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Topic: Torus knot

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Knot theory

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	Gordian Knot
		The Gordian knot can be made on a Torus tube which looks like a donut or a sphere that turns in from one side and comes out the other in a perpetual motion.
		It was further prophesied by an oracle that the one to untie the knot would become the king of Asia.
		The Gordian knot is the simplest of all Torus knots.
	www.ka-gold-jewelry.com /p-articles/gordian-knot.php (477 words)

	Hyperseeing, Hypersculptures, Knots, and Minimal Surfaces
		The modified knot is shown in Figure 10(a) and the corresponding minimal surface is shown in (b).
		The knot in Figures 12 and 13 is the representation of a trefoil knot as the 2-3 torus knot.
		In particular, the 3-2 torus knot is shown in Figure 4(a) and it has a minimal surface as in Figure 8(a), which is a triple twist Möbius band.
	members.tripod.com /vismath7/nat/index.html (2597 words)

	Hyperseeing, Knots, and Minimal Surfaces (Site not responding. Last check: 2007-10-21)
		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)

	Knots and Links (Site not responding. Last check: 2007-10-21)
		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)

	Calc III Lab #15: Torus Knots
		In general, for nonzero integers m,n, a torus knot of type (m,n) is the image of the line ns=mt + const.
		If either m=0 or n=0, then the definition changes slightly: a torus knot of type (1,0) is the image of the line t=constant whereas a knot of type (0,1) is the image of a line s=constant.
		For each of the torus knots below, imitate Figure 1 to sketch a line in parameter space and the image of that line on the torus.
	www.geom.uiuc.edu /~fjw/Labs/Lab15/Knots.html (335 words)

	Knots and Surfaces: Torus Knots (Site not responding. Last check: 2007-10-21)
		Recall that a torus may be parametrized by rotating a circle of radius r about another circle of radius R.
		A torus knot is a closed curve that winds around the torus.
		For each of the torus knots below, imitate Figure 1 to sketch a line in "parameter space" (the domain of T) and the image under T of that line on the torus.
	www.geom.uiuc.edu /~fjw/calcIII/Lab19/Knots.html (382 words)

	Torus Knot Splice Base - Knot Atlas (Site not responding. Last check: 2007-10-21)
		{ Data:Torus Knot Splice Base/Determinant, Data:Torus Knot Splice Base/Signature }
		{ Data:Torus Knot Splice Base/Alexander Polynomial, Data:Torus Knot Splice Base/Jones Polynomial }
		(Data:Torus Knot Splice Base/V 2, Data:Torus Knot Splice Base/V 3)
	katlas.math.toronto.edu /wiki/Torus_Knot_Splice_Base (446 words)

	[No title]
		A knot which is prime on two strings is said to be {\it doubly prime}.
		The new knot $h(E)$ is the {\it q-twist satellite} of $K$ with {\it embellishment} $E$.
		Consider a Seifert fibered solid torus with an exceptional fiber at the core of index $n$, as in Figure 12, then this solid torus with an open regular neighborhood of a regular fiber removed is homeomorphic to the exterior of the other string in $N$, i.e.
	spot.pcc.edu /~jbradfor/knots2003.txt (5519 words)

	Why knot: knots, molecules and stick numbers
		The trefoil knot and the figure-eight knot are the two simplest nontrivial knots, the first having a picture with three crossings and the second with four.
		Since there are infinitely many possible distinct knots, it appears that a single sequence of atoms bonded in a certain order can generate an infinite number of different molecules, one for each of the different knots in the table of knots.
		Since the trefoil knot is a (q,q-1)-torus knot for q =3, we obtain 6 for the number of sticks to construct it, as we expected.
	pass.maths.org.uk /issue15/features/knots (2291 words)

	On Composite Twisted Unknots (Site not responding. Last check: 2007-10-21)
		A twisting of a knot K is parametrized by a choice of oriented twisting torus V and number of full twists d (=delta in paper).
		If a composite knot is given two (non-trivial) full twists and the result is composite, we began with a granny knot and obtained the granny knot of opposite handedness.
		The granny knot case is due to Motegi and Hayashi, and required very special attention in the proof (an additional strand had to be woven through the induction.
	comp.uark.edu /~cgstraus/papers/ctu.html (600 words)

	Double Torus Knot
		It is a (3, 2) torus knot, meaning that the knot thread goes around the torus 2 times and threads through the torus hole 3 times.
		As shown above, the (2,3) knot, the "Mereon knot" is a knot which fits on this surface and which satisfies Rule 1 above.
		Introducing 3 holes through the original sphere, and looking at knots which can be formed on the resulting 3-hole "torus", we find that the (3, 2) "simplest" Trefoil knot fits onto this surface.
	www.rwgrayprojects.com /Lynn/DoubleTorus/dt01.html (357 words)

