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Topic: Hyperbolic link

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	Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
		The discovery of hyperbolic space in the 1820s and 1830s by the Hungarian mathematician Janos Bolyai and the Russian mathematician Nicholay Lobatchevsky marked a turning point in mathematics and initiated the formal field of non-Euclidean geometry.
		A hyperbolic plane is a surface in which the space curves away from itself at every point.
		DH: The discovery of the hyperbolic plane came from the attempt to prove Euclid's fifth postulate, which is also known as the parallel postulate.
	www.cabinetmagazine.org /issues/16/crocheting.php (2537 words)

	Mathematics of The EPINET Project
		We discuss and illustrate tilings of the hyperbolic plane here, but the concepts apply to the sphere and euclidean plane with almost no adjustment, and the underlying theory generalises to higher-dimensional spaces.
		The dual of our example hyperbolic tiling by hexagons and squares is therefore a tiling by squares which have two distinct vertices, one of degree 6, and the other of degree 4.
		Hyperbolic geometry and the Poincaré disc model from The Institute for Figuring.
	epinet.anu.edu.au /mathematics/delaney_dress (1884 words)

	Crafty Geometry: Science News Online, Dec. 23, 2006
		One mathematician's crocheted models of a counterintuitive shape called a hyperbolic plane are enabling her students and fellow mathematicians to gain new insight into startling properties.
		Taimina realized that she could crochet a durable model of the hyperbolic plane using a simple rule: Increase the number of stitches in each row by a fixed factor, by adding a new stitch after, for instance, every two (or three or four or n) stitches.
		In addition to Taimina's hyperbolic planes and a Lorenz surface crocheted by Yackel, the exhibit featured Möbius strips, which are twisted rings that have only one side, and Klein bottles, which are closed surfaces that have no inside.
	www.sciencenews.org /articles/20061223/bob10.asp (2423 words)

	Semi-Regular Tilings of the Plane Part 1: Introduction and Historical Background
		The hyperbolic plane is amenable to an infinite number of different tilings by regular polygons.
		A similar illustration of triangulated regular hyperbolic hexagons meeting four at vertex (see the cover of [G]) was the inspiration for M.
		Hyperbolic Results describe some relatively simple geometric processes that allow one to construct several such tilings.
	people.hws.edu /mitchell/tilings/part1.html (871 words)

	Jessica Purcell: Research
		The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link.
		Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram.
		For every knot or link with hyperbolic complement, each cusp of the complement has a geometric shape given by the Euclidean similarity class of structures on horoball neighborhoods of the cusp.
	www.ma.utexas.edu /users/jpurcell/papers.html (618 words)

	Link (knot theory) - Wikipedia, the free encyclopedia
		More formally, a link is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components are homeomorphic to circles.
		Links and knots are studied in a branch of mathematics called knot theory.
		The simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once.
	en.wikipedia.org /wiki/Link_(knot_theory) (188 words)

	Publications
		In his paper, Marc Lackenby proves that the volume of the complement of a hyperbolic alternating link is bounded above and below by linear functions of the twist number, the number of non-parallel crossings.
		In the appendix, Ian Agol and I improve the upper bound and show that it is asymptotically sharp by constructing an explicit chain-link fence link.
		We show that there are finite type link homotopy invariants for links with 9 or more components, but none for links with 5 or fewer components.
	www.math.columbia.edu /~dpt/writing.html (1199 words)

	New Zionist » 2006 » March (Site not responding. Last check: 2007-11-07)
		His interest is the academic study of Zionism and the State of Israel, as well as Jewry in general.
		Our pals at Jewschool posted a fun little link to a recent string of the numerous political campaign ads going on in Israel right now in preparation for the elections happening at the end of the month.
		They are shocking because the content is a no holds barred all out gut fest of blatant stabs at exposing the worst in the competition.
	www.newzionist.com /2006/03 (3075 words)

	Design and Analysis of Queue Control Functions for Explicit Rate Switch Schemes
		The behavior of the system is the same as in the case of the hyperbolic function for underload and the steady state range of queue lengths.
		All links are of length 1000 km, which corresponds to a propagation delay of 5 ms.
		The value of the link distance D was chosen to be 1000 km.
	www.cs.wustl.edu /~jain/papers/ftp/qctrl_bv/index.html (3518 words)

	Hyperbolic link - Wikipedia, the free encyclopedia
		In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e.
		A hyperbolic knot is a hyperbolic link with one component.
		Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco.
	en.wikipedia.org /wiki/Hyperbolic_link (177 words)

