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Satellite Knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite ''link'' is one that orbits a companion knot ''K'' in the sense that it lies inside a regular neighborhood of the companion. A satellite knot K can be picturesquely described as follows: start by taking a nontrivial knot K' lying inside an unknotted solid torus V. Here "nontrivial" means that the knot K' is not allowed to sit inside of a 3-ball in V and K' is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot. This means there is a non-trivial embedding f\colon V \to S^3 and K = f\left(K'\right). The central core curve of the solid torus V is sent to a knot H, which is called the "companion knot" a ... [...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] |
B Sat4
B, or b, is the second letter of the Latin-script alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is '' bee'' (pronounced ), plural ''bees''. It represents the voiced bilabial stop in many languages, including English. In some other languages, it is used to represent other bilabial consonants. History Old English was originally written in runes, whose equivalent letter was beorc , meaning "birch". Beorc dates to at least the 2nd-century Elder Futhark, which is now thought to have derived from the Old Italic alphabets' either directly or via Latin . The uncial and half-uncial introduced by the Gregorian and Irish missions gradually developed into the Insular scripts' . These Old English Latin alphabets supplanted the earlier runes, whose use was fully banned under King Canute in the early 11th century. The Norman Conquest popularised the Carolingian half-uncial forms whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Knot
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they manifest hyperbolas, not because something about them is exaggerated. * Hyperbolic angle, an unbounded variable referring to a hyperbola instead of a circle * Hyperbolic coordinates, location by geometric mean and hyperbolic angle in quadrant I *Hyperbolic distribution, a probability distribution characterized by the logarithm of the probability density function being a hyperbola * Hyperbolic equilibrium point, a fixed point that does not have any center manifolds * Hyperbolic function, an analog of an ordinary trigonometric or circular function * Hyperbolic geometric graph, a random network generated by connecting nearby points sprinkled in a hyperbolic space * Hyperbolic geometry, a non-Euclidean geometry * Hyperbolic group, a finitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrization Conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries ( Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by , and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then sever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
JSJ Decomposition
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: :Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered. The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently. The characteristic submanifold An alternative version of the JSJ decomposition states: :A closed irreducible orientable 3-manifold ''M'' has a submanifold Σ that is a Seifert manifold (possibly disconnected and with boundary) whose complement is atoroidal (and possibly disconnected). The submanifold Σ with the smallest number of boundary tori is called the characteristic submanifold of ''M''; it is unique (up to iso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horst Schubert
Horst Schubert (11 June 1919 – 2001) was a German mathematician. Schubert was born in Chemnitz and studied mathematics and physics at the Universities of Frankfurt am Main, Zürich and Heidelberg, where in 1948 he received his PhD under Herbert Seifert with thesis ''Die eindeutige Zerlegbarkeit eines Knotens in Primknoten''. From 1948 to 1956 Schubert was an assistant in Heidelberg, where he received in 1952 his habilitation qualification. From 1959 he was a ''professor extraordinarius'' and from 1962 a ''professor ordinarius'' at the University of Kiel. From 1969 to 1984 he was a professor at the University of Düsseldorf. In 1949 he published his proof that every oriented knot in S^3 decomposes as a connect-sum of prime knots in a unique way, up to reordering. After this proof he found a new proof based on his study of incompressible tori in knot complements; he published this work ''Knoten und Vollringe'' in ''Acta Mathematica'', where he defined satellite and companio ... [...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 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, 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 a 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 logo of Ballantine beer, giving them the alternative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot With Borromean Rings In Jsj Decomp
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B Sat1
B, or b, is the second letter of the Latin-script alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is '' bee'' (pronounced ), plural ''bees''. It represents the voiced bilabial stop in many languages, including English. In some other languages, it is used to represent other bilabial consonants. History Old English was originally written in runes, whose equivalent letter was beorc , meaning "birch". Beorc dates to at least the 2nd-century Elder Futhark, which is now thought to have derived from the Old Italic alphabets' either directly or via Latin . The uncial and half-uncial introduced by the Gregorian and Irish missions gradually developed into the Insular scripts' . These Old English Latin alphabets supplanted the earlier runes, whose use was fully banned under King Canute in the early 11th century. The Norman Conquest popularised the Carolingian half-uncial forms whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot (mathematics)
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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