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Topic: Alternating knot

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	Alternating knot - Wikipedia, the free encyclopedia
		Alternating links end up having an important role in knot theory and 3-manifold theory, due to their complements having useful and interesting geometric and topological properties.
		Crossing number of a reduced, alternating diagram is the crossing number of the knot.
		An alternating knot diagram is in one to one correspondence with a planar graph.
	en.wikipedia.org /wiki/Alternating_knot (516 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 Guide for Making Jewelry. Knotting Techniques (Site not responding. Last check: 2007-10-22)
		The square knot picot are square knots with (wings).
		To tie an alternating square knot pattern, use 8 cords and tie one square knot with each group of four cords, one knot along side of the other knot.
		Tie a second row of square knots, using four cords, 3 and 4 from the square knot on the left and cords 1 and 2 from the square knot on the right.
	www.jewelrycord.com /Knot_Guide_for_Cords_Page01.htm (338 words)

	tses Hanging Macrame Table
		Tie rows of Alternate square knots under the cords that were larkshead on.
		Alternating Square Knots can be used to form a basket: Skip 4 inches, tie a row of alternating square knots, skip 3 inches, tie another row of alternating square knots.
		With eight cords, using the center two as filler cords tie a square knot, now divide the eight cords in to two groups of four and with the outer cord and the filler cord from the above square knot tie a square knot on each side.
	www.angelfire.com /ca7/sierratses/hanging_table.html (1245 words)

	NEW KNOT TABLES
		Kirkman's geometrical system for the systematic derivation of knot projections, closely connected with the enumeration of polyhedra, represented at the same time the geometrical method for the classification of knot projections [3].
		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/index.html (1879 words)

	Discovering symmetry of knots (Site not responding. Last check: 2007-10-22)
		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)

	Preprint Page for Stuart Rankin
		In the first paper in the list, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossings is obtained.
		The master array can be used to construct an ideal knot configuration for a prime alternating knot, by which we mean that two prime alternating knots, each in their ideal configuration, are equivalent if and only if they are identical.
		It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (namely T and OTS) to the selected seed link, one obtains all prime alternating link diagrams of the desired minimal crossing size.
	www.math.uwo.ca /~srankin/knotprint.html (1080 words)

	Motivate : Mathematics Videoconferences for Schools.
		An alternating knot is one having an alternating projection: as you walk around the knot the crossings alternate between under and over.
		A prime knot is one which cannot be represented as the join of two knots (unless one of them is the unknot).
		After that, knot theory was relatively quiet until Vaughan Jones discovered a completely new polynomial invariant in 1985; this was the catalyst for a burst of ideas and results.
	motivate.maths.org /conferences/conf28/c_28_talk.shtml (1182 words)

	Some horoball diagrams of knot complements (Site not responding. Last check: 2007-10-22)
		The knot group acts transitively on the entire collection of horoballs, but there is a (parabolic) subgroup, isomorphic to Z+Z, which preserves the "infinite" horoball z=1, and which acts on the remaining horoballs via Euclidean translations.
		We can take a fundamental region for the Z+Z action to be a rectangle whose vertices are the points of tangency with z=0 of two adjacent full-size (pink-maroon) spheres on the left edge of the picture, together with the points of tangency of two adjacent full-size spheres on the right edge of the picture.
		The symmetries of the knot lift to symmetries of the horoball packing, which are quite easy to spot in this example: for example, the order 8 symmetry lifts to a glide reflection along a horizontal axis.
	www.math.utk.edu /~morwen/3d_pics/horoball.html (397 words)

	Knot diagrams
		The other point of view to the classification of such perfect curves is that of the knot theory.
		Two such projections or knot diagrams are equal if they are isotopic in projection plane as graphs, where the isotopy is required to respect overcrossing respectively undercrossing.
		According to the last three properties, in the set of all polynomials d(t) we may distinguish even functions (d(t)=d(-t)), containing only even degrees of t, corresponding to amphicheiral knot projections, and odd functions (d(t)=-d(-t)), containing only odd degrees of t, which are invariant to a change of orientation of the knot projection.
	members.tripod.com /~modularity/gknot.htm (530 words)

