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Gordon–Luecke Theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian. The theorem is usually stated as "knots are determined by their complements"; however this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected. Often two knots are considered equivalent if they are '' isotopic''. The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic. These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere. The theorem was proved by Cameron Gordon and John Luecke. Essential ingredients of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Surgery Theorem
In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold ''M'' whose boundary is a torus ''T'', if ''M'' is not a Seifert-fibered space A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ... and ''r,s'' are slopes on ''T'' such that their Dehn fillings have cyclic fundamental group, then the distance between ''r'' and ''s'' (the minimal number of times that two simple closed curves in ''T'' representing ''r'' and ''s'' must intersect) is at most 1. Consequently, there are at most three Dehn fillings of ''M'' with cyclic fundamental group. The theorem appeared in a 1987 paper written by Marc Culler, Cameron Gordon, John Luecke and Peter Shalen.M. Culler, C. Gordon, J. Luecke, P. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
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 be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * Mathematical Reviews * Zentralblatt MATH * * ISI Alerting Services * CompuMath Citation Index * |
Whitehead Link
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop. Structure A common way of describing this knot is formed by overlaying a figure-eight shaped loop with another circular loop surrounding the crossing of the figure-eight. The above-below relation between these two unknots is then set as an alternating link, with the consecutive crossings on each loop alternating between under and over. This drawing has five crossings, one of which is the self-crossing of the figure-eight curve, which does not count towards the linking number. Because the remaining crossings have equal numbers of under and over crossings on each loop, its linking number is 0. It is not isotopic to the unlink, but it is link homotopic to the unlink. Although this construction of the knot treats its two loops differently from each other, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
JHC Whitehead
John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princeton, New Jersey, in 1960. Life J. H. C. (Henry) Whitehead was the son of the Right Rev. Henry Whitehead, Bishop of Madras, who had studied mathematics at Oxford, and was the nephew of Alfred North Whitehead and Isobel Duncan. He was brought up in Oxford, went to Eton and read mathematics at Balliol College, Oxford. After a year working as a stockbroker, at Buckmaster & Moore, he started a PhD in 1929 at Princeton University. His thesis, titled ''The representation of projective spaces'', was written under the direction of Oswald Veblen in 1930. While in Princeton, he also worked with Solomon Lefschetz. He became a fellow of Balliol in 1933. In 1934 he married the concert pianist Barbara Smyth, great-great-granddaughter of Elizabeth Fry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thin Position
Thin may refer to: * a lean body shape. ''(See also: emaciation, underweight)'' * ''Thin'' (film), a 2006 HBO documentary about eating disorders * Paper Thin (other), referring to multiple songs * Thin (web server), a Ruby web-server based on Mongrel * Thin (name) See also * * * Thin client, a computer in a client-server architecture network. * Thin film, a material layer of about 1 μm thickness. * Thin-film deposition, any technique for depositing a thin film of material onto a substrate or onto previously deposited layers * Thin film memory, high-speed variation of core memory developed by Sperry Rand in a government-funded research project * Thin-film optics, the branch of optics that deals with very thin structured layers of different materials * Thin layer chromatography (TLC), a chromatography technique used in chemistry to separate chemical compounds * Thin layers (oceanography), congregations of phytoplankton and zooplankton in the water column * Thin le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Shalen
Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition. Life He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard College in 1966 and his Ph.D. from Harvard University in 1972. After posts at Columbia University, Rice University, and the Courant Institute, he joined the faculty of the University of Illinois at Chicago. Shalen was a Sloan Foundation Research Fellow in mathematics (1977—1979). In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley, California. He was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to three-dimensional topology and for exposition". ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the 3-sphere). Let ''N'' be a tubular neighborhood of ''K''; so ''N'' is a solid torus. The knot complement is then the complement of ''N'', :X_K = M - \mbox(N). The knot complement ''XK'' is a compact 3-manifold; the boundary of ''XK'' and the boundary of the neighborhood ''N'' are homeomorphic to a two-torus. Sometimes the ambient manifold ''M'' is understood to be 3-sphere. Context is needed to determine the usage. There are analogous definitions of link complement. Many knot invariants, such as the knot group, are really invariants of the complement of the knot. When the ambient space is the three-sphere no information is lost: the Gordon–Luecke theorem state ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Culler
Marc Edward Culler (born November 22, 1953) is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the University of Illinois at Chicago. Culler is the son of Glen Jacob Culler who was an important early innovator in the development of the Internet. Work Culler specializes in group theory, low dimensional topology, 3-manifolds, and hyperbolic geometry. Culler frequently collaborates with Peter Shalen and they have co-authored many papers. Culler and Shalen did joint work that related properties of representation varieties of hyperbolic 3-manifold groups to decompositions of 3-manifolds. In particular, Culler and Shalen used the Bass–Serre theory, applied to the function field of the SL(2,C)- Character variety of a 3-manifold, to obtain information about inco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Luecke (mathematician)
John Edwin Luecke is an American mathematician who works in topology and knot theory. He got his Ph.D. in 1985 from the University of Texas at Austin and is now a professor in the department of mathematics at that institution. Work Luecke specializes in knot theory and 3-manifolds. In a 1987 paper Luecke, Marc Culler, Cameron Gordon, and Peter Shalen proved the cyclic surgery theorem. In a 1989 paper Luecke and Cameron Gordon proved that knots are determined by their complements, a result now known as the Gordon–Luecke theorem. Dr Luecke received a NSF Presidential Young Investigator Award in 1992 and Sloan Foundation fellow in 1994. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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