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Topic: Knot complement

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complement set theory

complement system

Phonetic complements

relative complement

complement good

Method of complements

Schur complement

Relatively complemented lattice

complement music

Complement set email filtering

complement mathematics

complement fixation

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	NationMaster - Encyclopedia: Knot invariant (Site not responding. Last check: 2007-10-05)
		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.
		In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere.
		This is a homology theory whose Euler characteristic is the Alexander polynomial of the knot.
	www.nationmaster.com /encyclopedia/Knot-invariant (875 words)

	Science Fair Projects - Knot complement
		Note that the knot complement is a compact 3-manifold with boundary homeomorphic to a torus.
		Sometimes "knot complement" means the complement in the 3-sphere of a knot (whether tame or not), in which case the knot complement is not compact.
		Many knot invariants, such as the knot group, are really invariants of the complement of the knot.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Knot_complement (351 words)

	[No title] (Site not responding. Last check: 2007-10-05)
		The homeomorphism problem for knot complements, and in fact for all "Haken 3-manifolds" was solved by Haken and Hemion over 20 years ago.
		The issue of whether Kontsevich's knot invariants are the "best" or not is also debatable, but in any case, they are very closely related to the Vassiliev invariants, which from the point of view of distinguishing knots, are enormously easier to compute.
		BTW, "Kontsevich" is an anagram of "knot chives".
	www.math.niu.edu /Papers/Rusin/known-math/98/knots (289 words)

	[No title] (Site not responding. Last check: 2007-10-05)
		A reference for the de Rham theory approach to knot spaces is math.GT/9910139, where the authors define a map from complexes of Feynman diagrams to the de Rham complex of a knot space.
		He finishes determining the homotopy type of the space of long knots in dimension three in math.GT/0506524, using a detailed understanding of the JSJ decomposition of a knot complement.
		A nice introductory paper to the connection between knot spaces and calculus of the embedding functor is D. Sinha, "Topology of spaces of knots", available at math.AT/0202287.
	www.aimath.org /WWN/spaceofknots (460 words)

	Definition of Knot (mathematics)
		This is basically equivalent to a conventional knot with the ends of the string joined together to prevent it from becoming undone.
		The simplest nontrivial knots are the trefoil knot and the figure-eight knot.
		Since a knot itself is just a tangled piece of metaphorical circular twine, the interesting part about studying it is looking at how its tangling affects the shape of the space it is embedded in.
	www.wordiq.com /definition/Knot_%28mathematics%29 (333 words)

	Clay Mathematics Institute
		The figureight knot is presented as an Esker curve winding with no self-intersections on a double torus.
		The complement of the figureight knot has the structure of a double quotient group, one of the groups being discrete.
		The mathematical object which the sculpture represents is the orbifold X given as a quotient of three-dimensional hyperbolic space by a discrete group action, as described by the following equations, permanently inscribed on the larger granite sculpture:
	www.claymath.org /helaman (197 words)

	[No title]
		Now, the group presentations which result from a knot projection do not include all group presentations, so it is possible that one _could_ develop a procedure to decide unfailingly whether or not the corresponding fundamental group is trivial (or perhaps whether two of them are isomorphic).
		It is true (I think) that a string is unknotted iff its knot group is Z (the simplest possible), but it's also true that there are pairs of knots with the same groups but which are really _distinct_ knots.
		The algorithm is based on triangulating the exterior of the knot, and showing that a compressing disk for the knot exterior (if it were an unknot) would necessarily intersect the triangulation in specific way (it is a normal surface).
	www.math.niu.edu /Papers/Rusin/known-math/96/knothy (576 words)

	Knot Table: A-Polynomial (Site not responding. Last check: 2007-10-05)
		Thus, if the complement of a knot is decomposed into tetrahedra, the set of glueings that yield hyperbolic structures on the knot complement is determined by the solutions to glueing equations.
		For an exposition of this alternative viewpoint of A-polynomials, see the appendix by N. Dunfield to Mahler's Measure and the Dilogarithm by Boyd, Rodrigues-Villegas, and Dunfield.
		Warning: a change of orientation, from a knot to its mirror image, changes the A-polynomial.
	www.indiana.edu /~knotinfo/descriptions/a_polynomial.html (423 words)

