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Topic: Knot invariants

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	PlanetMath: knot theory
		A knot diagram is a projection of a link onto a plane such that no more than two points of the link are projected to the same point on the plane and at each such point it is indicated which strand is closest to the plane (usually by erasing part of the lower strand).
		Knot theorists have accumulated a large number of knot invariants, values associated with a knot diagram which are unchanged when the diagram is modified by a Reidemeister move.
		Knot theorists also study ways in which a complex knot may be described in terms of simple pieces -- for example every knot is the connected sum of non trivial prime knots and many knots can be described simply using Conway notation.
	planetmath.org /encyclopedia/Knot.html (960 words)

	Knot theory - Wikipedia, the free encyclopedia
		Knot theory concerns itself with abstract properties of theoretical knots — the spatial arrangements that in principle could be assumed by a loop of string.
		Knot theory originated in an idea of Lord Kelvin's (1867), that atoms were knots of swirling vortices in the æther.
		Knot invariants can be defined by demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves.
	en.wikipedia.org /wiki/Knot_theory (1070 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 theory
		Knot theory concerns itself with abstract properties of theoretical knots--the spatial arrangements that in principle could be assumed by a loop of string.
		Knot theory originated in an idea of Lord Kelvin's (1867), that atoms were knots of swirling vortices in the æther;, and that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do (i.e.
		He did this by inventing the first knot invariant, demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves.
	www.brainyencyclopedia.com /encyclopedia/k/kn/knot_theory.html (628 words)

	The Proof
		Since the value for knot 4-1 is 1 and for knot 3-1 is 3, the value for knot (4-1)#(3-1) is 3.
		Since the value for knot 4-1 is 1 and for knot 3-1 is 5, the value for knot (3-1)#(3-1) is 5.
		The 2-1-1 invariants for the remaining unlinked unions are all equal to 116=6, while the 2-1-1 invariant for 8-25 is 10.
	www.inst.bnl.gov /~wei/link22.htm (892 words)

	Knot invariant -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-10)
		Some knot invariants are worked out from a knot diagram, in which case they must be unchanged (that is to say, invariant) under the (Click link for more info and facts about Reidemeister moves) Reidemeister moves; (Click link for more info and facts about knot polynomial) knot polynomials are examples of this.
		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 (Click link for more info and facts about unknot) unknot from all other knots.
		Some invariants associated with the knot complement include the (Click link for more info and facts about knot group) knot group which is just the (Click link for more info and facts about fundamental group) fundamental group of the complement.
	www.absoluteastronomy.com /encyclopedia/k/kn/knot_invariant.htm (453 words)

	Untitled Document (Site not responding. Last check: 2007-10-10)
		The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of state-sums.
		"Extensions of Quandles and cocycle knot invariants," (with Marina A. Nikiforou, J.Scott Carter, and Masahico Saito), J. Knot Theory and its Ramifications, vol 12, n 6 (2003), pp 725-738.
		Abstract: Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña.
	www.math.usf.edu /~emohamed/articles.htm (613 words)

	Knots and Links (Site not responding. Last check: 2007-10-10)
		Some say he struck it with his sword, cut the knot, and said it was now untied-but Aristobulus says that he took out the pole-pin, a bolt driven right through the pole, holding the knot together, and so removed the yoke from the pole.
		Knots were not treated as mathematical objects until the eighteenth and nineteenth centuries, when mathematicians including Alexandre-Theophile Vandermonde (1735-1796) and Carl Friedrich Gauss (1777-1855) introduced knots as subjects for a "geometry of position" (a concept first proposed by Leibniz in 1679).
		Composite knots are not given minimal crossing or unknotting numbers, as they can be better described in terms of their component, or prime, knots.
	www.blight.com /~cr/knots.html (1505 words)

	KNOTS
		The knot theorist’s usual convention for preventing this is to assume that the knot is formed in a closed loop of string.
		The Alexander polynomial is an algebraic modulus for the knot.
		The knot K* obtained by reversing all the crossings of K is called the mirror image of K. K* is the mirror image of the knot that would ensue if the plane on which the knot is drawn were a mirror.
	www2.math.uic.edu /~kauffman/Tots/Knots.htm (16146 words)

	Knottedness of Compact Lattice Chains
		Each invariant is guaranteed not to change value or form as the knot is deformed without cutting or letting the strands pass through each other (these 'destructive' operations may change the knot type).
		Once the knot invariants for a compact lattice chain are determined, that chain may be assigned a knot type.
		The trivial knot (unknot) is equivalent to a circle.
	webusers.physics.umn.edu /~rlua/knots/paper (3394 words)

