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Hyperbolic Link
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. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component. As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links. As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3-manifolds. Examples *Borromean rings are hyperbolic. *Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco. * 41 knot (the figure-eight knot) * 52 knot (the three-twist knot) * 61 knot (the stevedore knot) * 62 knot * 63 knot * 74 knot * 10 161 knot (the "Perko ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blue Figure-Eight Knot
Blue is one of the three primary colours in the RYB colour model (traditional colour theory), as well as in the RGB (additive) colour model. It lies between violet and cyan on the spectrum of visible light. The eye perceives blue when observing light with a dominant wavelength between approximately 450 and 495 nanometres. Most blues contain a slight mixture of other colours; azure contains some green, while ultramarine contains some violet. The clear daytime sky and the deep sea appear blue because of an optical effect known as Rayleigh scattering. An optical effect called Tyndall effect explains blue eyes. Distant objects appear more blue because of another optical effect called aerial perspective. Blue has been an important colour in art and decoration since ancient times. The semi-precious stone lapis lazuli was used in ancient Egypt for jewellery and ornament and later, in the Renaissance, to make the pigment ultramarine, the most expensive of all pigments. In the ei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split Link
In the mathematical field of knot theory, a split link is a link that has a (topological) 2-sphere in its complement separating one or more link components from the others. A split link is said to be splittable, and a link that is not split is called a non-split link or not splittable. Whether a link is split or non-split corresponds to whether the link complement is reducible or irreducible as a 3-manifold. A link with an alternating diagram, i.e. an alternating link, will be non-split if and only if this diagram is connected. This is a result of the work of William Menasco William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory. Biography Menasco received his B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the Univ ..... A split link has many connected, non-alternating link diagrams. References Links (knot theory) {{knottheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SnapPea
SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version as part of his doctoral thesis, supervised by William Thurston. It is not to be confused with the unrelated android malware with the same name. The latest version is 3.0d3. Marc Culler, Nathan Dunfield and collaborators have extended the SnapPea kernel and written Python extension modules which allow the kernel to be used in a Python program or in the interpreter. They also provide a graphical user interface written in Python which runs under most operating systems (see external links below). The following people are credited in SnapPea 2.5.3's list of acknowledgments: Colin Adams, Bill Arveson, Pat Callahan, Joe Christy, Dave Gabai, Charlie Gunn, Martin Hildebrand, Craig Hodgson, Diane Hoffoss, A. C. Manoharan, Al Marden, Dick McGehee, Rob Meyerhoff, Lee Mosher, Wal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(−2,3,7) Pretzel Knot
In geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery constructions. Mathematical properties The (−2, 3, 7) pretzel knot has 7 ''exceptional'' slopes, Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight knot The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ..., which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes. References Further reading * Kirby, R., ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perko Pair
In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot. In Dale Rolfsen's knot table, this supposed pair of distinct knots is labeled 10161 and 10162. In 1973, while working to complete the classification by knot type of the Tait–Little knot tables of knots up to 10 crossings (dating from the late 19th century), Perko found the duplication in Charles Newton Little's table. This duplication had been missed by John Horton Conway several years before in his knot table and subsequently found its way into Rolfsen's table. The Perko pair gives a counterexample to a "theorem" claimed by Little in 1900 that the writhe In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ... of a red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
74 Knot
In mathematical knot theory, 74 is the name of a 7-crossing knot which can be visually depicted in a highly-symmetric form, and so appears in the symbolism and/or artistic ornamentation of various cultures. Visual representations The interlaced version of the simplest form of the Endless knot symbol of Buddhism is topologically equivalent to the 74 knot (though it appears to have nine crossings), as is the interlaced version of the unicursal hexagram of occultism. (However, the endless knot symbol has more complex forms not equivalent to 74, and both the endless knot and unicursal hexagram can appear in non-interlaced versions, in which case they are not knots at all.) File:EndlessKnot03d.png, One form of the Endless knot of Buddhism File:Interwoven unicursal hexagram.svg, Interwoven unicursal hexagram. File:Celtic-knot-linear-7crossings.svg, 74 knot in Celtic artistic form, also found in some Hausa embroideries.''Celtic Art: The Methods of Construction'' by George Bain, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