	3. Knot and Manifold Groups - Residually Finite Groups
		Thus the fundamental group of a fibered knot is an extension of a finitely generated free group (the fundamental group of a punctured surface) by an infinite cyclic group.
		In the 1970's E. Mayland first proved that twist knots (ie twisted Whitehead doubles of the unknot) and generalized twist knots (where one is allowed to twist the doubling clasp) have residually finite groups [Mayl72].
		Finally, using special techniques to handle six exceptional knots, he was able to show that all of the knots in Reidemeister's knot table (the knots of up to 9 crossings, which he referred to as the "classical knots") had residually finite groups [Mayl75].
	www.math.umbc.edu /~campbell/CombGpThy/RF_Thesis/3_Knot_Manifold_Groups.html (1205 words)

	Knot - Information from Reference.com
		Knot is sometimes mistakenly used to refer to the nautical mile itself,...
		Because a knot is defined as a nautical mile/hour, the expression "knots per...
		Even with secure knots some slippage may occur as the knot is first put...
	www.reference.com /search?q=Knot&db=web (192 words)

	Figure-eight knot (mathematics) (Site not responding. Last check: 2007-10-05)
		In knot theory, a figure-8 knot is the unique knot with a crossing number of four, the smallest possible except for the unknot and trefoil knot.
		The name is given because joining the ends of a string with a normal figure-8 knot tied in it, in the most natural way, gives a model of the mathematical knot.
		The knot is alternating, rational with an associated value of 5/2, and is achiral.
	www.fact-index.com /f/fi/figure_eight_knot__mathematics_.html (123 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)

	CSI Math
		Knot theory is one of the most active research areas of mathematics today.
		A revolution in knot theory was ushered in with the discovery of the Jones polynomial in 1984, which led to vast families of quantum invariants.
		My aim is to establish the relationship between quantum invariants of a knot, such as the Jones polynomial, and the geometry of the knot complement
	wiener.math.csi.cuny.edu /Faculty/Kofman_ (217 words)

	Amazon.com: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots: Books: Colin Conrad Adams (Site not responding. Last check: 2007-10-05)
		Knot theory has been a branch of mathematics that has been around for over a century, and now is finding applications in mnay areas, some of these being electrical circuit analysis, genetics, dynamical systems, and cryptography.
		The assignment of polynomials to knots goes back to the early 20th century, but it took the work of Vaughan Jones and his use of ideas from operator theory and statistical mechanics to provide polynomial invariants of knots that were much finer than the Alexander polynomial of the 1930s.
		Chapters 9 and 10 are an introduction to knot theory as it relates to research in the topology of 3-dimensional manifolds and the existence of knots in dimensions higher than 3.
	www.amazon.com /Knot-Book-Elementary-Introduction-Mathematical/dp/071672393X (2785 words)

	21世紀COEプログラム Friday Seminar on Knot Theory（2005年度） /Math of OCU(J)
		Then $K$ is a fibered knot such that the monodromy is decomposed into periodic maps and the JSJ-family of the complement consists of Seifert fibered spaces.
		Every symmetric union is a ribbon knot and it has been shown that a ribbon knot with crossing number less than or equal to ten is a symmetric union.
		This relation comes from a sliding of the trivial knot in a genus $2$ handlebody along a simple closed curve (attaching slope) along which a $2$-handle is attached to obtain the exterior of a $2$-bridge knot, and we show that the relation is essential.
	math01.sci.osaka-cu.ac.jp /21COE/friday_seminar/friday_seminar_05.html (2819 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)

	1
		The accompanying diagram shows a portion of Tait's study---an enumeration of knots and links in terms of the crossing number of a plane projection.
		But for knots a seminal result of Cameron Gordon and John Luecke showed that two knot are homeomorphic if and only if they are isotopic.
		In the vernacular of the knot theorist, a knot determines its complement.
	www.math.buffalo.edu /~menasco/Knottheory.html (1176 words)

	Figure-8 Knot (2) (Site not responding. Last check: 2007-10-05)
		Complement is a space excluding the figure-8 knot that has lived in the space, namely all of space minus figure-8 knot.
		It is a knot with a projection that has crossings that alternate between over and under as one travels around the knot in a fixed direction.
		Because the boundary between the knot K and its complement belongs to the knot K, and the open end point of the cusp must stay at infinity.
	www1.kcn.ne.jp /~iittoo/usDraft_B3.htm (8899 words)

	Definition of Knot group
		The knot group of a knot K is defined as the fundamental group of the knot complement of K in R
		Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between inequivalent knots.
		The unknot has a knot group isomorphic to Z, the infinite cyclic group.
	www.wordiq.com /definition/Knot_group (195 words)

