|
Flype
In the mathematical theory of knots, a flype is a kind of manipulation of knot and link diagrams used in the Tait flyping conjecture. It consists of twisting a part of a knot, a tangle T, by 180 degrees. Flype comes from a Scots word meaning ''to fold'' or ''to turn back'' ("as with a sock").. Tait used the term to mean, "a change of infinite complementary region"). Two reduced alternating diagrams of an alternating link can be transformed to each other using flypes. This is the Tait flyping conjecture, proven in 1991 by Morwen Thistlethwaite and William Menasco. See also * Reidemeister move Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...s are another commonly studied kind of manipulation to knot diagrams. References Knot operations {{knottheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flype
In the mathematical theory of knots, a flype is a kind of manipulation of knot and link diagrams used in the Tait flyping conjecture. It consists of twisting a part of a knot, a tangle T, by 180 degrees. Flype comes from a Scots word meaning ''to fold'' or ''to turn back'' ("as with a sock").. Tait used the term to mean, "a change of infinite complementary region"). Two reduced alternating diagrams of an alternating link can be transformed to each other using flypes. This is the Tait flyping conjecture, proven in 1991 by Morwen Thistlethwaite and William Menasco. See also * Reidemeister move Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...s are another commonly studied kind of manipulation to knot diagrams. References Knot operations {{knottheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tait Conjectures
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture. Background Tait came up with his conjectures after his attempt to tabulate all knots in the late 19th century. As a founder of the field of knot theory, his work lacks a mathematically rigorous framework, and it is unclear whether he intended the conjectures to apply to all knots, or just to alternating knots. It turns out that most of them are only true for alternating knots. In the Tait conjectures, a knot diagram is called "reduced" if all the "isthmi", or "nugatory crossings" have been removed. Crossing number of alternating knots Tait conjectured that in certain circumstances, crossing number was a knot invariant, specifically: Any reduced diagram of an alterna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tait Flyping Conjecture
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture. Background Tait came up with his conjectures after his attempt to tabulate all knots in the late 19th century. As a founder of the field of knot theory, his work lacks a mathematically rigorous framework, and it is unclear whether he intended the conjectures to apply to all knots, or just to alternating knots. It turns out that most of them are only true for alternating knots. In the Tait conjectures, a knot diagram is called "reduced" if all the "isthmi", or "nugatory crossings" have been removed. Crossing number of alternating knots Tait conjectured that in certain circumstances, crossing number was a knot invariant, specifically: Any reduced diagram of an alterna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), 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, Unknot, 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 descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Diagram
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 di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangle (knot Theory)
In mathematics, a tangle is generally one of two related concepts: * In John Conway's definition, an ''n''-tangle is a proper embedding of the disjoint union of ''n'' arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2''n'' marked points on the ball's boundary. * In link theory, a tangle is an embedding of ''n'' arcs and ''m'' circles into \mathbf^2 \times ,1/math> – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance. (A quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, ''Journal of Combinatorial Theory'' B 52 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to matroids.) The balance of this article discusses Conway's sense of tangles; fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scots Language
Scots ( endonym: ''Scots''; gd, Albais, ) is an Anglic language variety in the West Germanic language family, spoken in Scotland and parts of Ulster in the north of Ireland (where the local dialect is known as Ulster Scots). Most commonly spoken in the Scottish Lowlands, Northern Isles and northern Ulster, it is sometimes called Lowland Scots or Broad Scots to distinguish it from Scottish Gaelic, the Goidelic Celtic language that was historically restricted to most of the Scottish Highlands, the Hebrides and Galloway after the 16th century. Modern Scots is a sister language of Modern English, as the two diverged independently from the same source: Early Middle English (1150–1300). Scots is recognised as an indigenous language of Scotland, a regional or minority language of Europe, as well as a vulnerable language by UNESCO. In the 2011 United Kingdom census, 2011 Scottish Census, over 1.5 million people in Scotland reported being able to speak Scots. As there are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Link
Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alternating knot, a knot or link diagram for which the crossings alternate under, over, under, over, as one travels along each component of the link * Alternating map, a multilinear map that is zero whenever any two of its arguments are equal * Alternating operator, a multilinear map in algebra * Alternating permutation, a type of permutation studied in combinatorics * Alternating series, an infinite series in which the signs of the general terms alternate between positive and negative Electronics * Alternating current, a flow of electric charge that periodically reverses direction Other * Alternating turns, the process by which people in a conversation decide who is to speak next See also * Alternate bass * Alternative (other) Al ... [...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] |
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] |
Reidemeister Move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttingen. In 1920, he got the staatsexamen (master's degree) in mathematics, philosophy, physics, chemistry, and geology. He received his doctorate in 1921 with a thesis in algebraic number theory at the University of Hamburg under the supervision of Erich Hecke. He became interested in differential geometry; he edited Wilhelm Blaschke's 2nd volume about that issue, and both made an acclaimed contribution to the Jena DMV conference in Sep 1921. In October 1922 (or 1923) he was appointed assistant professor at the University of Vienna. While there he became familiar with the work of Wilhelm Wirtinger on knot theory, and became closely connected to Hans Hahn and the Vienna Circle. Its manifesto (1929) lists one of Reidemeister's publications i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