	Theses from Uppsala University : 90 - Invariants of knot diagrams and diagrammatic knot invariants
		In this dissertation classes of knot diagrams, Reidemeister moves, and relations between sequences of Reidemeister moves are investigated, in a manner inspired by V. Arnold's theory of plane curves.
		The local knot diagram invariants are classified, and the concept of knot diagram invariants of nite degree is introduced.
		Invariants of knot diagrams and diagrammatic knot invariants.
	publications.uu.se /theses/abstract.xsql?lang=en&isbn=91-506-1462-2 (398 words)

	It:knots - ChemWiki
		The knot theory is a well established part of topology, and from such a viewpoint, representing a particular knot or link with a rope, or necklace-like string with beads has become entirely unimportant(shown on the right).
		It is believed that the curvature in molecular knotting directs the interaction of intracellular filaments and their associated proteins and ligands.
		Knotting can also be used as a tool to probe the mechanics of biological phenomena on the molecular level.
	www.ch.ic.ac.uk /wiki/index.php/It:knots (811 words)

	University of Oregon: Department of Mathematics
		In this paper we give a new interpretation of the simplest finite-type invariant of knots in terms of counting collinearities of four points on the knot.
		We show that this invariant is the only knot invariant which arises in the Goodwillie tower for classical knots through degree three.
		Conant and I have shown that all knot invariants obtained from the embedding calculus are of finite type, and we are presently writing up that result.
	www.uoregon.edu /~dps/knotspaces.php (709 words)

	Amazon.ca: Books: Quantum Invariants: A Study of Knot, 3-Manifolds, and Their Sets (Site not responding. Last check: 2007-10-10)
		Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e.
		With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized.
		Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed.
	www.amazon.ca /exec/obidos/ASIN/9810246757 (298 words)

	Virtual Knots
		A knot diagram is a projection or shadow of a knot on the plane, with the restriction that each point on the diagram is the shadow of no more than two points on the knot.
		Moving a knot around in space does not change what kind of knot it is -- in order to get a different knot, we have to cut the strand, tie a different knot in it, and rejoin the ends.
		The usual technique that knot theorists use is to devise knot invariants, which are numerical or algebraic quantities associated to particular knot diagrams in such a way that doing Reidemeister moves on the diagram does not change the numerical quantity.
	www.math.lsu.edu /~nelson/vknots.html (2679 words)

	Knot Invariant -- from Wolfram MathWorld
		A knot invariant is a function from the set of all knots to any other set such that the function does not change as the knot is changed (up to isotopy).
		In other words, a knot invariant always assigns the same value to equivalent knots (although different knots may have the same knot invariant).
		Standard knot invariants include the fundamental group of the knot complement, numerical knot invariants (such as Vassiliev invariants), polynomial invariants (knot polynomials such as the Alexander polynomial, Jones polynomial, Kauffman polynomial F, and Kauffman polynomial X), and torsion invariants (such as the torsion number).
	mathworld.wolfram.com /KnotInvariant.html (144 words)

	KNOTS
		The knot theorist’s usual convention for preventing this is to assume that the knot is formed in a closed loop of string.
		The Alexander polynomial is an algebraic modulus for the knot.
		The knot K* obtained by reversing all the crossings of K is called the mirror image of K. K* is the mirror image of the knot that would ensue if the plane on which the knot is drawn were a mirror.
	www.math.uic.edu /~kauffman/Tots/Knots.htm (16146 words)

	Abstract (Site not responding. Last check: 2007-10-10)
		ABSTRACT: A knot is an embedding of a circle in 3-space.
		A knot invariant is an algebraic object corresponding to the knot that doesn't change when the knot is isotoped into its various equivalent forms.
		The polynomial and classical numerical invariants are easy to compute, but incomplete, while the completeness of the Vassiliev invariant is unknown.
	infohost.nmt.edu /~iavramid/seminar/100303abs.html (107 words)

	NEW KNOT TABLES
		All knot tables are followed by the corresponding polynomial knot invariants: Alexander polynomials, Jones polynomials [8], Laurent polynomials [7], and data about some other knot invariants and properties - hyperbolic volumes [8], signatures [6,9], unknotting numbers [9], chirality and invertibility [6,9], symmetry groups of knots [9], etc.).
		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.
	members.tripod.com /vismath7/knotab (1879 words)

	Publications
		The contents of the appendix are now found here and the results about fibered knots are proved in [17].
		of Knot Theory and its Ramifications, Vol 7 (2), 173-185 (1998).
		of Knot Theory and its Ramifications, Vol 2 (4), 431-451 (1993).
	www.math.msu.edu /~kalfagia/publications.htm (220 words)

	Abstract (Site not responding. Last check: 2007-10-10)
		Each knot in 3-space gives rise to a family of knots in the solid torus.
		We associate to each such knot a loop in the space M of all knots in the solid torus.
		The new invariants, called character invariants, are no longer of finite type.
	www.math.columbia.edu /~ikofman/abstracts/fiedler.html (139 words)

