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Alternating Diagram
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram. Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions. The simplest non-alternating prime knots have 8 crossings (and there are three such: 819, 820, 821). It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly. 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. This led Ralph Fox to ask, "What is an alternating knot?" By this he was asking what non-diag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot 8sb19
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ''bend'' fastens two ends of a rope to each another; a ''loop knot'' is any knot creating a loop; and ''splice'' denotes any multi-strand knot, including bends and loops. A knot may also refer, in the strictest sense, to a stopper or knob at the end of a rope to keep that end from slipping through a grommet or eye. Knots have excited interest since ancient times for their practical uses, as well as their topological intricacy, studied in the area of mathematics known as knot theory. History Knots and knotting have been used and studied throughout history. For example, Chinese knotting is a decorative handicraft art that began as a form of Chinese folk art in the Tang and Song Dynasty (960–1279 AD) in China, later popularized in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frise
Frise may refer to: * Frise (department), the French name of Friesland as a ' of the First French Empire * Frise, Somme, a commune of the Somme department in France * Leslie Frise (1895-1979), British aerospace engineer and aircraft designer ** Aileron#Frise ailerons, a type of aircraft part See also * {{dab, geodis, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Lackenby
Marc Lackenby is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory. Lackenby studied mathematics at the University of Cambridge beginning in 1990, and earned his Ph.D. in 1997, with a dissertation on ''Dehn Surgery and Unknotting Operations'' supervised by W. B. R. Lickorish. After positions as Miller Research Fellow at the University of California, Berkeley and as Research Fellow at Cambridge, he joined Oxford as a Lecturer and Fellow of St Catherine's in 1999. He was promoted to Professor at Oxford in 2006. Lackenby's research contributions include a proof of a strengthened version of the 2 theorem on sufficient conditions for Dehn surgery to produce a hyperbolic manifold, a bound on the hyperbolic volume of a knot complement of an alternating knot, and a proof that every diagram of the unknot can be transformed into a diagram without crossings by only a polynomial number of Reidemeister moves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Link
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers ''p'' and ''q''. A torus link arises if ''p'' and ''q'' are not coprime (in which case the number of components is gcd(''p, q'')). A torus knot is trivial (equivalent to the unknot) if and only if either ''p'' or ''q'' is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot. Geometrical representation A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following. The (''p'',''q'')-torus knot winds ''q'' times around a circle in the interior of the torus, and ''p'' times around its axis of rotational symmetry.. If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Haken Manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface. A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken. This conjecture was proven by Ian Agol. Haken manifolds were introduced by . proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. gave an algorithm to determine if a 3-manifold was Haken. Normal surfaces are ubiquitous in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolization Theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. Statement One form of Thurston's geometrization theorem states: If ''M'' is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of ''M'' has a complete hyperbolic structure of finite volume. The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique. The conditions that the manifold ''M'' should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C. to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, he r ... [...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] |
Louis Kauffman
Louis Hirsch Kauffman (born February 3, 1945) is an American mathematician, topologist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the introduction and development of the bracket polynomial and the Kauffman polynomial. Biography Kauffman was valedictorian of his graduating class at Norwood Norfolk Central High School in 1962. He received his B.S. at the Massachusetts Institute of Technology in 1966 and his Ph.D. in mathematics from Princeton University in 1972 (with William Browder as thesis advisor). Kauffman has worked at many places as a visiting professor and researcher, including the University of Zaragoza in Spain, the University of Iowa in Iowa City, the Institut des Hautes Études Scientifiques in Bures Sur Yevette, France, the Institut Henri Poincaré in Paris, France, the University of Bologna, Italy, the Federal University of Pernambuco in Recife, Brazil, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morwen Thistlethwaite
Morwen Bernard Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory. Biography Morwen Thistlethwaite received his BA from the University of Cambridge in 1967, his MSc from the University of London in 1968, and his PhD from the University of Manchester in 1972 where his advisor was Michael Barratt. He studied piano with Tanya Polunin, James Gibb and Balint Vazsonyi, giving concerts in London before deciding to pursue a career in mathematics in 1975. He taught at the North London Polytechnic from 1975 to 1978 and the Polytechnic of the South Bank, London from 1978 to 1987. He served as a visiting professor at the University of California, Santa Barbara for a year before going to the University of Tennessee, where he currently is a professor. His wife, Stella Thistlethwaite, also teaches at the University of Tennessee-Knoxville. Thistlet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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