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Hyperbolic Volume (knot)
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture. Knot and link invariant A hyperbolic link is a link in the 3-sphere whose complement (the space formed by removing the link from the 3-sphere) can be given a complete Riemannian metric of constant negative curvature, giving it the structure of a hyperbolic 3-manifold, a quotient of hyperbolic space by a group acting freely and discontinuously on it. The components of the link will become cusps of the 3-manifold, and the manifold itself will have finite volume. By Mostow rigidity, when a link complement has a hyperbolic structure, this structure is uniquely determined, and any geometric invariants of the struct ... [...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 term ''blue'' generally describes colours perceived by humans observing light with a dominant wavelength that's 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 the 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 ultramar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutation (knot Theory)
In the mathematics, mathematical field of knot theory, a mutation is an knot operation, operation on a knot that can produce different knots. Suppose ''K'' is a knot given in the form of a knot diagram. Consider a disc ''D'' in the projection plane of the diagram whose boundary circle intersects ''K'' exactly four times. We may suppose that (after planar isotopy) the disc is geometrically round and the four points of intersection on its boundary with ''K'' are equally spaced. The part of the knot inside the disc is a tangle (mathematics), tangle. There are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections. A mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a mutant of ''K''. Mutants can be difficult to distinguish as they have a number of the same invariants. They have the same hyperbolic link, hyperbolic volume (by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borromean Rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are link (knot theory), topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternating link, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing. The Borromean rings are named after the Italian House of Borromeo, who used the circular form of these rings as an element of their coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan. They have been used in Christian symbolism as a sign of the Trinity, and in modern commerce as the ... [...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 ... [...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 of a reduced diagram of a knot is an invariant (see Tait conjectures), as the two diagrams for the pair have different writhes. In some later knot tables, the knots have been renumbered slightly (knots 10163 to 10166 are renumbered as 10162 to 10165) so t ... [...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.) The 74 knot is a Lissajous knot, representable for example by the parametric equation : \begin x &= \cos (2t+0.22)\\ y &= \cos (3t+1.10)\\ z &= \cos 7t \end File:EndlessKnot03d.png, One form of the Endless knot of Buddhism File:Interwoven unicursal hexagram.svg, Interwoven unicursal hexagram. File:Celtic-kn ... [...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 () is an ancient and simple knot used to form a fixed loop at the end of a rope. It has the virtues of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stevedore Knot (mathematics)
A dockworker (also called a longshoreman, stevedore, docker, wharfman, lumper or wharfie) is a waterfront manual laborer who loads and unloads ships. As a result of the intermodal shipping container revolution, the required number of dockworkers has declined by over 90% since the 1960s. Etymology The word ''stevedore'' () originated in Portugal or Spain, and entered the English language through its use by sailors. It started as a phonetic spelling of ''estivador'' ( Portuguese) or ''estibador'' (Spanish), meaning ''a man who loads ships and stows cargo'', which was the original meaning of ''stevedore'' (though there is a secondary meaning of "a man who stuffs" in Spanish); compare Latin ''stīpāre'' meaning ''to stuff'', as in ''to fill with stuffing''. In Ancient and Modern Greek, the verb στοιβάζω (stivazo) means pile up. In Great Britain and Ireland, people who load and unload ships are usually called ''dockers''; in Australia, they are called ''stevedores'', ''dock ... [...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^, \, since \begin1 & -1 \\ 0 & 2\end is a possible Seifert matrix, or because of its Conway polynomial, which 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 regions in 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, cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure-eight Knot (mathematics)
In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number (knot theory), crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot. Origin of name The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. Description A simple parametric representation of the figure-eight knot is as the set of all points (''x'',''y'',''z'') where : \begin x & = \left(2 + \cos \right) \cos \\ y & = \left(2 + \cos \right) \sin \\ z & = \sin \end for ''t'' varying over the real numbers (see 2D visual realization at bottom right). The figure-eight knot is Prime knot, prime, alternating knot, alternating, rational knot, rational with an associated value of 5/3, and is Chiral kn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * '' Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...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 Moshe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
