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	Trefoil - Wikipedia, the free encyclopedia
		Trefoil (from Latin trifolium, three-leaved plant, French trèfle, German Dreiblatt and Dreiblattbogen) is a term in Gothic architecture given to the ornamental foliation or cusping introduced in the heads of window-lights, tracery, panellings, etc., in which the center takes the form of a three-lobed leaf (formed from three partially-overlapping circles).
		A trefoil combined with an equilateral triangle was also a moderately common symbol of the Christian Trinity during the late middle ages in some parts of Europe.
		In mathematical knot theory, a trefoil refers to a trefoil knot.
	en.wikipedia.org /wiki/Trefoil (228 words)

	Svetlana Varchenko's Home Page
		Two knots are considered to be the same if, when they are made out of rope or some other material you can twist one of them around (without cutting) so that each looks exactly like the other one with all the over- and under-crossings in the same place.
		Some of the ways that knots are classified is by the arrangement of their crossings, the characteristics of their mirror images, and the braids from which they are formed.
		There are two knots with a crossing number of five, three with a crossing number of six, and seven knots with a crossing number of seven.
	www.unc.edu /~svetlana/knot.htm (669 words)

	Search ScienceWorld
		In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot).
		The square knot, also called the reef knot, is a composite knot of six crossings consisting of a knot sum of a trefoil knot and its mirror image (Rolfsen 1976, p.
		Given a knot diagram, it is possible to construct a collection of variables and equations, and given such a collection, a group naturally arises that is known as the group of the knot.
	scienceworld.wolfram.com /search/index.cgi?as_q=knot (538 words)

	Hyperseeing, Knots, and Minimal Surfaces
		The modified knot is shown in Figure 9(a) and the corresponding minimal surface is shown in (b).
		The knot in Figures 10 and 11 is the representation of a trefoil knot as the 2-3 torus knot.
		The p-q torus knot is equivalent to the q-p torus knot.
	arpam.free.fr /friedman.html (2777 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.
		One is that with all the specialization of tools and interests due to the growth of the body Mathematics, this science is unified in that the basic strains permeate virtually every branch of Mathematics and this is a regular occurrence to detect links between distinct mathematical theories.
	www.cut-the-knot.org /do_you_know/knots.shtml (714 words)

	Ivars Peterson's MathTrek - The Tangled Task of Distinguishing Knots
		In this context, a knot is a one-dimensional curve that winds through itself in three-dimensional space, finally catching its tail to form a closed loop.
		One approach to labeling knots is to use the arrangement of the crossings in a knot diagram to produce an algebraic expression for that knot.
		To solve the problem of distinguishing among knots, mathematicians have tried to find schemes for labeling them in such a way that two knots having the same label are really equivalent—even when their diagrams may appear different—and that two knots with different labels are truly different.
	www.maa.org /mathland/mathtrek_02_24_03.html (1038 words)

	Read This: Why Knot? An Introduction to the Mathematical Theory of Knots
		Knot theory is a hands-on subject which shouldn't just be read about: one would miss too much of the fun.
		After the trefoil knot is introduced, it is carefully noted that no amount of fiddling with the knot and not being able to untie it can show that it is not equivalent to the unknot.
		It uses the lovely notion of tricolorability: one colors the arcs of a projection of the trefoil with three colors in such a way that at 1) at least two colors are used, and 2) at every crossing, the three arcs meeting at that crossing are either three different colors or all the same color.
	www.maa.org /reviews/whyknot.html (868 words)

	Discovering symmetry of knots
		In the case of rational knots all amphicheiral knots are derived from the same source: from the figure-eight knot 2 2.
		If a knot K could be represented by an antisymmetrical vertex-bicolored graph on a sphere, it is achiral In this case, for the oriented knot K there exist a symmetry transposing orientations of vertices, i.e., mutually exchanging vertices with the signs +1 and –1.
		For the knot 2 2, the graph symmetry group is G = [2+, 4], and the knot symmetry group G' = [2+, 4+] is generated by the rotational reflection, with the axis defined by the midpoints of colored (i.e., double) edges of the tetrahedron.
	www.mi.sanu.ac.yu /vismath/visbook/jablan/jablan2.html (738 words)

	Illiteracy psychoanalysis and Topologu-Knot theory-Lituraterre.org
		Knot theory is a branch of algebraic topology where one studies what is known as the placement problem, or the embedding of one topological space into another.
		Thus a mathematical knot is somewhat different from the usual idea of a knot, that is, a piece of string with free ends.
		Knots that are equivalent to the unit circle are considered to be unknotted or trivial.
	www.lituraterre.org /illiteracy-knot_theory.htm (719 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)