	Read This: Knots: Mathematics with a Twist (Site not responding. Last check: 2007-10-10)
		Reidemeister (1928) shows that all knot deformations can be captured by three types of moves in the knot's planar projection.
		For instance, in knot diagrams it matters when one strand goes over rather than under another (indeed, that is the idea behind Vassiliev invariants!), and you might enjoy finding the crossing errors in diagrams on pages xi, 88 and 92.
		Although Knots expects a mathematically active reader, it is meant to be an interest-piquing and thought-provoking overview of elegant ideas in knot theory, rather than a text.
	www.maa.org /reviews/knotstwist.html (650 words)

	Modifying Knot Pages - Knot Atlas (Site not responding. Last check: 2007-10-10)
		At the moment there are two Knot Atlas experts and they do "bulk editing" (adding invariants or pictures to all knots at the same time) in two different ways (though the first of those is now preferred).
		The Jones polynomial of the knot [[7_5]] is {{Data:7_5/Jones_Polynomial}}.
		Each knot page calls a master template, assigning value to lots of parameters, and these values are the values of the various "invariants" of that knot, including honest knot-theoretic invariants, names of image files and even a long text field that renders a simulated Mathematica (http://www.wolfram.com) session.
	katlas.math.toronto.edu /wiki/Modifying_Knot_Pages (1153 words)

	Dror Bar-Natan: Classes: 2001-02: Knot Theory Seminar
		The fundamental group of a knot complement is an extremely strong invariant of the knot, it's easy to compute, but...
		These are knot invariants that behave as if they are polynomials on the the space of all knots.
		The theorem stating that every knot is the closure of a braid and Markov's theorem, a complete description of knots in terms of braids.
	www.math.toronto.edu /~drorbn/classes/0102/KnotTheory (462 words)

	Open Directory - Science: Math: Topology: Knot Theory
		A Circular History of Knot Theory - Starting with the flawed theory of Kelvin's knotted vortex to the work of Thurston, Jones and Witten, knot theory has circled back to its ancestral origins of theoretical physics.
		Knot Plot - A collection of knots and links, viewed from a (mostly) mathematical perspective.
		Knot Theory Invariants: The HOMFLY Polynomial - A brief article on the HOMFLY polynomial and how it is calculated.
	dmoz.org /Science/Math/Topology/Knot_Theory (668 words)

	Undergraduates study knot invariants as functions: What understandings are revealed ? (Site not responding. Last check: 2007-10-10)
		Mathematicians agree that the construct of function is a fundamental and unifying concept in mathematics.
		The curriculum made explicit connections between the term "invariant" and the mathematical construct of function.
		This study indicates a wide variation among students in the depth of understanding of functions and in their flexibility to move from their experience with real functions to a context in which function provides a framework for the study of advanced mathematics.
	www.math.sunysb.edu /~calendar/event.php?ID=381&Date=2004-01-28 (166 words)

	ARCC Workshop: Moduli spaces of knots (Site not responding. Last check: 2007-10-10)
		Pioneered by Vassiliev, this led to the study of finite-type knot invariants.
		Three-manifold techniques: In dimension three it is feasible to study the space of embeddings of a knot by comparing the space of diffeomorphisms of the ambient manifold with the space of diffeomorphisms of the knot’s complement.
		Such fundamental questions include characterizing the homology of the space of long knots in Euclidean space as a Poisson algebra, as well as giving new constructions and addressing the issue of completeness in the theory of finite-type knot invariants.
	www.aimath.org /ARCC/workshops/spaceofknots.html (395 words)

	Open Questions: Knot Theory
		Certain kinds of systems known as "fractional quantum Hall fluids" may be able to solve equations in the mathematical theory of braids, which could make possible the computation of the knot invariants known as "Jones polynomials", and consequently be able to solve manhy other hard problems as well.
		There are three themes: (1) Trying to understand the topological meaning of the new invariants, (2) The central role of braid theory in the subject, (3) Unifying principles provided by representations of simple Lie algebras and their universal enveloping algebras.
		The main focus is on the matematical description of knots, and especially the various invariants.
	www.openquestions.com /oq-ma012.htm (612 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		The definition of quantum Knot invariant associated to a compact semisimple Lie group and a finite dimensional representation of it can be done either by considering the corresponding DrinfelD-Jimbo algebra with its $R$-matrix and ribbon element, or via the Kontsevich integral.
		A formal Reshetikhin Turaev evaluation of knot invariants can be made in the first case for we know how the $R$-matrix and ribbon elements act.
		The second yields a family of knot invariants with values in $C[[h]]$ related with the finite dimensional ones.
	www.maths.qmw.ac.uk /~majid/maabs.html (244 words)

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