63 Knot
In knot theory, the 63 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 62 knot. It is alternating, hyperbolic, and fully amphichiral. It can be written as the braid word :\sigma_1^\sigma_2^2\sigma_1^\sigma_2. \, Symmetry Like the figure-eight knot, the 63 knot is fully amphichiral. This means that the 63 knot is amphichiral, meaning that it is indistinguishable from its own mirror image. In addition, it is also invertible, meaning that orienting the curve in either direction yields the same oriented knot. Invariants The Alexander polynomial of the 63 knot is :\Delta(t) = t^2 - 3t + 5 - 3t^ + t^, \, Conway polynomial is :\nabla(z) = z^4 + z^2 + 1, \, Jones polynomial is :V(q) = -q^3 + 2q^2 - 2q + 3 - 2q^ + 2q^ - q^, \, and the Kauffman polynomial is :L(a,z) = az^5 + z^5a^ + 2a^2z^4 + 2z^4a^ + 4z^4 + a^3z^3 + az^3 + z^3a^ + z^3a^ - 3a^2z^2 - 3z^2a^ - 6z^2 - a^3z - 2az - 2za^ - za^-3 + a^2 + a^ +3. \, The 63 kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
62 Knot
In knot theory, the 62 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 63 knot. This knot is sometimes referred to as the Miller Institute knot, because it appears in the logo of the Miller Institute for Basic Research in Science at the University of California, Berkeley. The 62 knot is invertible but not amphichiral. Its Alexander polynomial is :\Delta(t) = -t^2 + 3t -3 + 3t^ - t^, \, its Conway polynomial is :\nabla(z) = -z^4 - z^2 + 1, \, and its Jones polynomial is :V(q) = q - 1 + 2q^ - 2q^ + 2q^ - 2q^ + q^. \, The 62 knot is a hyperbolic knot, with its complement having a volume of approximately 4.40083. Surface File:Superfície - bordo Nó 6,2.jpg, Surface of knot 6.2 Example Ways to assemble of knot 6.2 File:6₂ knot.webm, Example 1 File:6₂ knot (2).webm, Example 2 If a bowline The bowline ( or ) is an ancient and simple knot used to form a fixed loop at the end of a rope. It has the virtues ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stevedore Knot (mathematics)
In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ..., and it can also be described as a twist knot with four twists, or as the (5,−1,−1) pretzel link, pretzel knot. The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper knot, stopper at the end of a rope. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted loop (topology), loop. The stevedore knot is invertible knot, invertible but not amphichiral knot, amphichi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-twist Knot
In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot. Properties The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is :\Delta(t) = 2t-3+2t^, \, its Conway polynomial is :\nabla(z) = 2z^2+1, \, and its Jones polynomial is :V(q) = q^ - q^ + 2q^ - q^ + q^ - q^. \, Because the Alexander polynomial is not monic, the three-twist knot is not fibered. The three-twist knot is a hyperbolic knot, with its complement having a volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ... of approximately 2.82812. If the fibre of the knot in the initial ima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure-eight Knot (mathematics)
Figure 8 (figure of 8 in British English) may refer to: * 8 (number), in Arabic numerals Entertainment * ''Figure 8'' (album), a 2000 album by Elliott Smith * "Figure of Eight" (song), a 1989 song by Paul McCartney * '' Figure Eight EP'', a 2008 EP by This Et Al * "Figure 8" (song), a 2012 song by Ellie Goulding from ''Halcyon'' * "Figure Eight", an episode and song from the children's educational series ''Schoolhouse Rock!'' * "Figure of Eight", song by Status Quo from ''In Search of the Fourth Chord'' * "Figure 8", a song by FKA Twigs from the EP ''M3LL155X'' Geography * Figure Eight Island, North Carolina, United States * Figure Eight Lake, Alberta, Canada * Figure-Eight Loops, feature of the Historic Columbia River Highway in Guy W. Talbot State Park Mathematics and sciences * Figure-eight knot (mathematics), in knot theory * ∞, symbol meaning infinity * Lemniscate, various types of mathematical curve that resembles a figure 8 * Figure 8, a two-lobed Lissajous c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Menasco
William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory. Biography Menasco received his B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the University of California, Berkeley in 1981, where his advisor was Robion Kirby. He served as assistant professor at Rutgers University from 1981 to 1984. He then taught as a visiting professor at the University at Buffalo where he became an assistant professor in 1985, an associate professor in 1991. In 1994 he became a professor at the University at Buffalo where he currently serves. Work Menasco proved that a link with an alternating diagram, such as an alternating link, will be non-split if and only if the diagram is connected. Menasco, along with Morwen Thistlethwaite proved the Tait flyping conjecture, which states that, given any two reduced alternating diagrams D1,D2 of an oriented, prime alternating link, D1 may be transformed to D2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