	Dave Futer: Research Papers
		We use the diagram of a link K to obtain a Dehn surgery description of K from a hyperbolic link L.
		When K is an arborescent link, we use the correspondence between the link and a weighted tree to simplify the projection diagram into a particularly nice form.
		The combinatorial argument also proves that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link..
	math.stanford.edu /~dfuter/research/papers.html (472 words)

	Using Quick Searches (CA) (Site not responding. Last check: 2007-11-07)
		The Links pane shows only links used to embed inline resources in a page (links used to create hyperlink jumps to other resources are not shown).
		Links to these resources are truly broken because the resource is missing from the site.
		In the Links pane, examine the Link URL to determine whether or not it is correct—it may contain a syntax error.
	www.ppg.com /siteserver/docs/ssa_track_gkfk.htm (2306 words)

	Hyperbolic Geometry - Triangles (Site not responding. Last check: 2007-11-07)
		The Poincare half-plane model is conformal, which means that hyperbolic angles in the Poincare half-plane model are exactly the same as the Euclidean angles (with the angles between two intersecting circles being the angle between their tangent lines at the point of intersection.
		A hyperbolic triangle is just three points connected by (hyperbolic) line segments.
		Since the hyperbolic line segments are (usually) curved, the angles of a hyperbolic triangle add up to strictly less than 180 degrees.
	www.math.ksu.edu /math572/tri.html (665 words)

	A new twist on pong... - GameDev.Net Discussion Forums
		Edit: For those unaware of the unique nature of hyperbolic geometry, this website I found a while ago has lots of nice Java applets to help visualization of what pong might be like in such a space.
		In the case of the hyperbolic disk, the ball would move straight wihtin the system, but when you try to visualize it to the player, it would have curved movements on the screen.
		If you think about it, hyperbolic pong would probably play out a lot like the bounding circle was gravitationally attracting the ball.
	www.gamedev.net /community/forums/viewreply.asp?ID=2214059 (1585 words)

	tilings.org -- home page
		Here is an example of an unwrapped tiled surface giving a tiling of the hyperbolic plane by (4,3,3) triangles (for more examples go to the images page).
		The Rose-Hulman Tilings, Hyperbolic Geometry and Computational Group Theory website serves as a resource to the REU participants and others who have contributed, and as a dissemination site for their work, available to anyone who is interested.
		Links to the pdf versions of these papers are supplied.
	www.tilings.org (1271 words)

	Radar, Cold War and Service related links (Site not responding. Last check: 2007-11-07)
		The "Signals Collection '40-'45" is a Dutch non-profit foundation whose aim is to conserve and preserve allied army, navy and air force radio and radar equipment which was used and or built by the allies during the second world war.
		In addition to the well researched articles there are links to a wide variety of other web sites dealing with these matters.
		There are lots of links to web sites built by ex "Brats", an E-mail centre where you can look for old pals and lots of other interesting items.
	www.radarpages.co.uk /links/links.htm (1789 words)

	Atlas: More hyperbolic link complements in the 4-sphere by Dubravko Ivansic (Site not responding. Last check: 2007-11-07)
		Many noncompact hyperbolic 3-manifolds are known to be complements of links in the 3-sphere.
		Generalizing to dimension 4, an example of a hyperbolic 4-manifold that is a complement of 5 tori in the 4-sphere was previously obtained by the speaker.
		We enrich the catalog of examples of hyperbolic ``link'' complements in the 4-sphere by showing that other Ratcliffe-Tschantz manifolds (about 10 so far) also have double covers that are complements either of several tori or a combination of several tori and Klein bottles.
	atlas-conferences.com /cgi-bin/abstract/camc-88 (174 words)

	The Math Forum - Math Library - Hyperbolic Geom. (Site not responding. Last check: 2007-11-07)
		Hyperbolic tessellations shown in various stages of truncation, and represented by their Schlafli symbols.
		Krickl's diploma thesis: a list of all pentahedra that tessellate hyperbolic 3-space in the sense that they are a fundamental domain to their discrete reflection group.
		One illustrates that hyperbolic reflection in the Poincare disk corresponds to Euclidean inversion.
	mathforum.org /library/topics/hyperbolic_g (1487 words)

	Alternating knot - Wikipedia, the free encyclopedia
		In knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link.
		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.
		the link complement has a hyperbolic geometry, unless the link is a torus link.
	en.wikipedia.org /wiki/Alternating_knot (503 words)

	NPR : Mathematicians Get Crafty with Geometry
		A model of the hyperbolic plane crocheted by Daina Taimina.
		Those curves -- an example of a high-level geometry concept called the hyperbolic plane -- were not even defined by geometry theorists until the 19th century.
		In 1997, Taimina, of Cornell University, found a way to crochet her way into "hyperbolic space." Her woolen creations, which resemble crenulated flowers and hair scrunchies, became the first physical models of the hyperbolic plane.
	www.npr.org /templates/story/story.php?storyId=4531695 (302 words)