	Knot (mathematics) - Wikipedia, the free encyclopedia
		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.
		In knot theory and 3-manifold theory, oftentimes the adjective "tame" is omitted.
	en.wikipedia.org /wiki/Knot_(mathematics) (280 words)

	Knot Enumeration (Site not responding. Last check: 2007-09-02)
		However, if the knot's material properties are taken into consideration when doing the energy minimization calculations, in particular, the twist energy in the material, this extra twist-originating lobe may occur as an energy minimum configuration.
		The ability of Lynncalaire's knot to transform into a Tetrahedron configuration is one of the principle properties of her knot.
		The "Lynnclaire" and the "Lou" knots are Trefoil knots.
	www.rwgrayprojects.com /Lynn/KnotTypes/kt01.html (1232 words)

	Knot Theory
		Knot theory studies the placement of one-dimensional objects called strings [23,24,25] in a three-dimensional space.
		Figure 5.6: Planar diagrams of knots: (a) the trivial or unknot ; (b) figure-eight knot ; (c) left-handed trefoil ; (d) right-handed trefoil; (e) square knot ; (f) granny knot.
		A mathematician's knot is a non-self-intersecting smooth closed curve (a string) embedded in three-space.
	cnls.lanl.gov /People/nbt/Book/node141.html (552 words)

	Discrete Knots
		An orthogonal discrete knot is a path through the lattice of integer-valued points and orthogonal edges between them, which never visits the same point twice and ends where it starts.
		Two knots are equivalent if, when they are realized with pieces of stretchy string, we can move one around to look like the other.
		The picture above shows a knot of length 64 which turns out to be equivalent to the trefoil knot.
	www.jansteckel.com /Hew/WebSite/DiscreteKnots/index.html (1613 words)

	[No title] (Site not responding. Last check: 2007-09-02)
		A Fourier knot is a knot that is described by equations in cartesian coordinates x,y,z of the form
		Thus the trefoil is not a Lissajous knot.
		The Pattern is a trefoil knot seen as an orbit about a central sphere.
	www2.math.uic.edu /~kauffman/Fourier.html (401 words)

	Untangling the Mathematics of Knots (Site not responding. Last check: 2007-09-02)
		Have the students look at the knot in the upper left hand corner of the collection of knot diagrams it is a trefoil knot.
		Be especially careful that the students have not made the other trefoil knot --the one to the right of it in the collection of knot diagrams--by mistake.
		If the knots appear different, have them study the knots and the diagram carefully to determine what the problem is. When one group has completed a set of knots, they should ask members of another group to verify that they have made the knots exactly as they are in the diagrams.
	www.cs.uidaho.edu /~casey931/mega-math/workbk/knot/knmenag.html (735 words)

	Carving for Mathematical Understanding of Surfaces
		The twisted trefoil knot surface shown was created on a 3D Systems stereolithographic machine along with a wood carving of the new collapsed surface.
		Returning briefly to the spanning surface of the trefoil knot, it is clear that this surface is not a minimal surface as it bulges out.
		The eight knot basis for this non-orientable sculpture was formed using piecewise special Bezier quartic curves [4] defined as an interpolating curve for the eight points shown here, which stem from lines drawn from the center to the vertices of a regular tetrahedron.
	www.wcnet.org /~clong/carving/carving.html (841 words)

	Knot Theory Online - The Web Site for Learning More about Mathematical Knot Theory
		Once we simplify the knot so that we cannot remove any further crossings, the knot is classified by the number of crossings that remain.
		Knots have some properties that depend only on the knot itself and not on how it is looking at any particular moment.
		The trefoil knot can be unknotted by changing only one crossing from under to over, so the trefoil knot has an unknotting number of 1 (try this to verify it).
	www.freelearning.com /knots/intro.htm (1095 words)

	Ideas, Concepts and Definitions (Site not responding. Last check: 2007-09-02)
		It seems like you should be able to flip or re-twist one trefoil knot and turn it into the other.
		It was a long time before this was actually proved, however, and that proof -- the only one that is known -- is far more complicated than a pair of trefoil knots appears to be.
		Maybe you will be the one to say "aha!" and find a simpler proof that the two trefoil knots are not the same.
	www.c3.lanl.gov /mega-math/gloss/knots/kntref.html (190 words)

