot Theory (N.D.Gilbert) (Site not responding. Last check: 2007-10-21)
		These are easy to calculate from an interlacing pattern, and yet can be used to understand some of the profound problems about knots that have influenced the subject from the beginning.
		The project will look at the definition and calculation of some of these knot polynomials, and investigate how they can be used as a means of classification and problem solving.
		A further possible direction is to look at the interaction between these recent ideas in knot theory and some aspects of quantum theory.
	www.ma.hw.ac.uk /~andrewl/F14PA2/list/node15.html (128 words)

	PolynomialSplineFunction xref
		The polynomials are assumed to 30 * have been computed to match the values of another function at the knot 31 * points.
		The value consistency constraints are not currently enforced by 32 * `PolynomialSplineFunction` itself, but are assumed to hold among 33 * the polynomials and knot points passed to the constructor.
		If `x` is less than the smallest knot point or greater 47 * than the largest one, an `IllegalArgumentException` 48 * is thrown. 49 * Let `j` be the index of the largest knot point that is less 50 * than or equal to `x`.
	jakarta.apache.org /commons/math/xref/org/apache/commons/math/analysis/PolynomialSplineFunction.html (813 words)

	knots
		I borrowed A Survey of Knot Theory (by Akio Kawauchi) and Knot Theory and its Applications (by Kunio Murasugi) from the local college library this weekend, and learned a lot.
		Pictures of all of these knots can be seen in the CRC Encyclopedia of Mathematics under the knot listing.
		It would be interesting for a computer to crank through all 2-bridge knots with bridges of length 1000 or less, calculate the Alexander, Jones, and HOMFLY polynomials for each, and sort through them for equivalencies.
	www.mathpuzzle.com /knots.html (418 words)

	Intro Page: Knot Theory (Site not responding. Last check: 2007-10-21)
		Knot theory is a relatively young mathematical field, which means there are still many open questions to explore.
		This course will be an introduction to the theory of knots where we will explore both the theory of knots as well as look at some of the open problems.
		Scientists are applying knot theory to the study of DNA, molecular structure, and statistical mechanics.
	it.stlawu.edu /~mbos/knot/Intro_knot.html (178 words)

	Jozef Przytycki's publications (Site not responding. Last check: 2007-10-21)
		Knots: combinatorial approach to the knot theory, Script, Warsaw, August 1995, (in Polish, English translation (extended) in preparation; to be published by Cambridge University Press).
		Knot polynomials and generalized mutation (with R.P.Anstee and D.Rolfsen) Topology and its appl.
		Knots and links, revisited, (with J.Kania-Bartoszy\'nska) Delta, Warsaw, June 1985, 10-12, in Polish (expository article on generalizations of the Jones polynomial).
	home.gwu.edu /~przytyck/publications (1573 words)

	The Closed Self-Avoiding Walk (SAW) Notation (Site not responding. Last check: 2007-10-21)
		Since the topological type of a knot is extremely sensitive to the addition or removal of any crossing, this method can easily yield inaccurate results.
		On a more practical level, classifying knots through the SAW notation is much more time consuming and less efficient than through regular projections.
		While for instance the first two non-trivial knots possess 3 and 4 crossings respectively, their minimal lengths as closed Self-Avoiding Walks are 24 and 30 respectively.
	www.inst.bnl.gov /~wei/morecube.html (573 words)

	References beginning with M
		Invariants of knots and 3-manifolds (Kyoto 2001), Geometry and Topology Monographs 4 201-214,
		Invariants of knots and 3-manifolds (Kyoto 2001), Geometry and Topology Monographs 4 215-233,
		J. Murakami, The Casson invariant for a knot in a 3-manifold, in Proc.
	www.math.toronto.edu /~drorbn/VasBib/References_beginning_with_M.html (806 words)

	Geometric topology Article, Geometrictopology Information (Site not responding. Last check: 2007-10-21)
		In mathematics, geometric topology is the study of manifolds and their embeddings, withrepresentative topics being knot theory and braid groups.
		The solution by Smale of the Poincaré conjecture in higher dimensionschanged matters somewhat, in that it made dimensions three and four seem the hardest; and indeed they required new methods, whilethe freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory.
		A number of advances, on the Poincaré conjecture and the Thurston conjecture programme, and on knotinvariants such as new knot polynomials, had the effect ofchanging geometric topology.
	www.anoca.org /theory/conjecture/geometric_topology.html (279 words)