	A Chain of Flowers, Macramé Plant Hanger (Site not responding. Last check: 2007-10-22)
		On one of the square knot sinnets add your butterfly bead for the center knot of the sinnet.
		Tie a square knot sinnet for 3 knots with two sets of two cords (Use the alternating chain of cords).
		Loop the chain of alternating double half hitches up and under the 2 knot sinnet in step #9, forming a ring with the chain.
	www.knotingwork.com /Pages/flower_chain.html (467 words)

	PAKG - Prime Alternating Knot Generator
		PAKG generates the complete collection of prime alternating knots at the given crossing size by applying various operators to the complete collection of prime alternating knots at the preceding crossing size.
		As a result, an initial collection of prime alternating knots of 4 crossings has to be prepared for PAKG to begin its work.
		An h-Dowker code for a prime alternating knot is the Dowker code for the configuration of the knot that is obtained by taking the zero position of each group in the MA for the knot (and then incorporating any non-flyping information as described above).
	www.math.uwo.ca /~srankin/papers/knots/pakg.html (1959 words)

	NDSU Club Math Page (Site not responding. Last check: 2007-10-22)
		Among knots with few crossings, the alternating behavior prevails; in fact, the simplest non-alternating knot has 8 crossings.
		This means that we can tell these knots apart by the (hyperbolic) volume of their complement.
		For example, the volume of the complement of the figure-eight knot is approximately 2.02988321282.
	www.ndsu.nodak.edu /ndsu/coykenda/math/mathclub/calvo9.html (234 words)

	tses Knots
		Alternating double half hitch knots Alternate lark's head knots
		This is similar to the alternating larks head braid only you use four working cords.
		Starting with the cord on the left make a loop over your thumb and over the other cords, working clockwise loop each cord over the next, and bring the fourth cord through the hole where your thumb is and remove your thumb.
	www.angelfire.com /ca7/sierratses/knots.html (1167 words)

	Knots and Surfaces (Site not responding. Last check: 2007-10-22)
		From the perspective of knot theory, some of the more interesting bounded surfaces are those with a single boundary component where the boundary forms a knot.
		Given a knot K and a projection of that knot, introduce an orientation on K. At each crossing of the projection, there will be two incoming and two outgoing strands.
		Theorem: When Seifert's algorithm is applied to an alternating projection of an alternating knot, the resulting Seifert surface has minimal genus.
	cog.brown.edu:16080 /~sgwater/personal/math/knots (872 words)

	E. Denne Research
		The existence of essential alternating quadrisecants has implications for geometric properties such as the total curvature of a knot.
		We show that flat torus knots in S^3 are distortion critical and that the distortion grows with knot complexity.
		Alternating quadrisecants capture the knottedness of a knot.
	www.math.harvard.edu /~denne/research.html (717 words)

	NSDL Metadata Record -- Alternating Knot Diagram -- from MathWorld
		NSDL Metadata Record -- Alternating Knot Diagram -- from MathWorld
		A knot diagram which has alternating under- and overcrossings as the knot projection is traversed.
		The first knot which does not have an alternating diagram has 8 crossings.
	nsdl.org /mr/698143 (57 words)

	A Chain of Flowers, Macramé Plant Hanger (Site not responding. Last check: 2007-10-22)
		With the first group of two cords tie a chain of alternating double half hitches, for 8½in.
		Tie an alternating square knot pattern with a 2½ drop.
		Take the filler cords from each square knot and tie another chain of alternating double half hitches for 10in with each set of filler cords.
	www.knotingwork.com /Pages/projects/flower_chain.html (606 words)

	[No title]
		As an example, Delman and Roberts have shown that every non-trivial Dehn surgery along a non-torus alternating knot yields a manifold containing an essential lamination (or {\it laminar} manifold, for short).
		Knot theory is one of the subjects in 3-manifold topology that has benefited the most from recent research into essential laminations.
		Finding essential laminations in knot exteriors, which survive (i.e., remain essential after) non-trivial Dehn surgery, thus provides one approach to solving such conjectures, for many classes of knots.
	www.math.unl.edu /~mbrittenham2/personal/myresold.html (1253 words)