	Trefoil Knot -- from Wolfram MathWorld
		The trefoil knot 03-001, also called the threefoil knot or overhand knot, is the unique prime knot with three crossings.
		There are no other knots on 10 or fewer crossings sharing the same Alexander polynomial, BLM/Ho polynomial, or Jones polynomial.
		The knot group of the trefoil knot is
	mathworld.wolfram.com /TrefoilKnot.html (222 words)

	Knot Theory Online - The Web Site for Learning More about Mathematical Knot Theory
		A composite knot is a knot which can be formed by the composition (joining) of two or more nontrivial knots.
		The trefoil knot (drawn as a stick knot to the right) has a stick number of 6, meaning we need at least 6 sticks to form the trefoil.
		Since the trefoil is tri-colorable, and the unknot is not tri-colorable, we therefore conclude that the trefoil knot is not the unknot.
	www.freelearning.com /knots/advanced.htm (1419 words)

	All Tied Up In Knots (Site not responding. Last check: 2007-09-02)
		Simplifying knot #1 to equate it with the unknot is like simplifying a fraction, e.g., simplifying 3/6 to the equivalent fraction 1/2.
		Some invariants applicable to knots are crossing number, the unknotting number (the number of changes required in a knot in order to unknot it), coloring number, and bridge number.
		For example, many students find it difficult at first to determine whether a given diagram of a knot stands for a structure that is actually knotted, but they become quite adept at this task with a little practice.
	www.riverdeep.net /current/2002/09/091602_knots.jhtml (1383 words)

	Hyperseeing, Hypersculptures and Space Curves
		Two diagrams of the so-called trefoil knot are shown in Figure 12 (a) and (b).
		The point is that from one knot we obtain infinitely many sculptures depending on the size, material, and shape of the knot.
		In general, knots are classified according to the minimum number of crossings in a diagram of the knot.
	members.tripod.com /vismath5/friedman/index.html (3076 words)

	Ideas, Concepts and Definitions (Site not responding. Last check: 2007-09-02)
		Knot Theory is the mathematical study of knots.
		The central problem of knot theory is distinguishing between various knots and classifying them.
		The best way to learn about knots is to make some knots and play with them.
	www.c3.lanl.gov /mega-math/gloss/knots/knots.html (65 words)

	Knots and Their Polynomials-11 (Site not responding. Last check: 2007-09-02)
		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 polynomial of (left-trefoil) splice (right-trefoil) = square knot.
	e-math.ams.org /featurecolumn/archive/knots11.html (67 words)

	Molecular Möbius Strips and Trefoil Knots
		The Trefoil Knot and the Möbius Strip are two such topologies which span both boundaries.
		August Möbius in the 19th Century, and is easily constructed by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then re-attaching the two ends.
		Nature is also capable of knotting proteins (large molecules or biopolymers, made of long chains of connected amino acids).
	www.ch.ic.ac.uk /motm/trefoil (1146 words)

	Knots (Bioinformatics) (Site not responding. Last check: 2007-09-02)
		Trefoil protein knots in which the knot is composed of few (ten) amino acids have been found.
		However, such knots are composed of ten amino-acid polypeptide, thus they barely make a knot (are named shallow trefoil knots).
		A trefoil knot is reportedly found in the yggJ Protein (HI0303) of Haemophilus influenzae that is crucial to methyl transfer from S-adenosylmethionine to carbon, nitrogen, or oxygen in DNA, RNA, and proteins.
	www.cybcon.com /~plasmid/Knots.html (1626 words)

	Trefoil Knot
		-These two trefoil knots, made from recycled mahogany (probably African), are mirror images of each other.
		-If one were to build such a knot with as few cubes as possible (with adjacent cubes meeting at faces), this shape would result.
		These knots, however, were made with 1x1x2 blocks.
	home.cc.umanitoba.ca /~gunderso/pages/topology/trefoil_knot.htm (58 words)

	Research Page (Site not responding. Last check: 2007-09-02)
		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)

	The Geometry Junkyard: Knot Theory
		There is of course an enormous body of work on knot invariants, the 3-manifold topology of knot complements, connections between knot theory and statistical mechanics, etc. I am instead interested here primarily in geometric questions arising from knot embeddings.
		Atlas of oriented knots and links, Corinne Cerf extends previous lists of all small knots and links, to allow each component of the link to be marked by an orientation.
		With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon.
	www.ics.uci.edu /~eppstein/junkyard/knot.html (686 words)

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