	Knot Theory (Site not responding. Last check: 2007-10-21)
		Knot theory is the study of knotted loops in three dimensional space (or more simply: pieces of string with their ends stuck together).
		I studied knot theory in summers of 95 and 96 with Prof.
		Scheme that calculates some knot polynomials (HOMFLY, Kauffman, Jones, Alexander) and a presentation of the fundamental group.
	www.cis.upenn.edu /~mmcdouga/knot.html (135 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		(Only "overhand" knots are possible with six segments.) I'm posting this to two other groups.
		The computation requires determining which of the 15 pairs of line segments have crossed projections in the xy plane, and for each such pair determining which passes "over" and which passes "under".
		So one may decide algebraically whether the six segments are knotted.
	www.math.niu.edu /~rusin/known-math/01_incoming/PL_knots (109 words)

	Table of Contents for Reidemeister
		The first one is an introduction to knots and braids, including (a sketch of) the original proof that two knots are equivalent if and only if their projections are related by a finite sequence of the three so-called Reidemeister moves.
		The third chapter deals with knot groups: definition by generators and relations from a projection, invariance, equivalence with the fundamental group of the knot complement, calculation of the group of special families of knots.
		This landmark in the history of knot theory should have a choice place on the shelves of all knot theorists.
	www.harbornet.com /bcsassociates/toc_rei.html (337 words)

	Fall 2001 Seminar Series
		A mathematical knot is a loop in three space which has been allowed to weave around itself, like a shoelace with the ends sealed after a knot has been tied.
		Such knot polynomials have been heavily studied in the past twenty years.
		Their most important use is as a knot invariant - a way to tell knots apart.
	www.gvsu.edu /math/mcdaniel_9.02.htm (121 words)

	Everywhere 1-Trivial Knot Projections (ResearchIndex)
		A knot diagram is called everywhere n-trivial, if it turns into an unknot diagram by switching any set of n of its crossings.
		We show several partial cases of the conjecture that the knots with everywhere 1-trivial knot diagrams are exactly the trivial, trefoil and figure eight knots.
		66 Knot polynomials and Vassiliev's invariants (context) - Birman, Lin - 1993
	citeseer.ist.psu.edu /askitas00everywhere.html (429 words)

	knots (Site not responding. Last check: 2007-10-21)
		A knot is defined as a continuous one to one function from the circle into the 3-sphere.
		Here, one to one implies that a knot is not allowed to have a self-intersection points.
		Two knots are called ambient isotopic iff one may contiunously pass from one knot to the other.
	home.anadolu.edu.tr /~adeniz/knots.html (73 words)

	Topological Quantum Field Theory (291) graduate course, Fall 2004 (Site not responding. Last check: 2007-10-21)
		The subject began in 1984 with Jones' discovery of his famous polynomial invariant of knots in the three-sphere.
		This discovery proved to be the tip of the iceberg: several other knot related knot polynomials (HOMFLY, Kauffman,...) were quickly discovered, and in 1989 Witten explained how this family of invariants should extend naturally to give lots of invariants for three-manifolds.
		Problems on invariants of knots and 3-manifolds edited by Ohtsuki.
	math.ucsd.edu /~justin/TQFT.html (402 words)

	Research Page (Site not responding. Last check: 2007-10-21)
		Despite the misleading title, “Why knot polynomials,” the purpose of this talk is not to justify polynomials, but a chance to share an unexpected application of algebra in the branch of topology known as knot theory.
		A knot is defined as a closed curve in three-dimensional space.
		The simplest example of a knot, a single unknotted loop, is called an unknot.
	www.uwosh.edu /faculty_staff/pricek/Talks_files/Knots.html (198 words)

	Summer 2001 REU Participants
		I then showed that the property of n-coloration of a knot is also a basic property of the knot group, i.e.
		My project was to write a program that allowed a user to draw a knot with a mouse or other pointing device, and determine from the drawing if that knot could be reduced to the "UN-knot." This was written in Java and is available from my web site at http://www.cs.utah.edu/~rpalmer/knot/knot.jar.
		The paper describes how the colorability of a knot/link is determined via the crossing matrix of a knot, using special matrix forms in linear algebra; along with important consequences/properties of prime knots, composite knots, and links relating to nullity and multicolorability.
	www.math.utah.edu /reu/summer01reu.html (1671 words)

	Knot Polynomials: A state-space representation of HOMFLY (Site not responding. Last check: 2007-10-21)
		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)

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