	Hemp Necklace Instructions (Site not responding. Last check: 2007-10-22)
		Tie 16 Square Knots with the outside 4 cords (the strand that has the five beads on it).
		Tie a 6 Square Knot sinnet in the bottom strand (the one with the five beads).
		Using the 2 longest cords as knotting cords, tie a 3 Square Knot sinnet around the rest of the cords.
	www.whutaworld.com /hempjewelry2.html (487 words)

	Celtic Knots: Construction (Site not responding. Last check: 2007-10-22)
		The "final" knot lines from this method are close to where they should be, but sometimes need to be moved slightly, or smoothed out, for purely aesthetic reasons.
		A splitting is a line through which the knot is not allowed to pass.
		I usually make a quick sketch of the knot, count the number of components, and add or remove splittings to make it all one components before drawing the final knot -- hopefully without greatly altering whatever symmetries the knot was supposed to have.
	www.wallace.net /knots/howto (1417 words)

	Knots and Links in Braid notation
		For knots, this is n(x), where n is the crossing number and x is an index.
		It has the merit of defining a fairly large class knots and links in a manner which is independent of historical accidents, but the notation for the remainder becomes rapidly more cumbersome after 11 crossings.
		A link is alternating when it has a diagram in which the crossings alternate over and under along every component.
	www.scoriton.demon.co.uk /knots (1449 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		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)

	Unicursal curves and knot diagrams (Site not responding. Last check: 2007-10-22)
		A unicursal curve differs from the curves shown in knot diagrams in that there is no sense of the curve's crossing over or under itself at an intersection.
		You can convert a unicursal curve into a knot diagram by indicating (probably with the aid of an eraser), which strand crosses over and which strand crosses under at each of the intersections.
		Draw the 32 knot diagrams that arise from the unicursal curve underlying the diagram of knot 5-2 shown in the previous section, and identify the knots that these diagrams represent.
	geom.math.uiuc.edu /docs/education/institute91/handouts/node11.html (356 words)

	NSDL Metadata Record -- Almost Alternating Link -- from MathWorld
		Then an almost alternating link is a link with an almost alternating projection, but no alternating projection.
		A prime knot which is almost alternating is either a torus knot or a hyperbolic knot.
		Therefore, no satellite knot is an almost alternating knot.
	nsdl.org /mr/698002 (115 words)

	Lauren Haley's CS403-05 Final Project
		The square knot is the simplest and most basic knot in Hemp necklace making.
		The alternating spiral is another knot that is perfect for spicing up your hemp necklace designs.
		By strategically alternating the simple square knot and the spiral knot, the alternating spiral will result.
	pubpages.unh.edu /~led3/knot.html (262 words)

	AMCA: Achirality of Knots by M. Azram (Site not responding. Last check: 2007-10-22)
		It has been shown that in reduced alternating achiral knots, number of fl regions is same as the number of white regions and consequently, W(K)=0 iff B = W is a necessary condition for reduced alternating knot to be achiral.
		Consequently, the equivalence of the companion graphs, necessary and sufficient conditions for a reduced alternating knot to be achiral has been established.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/j/r/03.htm (246 words)

	Alternating Square Knot Necklace (Site not responding. Last check: 2007-10-22)
		By alternating which side you tie your square knot on, you can create an open loop chain.
		First, take the two outside strands in the three strand section and tie a square knot around the middle strand.
		When you are ready to attach the clasp just thread the hemp through the center of the clasp and squeeze it on tight with a pair of pliers.
	media.lethbridgecollege.ab.ca /doylca/training/alternating.htm (267 words)

	Hemp Jewelry Instructions (Site not responding. Last check: 2007-10-22)
		Drop down 1/3 inch from fold and tie 2 Square Knots using 2 of the longest cords for knotting cords (all others are filler cords).
		With the 2 cords on the left, tie a chain of 7 Alternating Half Hitches.
		Bring remaining two longer cords around the bead and tie 2 Square Knots, using the end of the necklace plus the filler cords as filler.
	www.whutaworld.com /hempjewelry6.html (402 words)

	KnotInfo
		By checking the boxes you can select knots and knot invariants of interest and generate a table of knots.
		DT notation and click "Find." For now, KnotFinder finds only prime knots with crossing number 13 or less.
		KnotFinder is powered by KnotScape, but gives the classical Name of the knot, rather than the DT Name returned by KnotScape.
	www.indiana.edu /~knotinfo (193 words)

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