	Celtic Knot
		The Celtic Knot Public House, located at 626 Church Street in the heart of Evanston, blends the atmosphere of a traditional Irish pub with the sophistication of a modern European restaurant.
		The Celtic Knot is a warm and inviting gathering place for friends from all walks of life.
		The Celtic Knot Public House is located across from the Evanston Public Library.
	www.celticknotpub.com (259 words)

	ARCC Workshop: Moduli spaces of knots (Site not responding. Last check: 2007-10-05)
		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.
	aimath.org /ARCC/workshops/spaceofknots.html (395 words)

	Helix Knot
		Lou Kauffman was the first to develop equations for the Pattern Knot formed by intersecting helices.
		It indicates that the formation of the Pattern Knot from intersecting helices may be a little "forced" and may not arise "naturally".
		And it is known that any knot formed on a torus has a mirror complement knot.
	www.rwgrayprojects.com /Lynn/HelixKnot/helixknot01.html (885 words)

	Chinese Knotting Bibliography
		The History and Science of Knots and ends with some examples of the new knots in use in a variety of materials.
		Mostly a basic knot book (although can it be called that without the double-connection or double coin knots?) with the first instance of instructions for a few of the bao knots actually illustrated.
		Combining real knot tying, knot theory and history, this is an interesting collection of essays including the "History of Chinese Knotting" by Lydia Chen.
	www.chineseknotting.org /book (2936 words)

	[No title] (Site not responding. Last check: 2007-10-05)
		We show that the number of twists and twist regions in a prime diagram of an alternating knot with hyperbolic complement give estimates on the geometric shape of the cusp of the knot complement, including estimates on the lengths of slopes on the cusp.
		Here by the geometric shape of the cusp we mean the shape of the Euclidean similarity class of structures on horoball neighborhoods of the cusp in the hyperbolic structure on the knot complement.
		We use this to show that a large class of alternating knots have the property that every non-trivial Dehn filling is hyperbolic.
	www.newton.cam.ac.uk /programmes/SKG/poster/purcell.html (107 words)

	Figure 8/Horoball (Site not responding. Last check: 2007-10-05)
		The program then investigates the existence of a geometric structure on the complement of the knot, and computes a number of different numerical invariants of the space, such as its volume.
		It also produces diagrams that describe pictorially the behavior of the space as one approaches "infinity." The diagram at right--in technical language, the horoball diagram--is associated with the complement of the figure eight knot.
		The associated horoball diagram has proved to be an important tool in that study, as one can see that symmetries of the knot are displayed as symmetries in the more rigid horoball diagram.
	www.indiana.edu /~rcapub/v17n2/10c.html (181 words)

	The Fortier Tie (Site not responding. Last check: 2007-10-05)
		As most people know, when a striped tie is made up the stripes of the knot go in the opposite direction as the stripes in the display of the tie.
		Bob asked himself, "Why can't the stripes of the knot go in the same direction as the stripes in the display of the tie?" After careful thought and many diagrams he came up with the solution.
		He reasoned, if he could accomplish this, why couldn't he choose a coordinating color from the tie's display and make a solid or designed knot to complement the display of the tie.
	www.fortiertie.com (168 words)

	Summer 2001 Projects
		I then showed that the property of n-coloration of a knot is also a basic property of the knot group, i.e.
		I discovered an equation to check wheter an n component pretzel knot is colorable and when (in the case of a three component pretzel knot {i, j, k} the equation is: ij + jk + i*k = m.
		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 /vigre/reu/summer01reu.html (1624 words)

	About page (Site not responding. Last check: 2007-10-05)
		This is exemplified by the background, a picture of a figure-8 knot, which is hard to see appropriately unless one sits back and takes the far view.
		It is not a coincidence that I chose a figure-8 knot to be the backdrop.
		In many cases, the complement is actually a hyperbolic 3-manifold, such as with the figure-8.
	www.math.ucdavis.edu /~suh/about/about.html (434 words)

	index of Pine Knot farms Hellebore site
		These special sale days are always on the last Saturday in February and the first Saturday in March from 10 am until 4 pm when traditionally the majority of plants are flowering.
		We also stock an assortment of companion plants which complement hellebores.
		Other than these two special days the nursery is open for retail sales on Friday and Saturday from March 15 until May 15 and by appointment.
	www.pineknotfarms.com (123 words)

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