	QuickLinks - e-Learning
		Clifford Stoll, an astronomer, computer expert and gadfly who punctured hyperbolic claims about the societal benefits of technology in his last book, "Silicon Snake Oil", rejects the idea that students need to use computers intensively and at an early age to become computer literate.
		A recent government study found tremendous growth over the last few years in both the number of distance education courses offered by colleges and universities and in the enrollment in those classes, which includes courses given over the Internet, video links and other means.
		Links to news items about legal and regulatory aspects of Internet and the information society, particularly those relating to information content, and market and technology.
	www.qlinks.net /quicklinks/educat2.htm (2393 words)

	Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links, Hitoshi Murakami, Jun Murakami, Miyuki ...
		Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links, Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, Yoshiyuki Yokota
		Kashaev conjectured that the asymptotic behavior of the link invariant he introduced, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement.
		We observe numerically that for knots {\small $6_3$, $8_9$ and $8_{20}$} and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern--Simons invariants and propose a complexification of Kashaev's conjecture.
	projecteuclid.org /getRecord?id=euclid.em/1057777432 (214 words)

	NonEuclid - Hyperbolic Geometry Article & Applet
		Hyperbolic Geometry also has practical aspects such as orbit prediction of objects within intense gradational fields.
		Hyperbolic Geometry is used in Einstein's General Theory of Relativity and Curved Hyperspace.
		Area: - Exaimation of A=½bh and A=s² in Hyperbolic Geometry, Properties Necessary for an Area Function, Altitudes of a Hyperbolic Triangle, Defect of a Triangle, Defect of a Polygon, and an Upper Bound to Area.
	cs.unm.edu /~joel/NonEuclid/NonEuclid.html (477 words)

	The Hyperbolic Surface Activity Page
		Hyperbolic space, which is three-dimensional, has more volume than ordinary Euclidean space!
		So, to make a hyperbolic plane surface, we need to arrange for there to be, in some sense, more surface around a point than usual.
		The Geometry Center at the University of Minnesota has a number of exhibits devoted to Hyperbolic Geometry, including a Java Applet for drawing Hyperbolic Triangles and some pictures from a Computer Generated Fly-through of Hyperbolic Space.
	members.tripod.com /professor_tom/hyperbolic/index.html (441 words)

	Smith's Navier-Stokes
		Smith's proposed solution to the Millenium Problem on the Navier-Stokes equation is a culmination of many years of work on a new technique she developed to prove the existence of viscosity solutions for a larger class of partial differential equations than they had been applied to in the past.
		The comparison principles then allow Smith to approach hyperbolic equations by adapting Perron's Method, a method used in the past to create viscosity solutions to elliptic and parabolic equations.
		It is seen to be a second order hyperbolic equation for the metric when one uses gauge harmonic coordinates.
	comet.lehman.cuny.edu /sormani/others/SmithNavierStokes.html (1882 words)

	How to Use Ontobroker- Demonstration -
		Second a user needs an overview over the whole hierarchy to allow a quick and easy navigation from one class in the hierarchy to another class (and at best this navigation should be continuous, so that there is no rapid change in the presentation).
		Figure 5): classes in the center are depicted with a large circle, whereas classes at the border of the surrounding circle are only marked with a small circle.
		When a user selects a class from the hyperbolic ontology view, the class name appears in the class field and the user can select one of the attributes from the attribute choice menu because the pre-selected class determines the possible attributes.
	ksi.cpsc.ucalgary.ca /KAW/KAW98/decker (3194 words)

	Homepage of William W. Menasco (continued)
		Incompressible surfaces in complement of alternating knots and links, Thesis, U.C. Berkeley, (1981)
		Determining incompressibility of surfaces in alternating knot and link complements, Pacific Journal of Mathematics, Vol.
		Thistlethwaite)A classification of alternating links, Annals of Mathematics, 138, (1993), 113-171.
	www.math.buffalo.edu /~menasco/menasco.html (530 words)

	LMS Proceedings Abstract, paper PLMS 1429 (Site not responding. Last check: 2007-11-07)
		If a hyperbolic link has a prime alternating diagram $D$, then we show that the link complement's volume can be estimated directly from $D$.
		We define a very elementary invariant of the diagram $D$, its twist number $t(D)$, and show that the volume lies between $v_3(t(D) - 2)/2$ and $v_3(10t(D) - 10)$, where $v_3$ is the volume of a regular hyperbolic ideal 3-simplex.
		As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite-volume hyperbolic 3-manifolds, in the geometric topology.
	www.lms.ac.uk /publications/proceedings/abstracts/p1429a.html (103 words)

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