	Galbydeia Torus
		The Torus is ten kilometers in diameter, composed of a quark matter pipe containing superfluid neutronium.
		As the field developed, various field inductors on the torus rim attempted to use it to coax a quantum wormhole to spontaneously inflate to fit the torus (not unlike the failed NoCoZo Ynity Gate experiments many millennia before).
		The torus orbits Galbydeia Due, a dim M star surrounded by a vast disc of metatechnological debris from the construction project and later eras (including the largest collection of artificial fl holes in any system; this collection is currently used by the Tarynt-Ometa Clade in an experiment with gravity transmission).
	www.orionsarm.com /worlds/Galbydeia_Torus.html (701 words)

	Torus Knot
		A (m,n)-torus-knot is a curve on the torus, which is specified by winding m times around the main axis of the torus and n times around the tube of the torus.
		In this applet are three geometries displayed: The torus, the torus knot as a polygon and a tube around the torus knot polygon.
		The tube is generated by carrying a frame parallel along the polygon; this is independent of the surface underlying the polygon, so there may be a distortion between the ends of this tube.
	www.javaview.de /vgp/tutor/torusknot/PaTorusKnot.html (164 words)

	The KnotPlot Site
		Interactive knot viewer (randomly chosen knot, requires Java): smooth knots minimal-stick knots equilateral minimal-stick knots crazy knots.
		Knots can be loaded from a database of more than 3,000 knots and links or sketched by hand in three dimensions.
		Also, knots may be constructed via the Conway notation or using the tangle calculator.
	knotplot.com (535 words)

	Mark Brittenham: papers and preprints
		As a corollary, any essential lamination in a torus knot exterior is (with a single exception) isotopic to one which is everywhere transverse to the foliation of the exterior by circles.
		The canonical genus of a knot K is the minimum of the genera of Seifert surfaces built by Seifert's algorithm, taken over all projections of the knot K. In this paper we show for any g there is a constant C(g) so that any hyperbolic knot with canonical genus g has volume less than C(g).
		In this paper we use the construction of knots with genus one free Seifert surfaces (again) to create families of hyperbolic knots which each have a unique minimal genus Seifert surface which cannot be the sole compact leaf of a depth one foliation.
	www.math.unl.edu /~mbrittenham2/personal/pprdescr.html (2020 words)

	Torus Knot of type (7,11)
		This is an image of a knot on a torus that winds seven times around one hole and 11 times around the other.
		The torus, which is a product of two circles of radius one, is embedded in the three sphere of radius the square root of two in the standard way.
		The picture you see is the image of the knot just described under stereographic projection from the three sphere in four space to three space.
	www.math.su.se /~lambe/public/tk7-11.html (91 words)

	Knots and Multiple Möbius Band Minimal Surfaces
		The 2-3 torus knot admits a minimal surface consisting of two Möbius bands that share edges and alternately cross over each other.
		The 3-4 torus knot admits a minimal surface consisting of three Möbius bands that share edges and alternately cross over each other.
		In Figure 3 there is a minimal surface on a 3-4 torus knot consisting of red, blue and green Möbius bands with a space in the center.
	members.tripod.com /vismath6/friedman1/index.html (221 words)

	Knot Table: Arc Index
		The knot is embedded in five half-planes numbered 1 to 5 in sequence around the binding axis (shown red); the knot meets the binding in five points, called vertices, labelled a to e in sequence along the axis.
		In one of the earliest papers on knot theory, Hermann Brunn asked whether it was possible to have a projection of any link with a single singular point of high multiplicity.
		The transparent tube is the companion torus and is a tubular neighbourhood of an arc presentation of the trefoil having the green line as its axis.
	www.indiana.edu /~knotinfo/descriptions/arc_index.html (969 words)

	Of Torus and Turk's-Head Knots... (Site not responding. Last check: 2007-10-21)
		The knots depicted may actually be tied utilizing the diagrams contained herein, although a good many of the knots may also be tied “in the hand,” as a sailor/ boater would say, because these knots are kept relatively simple for illustrating purposes.
		For a flat knot, if you count, the number is the same around the inside hole as around the outside rim of the disk formed by the knot.
		For knots which are tied of softer material, such as most cordage, this knot may be made without resort to nails, pins, or diagram, and may easily be tied in the hand.
	www.mi.sanu.ac.yu /vismath/pennock/index.html (1602 words)

	Paper strip
		A bagel is a solid body whereas the torus, as defined and used here, is a surface.
		I hope to return to the topic of shredding the torus at a later date.
		Living on a torus is not the same as living on a plane or even on a sphere.
	www.cut-the-knot.org /do_you_know/paper_strip.shtml (1589 words)

	Knots (Site not responding. Last check: 2007-10-21)
		This is a special case of the more general torus knot described in knot 3.
		This is an example of a torus knot which exists on the surface a torus.
		It is characterised by the number of time it wraps around the meridian and longitudinal axis of a torus.
	local.wasp.uwa.edu.au /~pbourke/surfaces_curves/knot/index.html (166 words)

	diff geo images page
		Curvature and torsion of torus knots 1 2
		The [2,5]-torus knot means a curve which wraps 'around' the torus twice as it 'spirals' five times over and under the torus.
		For this longer thinner frame we take the initial conditions to be the same as those for torus knot Frenet frame, at a point on the torus knot where curvature is small and torsion is large (right after the first 'hilltop' is crossed by the torus knot Frenet frame).
	www.math.uiowa.edu /~wseaman/DGImage53100.htm (3122 words)

	Sculpture Maths - Torus Knots (page1 of 2)
		It is called a (15,4) torus knot, because it is wrapped 15 times one way and 4 times the other.
		The GORDIAN KNOT is also an (8,3) torus knot, but with a thick tube, and an invisible inner torus!
		This material may be used freely for educational, artistic and scientific purposes, but may not be used for commercial purposes, for profit or in texts without the permission of the publishers.
	www.popmath.org.uk /sculpmath/pagesm/torus.html (269 words)

	explore_torus_knots_pub.nb
		For example - the (1,2) knot is a mobius band which is not-orientable.
		We are looking at how to generate the Seifert surface given a knot.
		This is an "n,m" torus knot, n and m must
	www.cs.indiana.edu /~sithakur/b689/explore_torus_knots/explore_knots.html (58 words)

	Knot Table: Crosscap Number (Site not responding. Last check: 2007-10-21)
		If a knot is of crosscap number 1, then it bounds a Mobius band, and thus is either a (2,n)-torus knot, or has a companion, and hence is not hyperbolic.
		There were three exceptions to this for knots with 11 or less crossings, for which Seifert's algorithm produced a genus greater than 2g+1, where g is the orientable genus.
		In the paper about Klein bottles by Teragaito it is shown that if a knot is of (orientable) genus 1 and of crosscap number 2, then it is a twist knot.
	www.indiana.edu /~knotinfo/descriptions/crosscap_number.html (301 words)

	Amazon.com: "knot diagram": Key Phrase page (Site not responding. Last check: 2007-10-21)
		Knots -7-~ CJ C((-, ~)~ 9 Figure 16.1.
		A knot diagram and a link diagram 16.1 Knots and Their Projections A knot is a piecewise-linear closed curve in IP~3.
		A knot diagram and a link diagram 16.1 Knots and Their Projectiolis A kiiot is a piccewise-linear closed curve in = :i.
	www.amazon.com /phrase/knot-diagram (450 words)

	Mathematical Imagery Presented by the American Mathematical Society - Search results
		This is an example of a torus knot.
		A torus knot can be thought of as looping around and through the torus.
		The symbol T(4,3) means that the string making the knot loops through the hole of the torus 4 times, making 3 revolutions.
	www.ams.org /mathimagery/thumbnails.php?album=search&search=bar-natan (143 words)

	Knots in Parametric Curves
		Knot analysis confirms that this is indeed a trefoil:
		Following the terminology used in the book `Ideal Knots', the parametric curves generated using sines and cosines are called Fourier knots or Harmonic knots.
		The knot is determined after having two successive subdivisions (starting from divs=10*n_max and then doubling thereafter) yield the same set of invariants.
	webusers.physics.umn.edu /~rlua/knots/curve (534 words)

	Meru Presentation Posters
		A chart relating tetrahelical column length with 3,x torus knots
		The 3,10 Torus Knot, Adam Kadmon, and the Tree of Life
		This animation of the 3,10 torus knot, displaying 3 pairs of First Hand
	www.meru.org /Posters.html (443 words)

	Visualizing Quaternions: Quaternion Maps Documentation
		Maps of simple, closed curves like a knot or a circle can be highly instructive in understanding the nature of a sequence of orientation-frames.
		Knot constants m,n - set m and n to be mutually prime to generate a knotted curve.
		Otherwise, an 'Un-knot' or a loop is generated that is not a knotted curve, i.e.
	www.cs.indiana.edu /~hanson/quatvis/Quaternion-Maps/index.html (1161 words)

	Torus knots (Site not responding. Last check: 2007-10-21)
		The 2nd view actually runs around a torus.
		It runs twice around the torus horizontally and three times in the other direction.
		I would like to try other examples on classifications of knots.
	www.math.cuhk.edu.hk /publect/lecture4/torus.html (33 words)

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