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Ribbon Knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ''ribbon singularities''. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part of the disk through the slit. More precisely, this type of singularity is a closed arc consisting of intersection points of the disk with itself, such that the preimage of this arc consists of two arcs in the disc, one completely in the interior of the disk and the other having its two endpoints on the disk boundary. Morse-theoretic formulation A slice disc ''M'' is a smoothly embedded D^2 in D^4 with M \cap \partial D^4 = \partial M \subset S^3. Consider the function f\colon D^4 \to \mathbb R given by f(x,y,z,w) = x^2+y^2+z^2+w^2. By a small isotopy of ''M'' one can ensure that ''f'' restricts to a Morse function on ''M''. One says \partial M \subset \partial D^4 = S^3 is a ribbon knot if f_\colon M \to \mathbb R has no interior local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ribbon Knot 8 20
A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, cotton, and jute and of synthetic materials, such as polyester, nylon, and polypropylene. Ribbon is used for useful, ornamental, and symbolic purposes. Cultures around the world use ribbon in their hair, around the body, and as ornament on non-human animals, buildings, and packaging. Some popular fabrics used to make ribbons are satin, organza, sheer, silk, velvet, and grosgrain. Etymology The word ribbon comes from Middle English ''ribban'' or ''riban'' from Old French ''ruban'', which is probably of Germanic origin. Cloth Along with that of fringes, and other smallwares, the manufacture of cloth ribbons forms a special department of the textile industry">textile industries. The essential feature of a ribbon loom is the simultaneous weaving in one loom frame of two or m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In 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, 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. 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 it 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 diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental p ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics ( critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts To illustrate, consider a mountainous landscape surface M (more generally, a manifold). If f is the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the boundary of an Embedding, embedded disk in the 4-ball B^4, which is Local flatness, locally flat or Smoothness, smooth, respectively. Here we use S^3 = \partial B^4: the 3-sphere S^3 = \ is the boundary (topology), boundary of the four-dimensional ball (mathematics), ball B^4 = \. Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat. Usually, smoothly slice knots are also just called slice. Both types of slice knots are important in 3- and 4-dimensional topology. Smoothly slice knots are often illustrated using knots diagrams of ribbon knots and it is an open question whether there are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′). Cone construction The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Fox
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played an important role in the modernization of knot theory and of bringing it into the mainstream. Biography Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a PhD degree from Princeton University in 1939. His doctoral dissertation, ''On the Lusternick–Schnirelmann Category'', was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of the Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bridge Number
In the mathematical field of knot theory, the bridge number, also called the bridge index, is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot. Definition Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an unbroken arc in this diagram a bridge if it includes at least one overcrossing. Then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot.. Bridge numbers were first studied in the 1950s by Horst Schubert. The bridge number can equivalently be defined geometrically instead of topologically. In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines. Equivalently, the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pretzel Link
In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number of tangles made of two intertwined circular helices. The tangles are connected cyclicly, and the first component of the first tangle is connected to the second component of the second tangle, the first component of the second tangle is connected to the second component of the third tangle, and so on. Finally, the first component of the last tangle is connected to the second component of the first. A pretzel link which is also a knot (that is, a link with one component) is a pretzel knot. Each tangle is characterized by its number of twists: positive if they are counter-clockwise or left-handed, negative if clockwise or right-handed. In the standard projection of the (p_1,\,p_2,\dots,\,p_n) pretzel link, there are p_1left-handed crossings in the first tangle, p_2in the second, and, in general, p_nin the nth. A pretzel link can also be described as a Montesinos link w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quanta Magazine
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science News'' in October 2012, but it was renamed to its current title in July 2013. It was founded by the former ''New York Times'' journalist Thomas Lin, who was the magazine's editor-in-chief until 2024. The two deputy editors are John Rennie and Michael Moyer, formerly of ''Scientific American'', and the art director is Samuel Velasco. In 2024, Samir Patel became the magazine's second editor in chief. Content The articles in the magazine are freely available to read online. ''Scientific American'', ''Wired'', ''The Atlantic'', and ''The Washington Post'', as well as international science publications like '' Spektrum der Wissenschaft'', have reprinted articles from the magazine. In November 2018, MIT Press The MIT Press is the uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In 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, 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. 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 it 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 diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knots And Links
A knot is a fastening in rope or interwoven lines. Knot or knots may also refer to: Other common meanings * Knot (unit), of speed * Knot (wood), a timber imperfection Arts, entertainment, and media Films * ''Knots'' (film), a 2004 film * ''Knots'', a 2011 film starring Kimberly-Rose Wolter Music * Rosette (music), soundhole decoration on string instruments * ''Knots'' (Sons of Noel and Adrian album), a 2012 album by Sons of Noel and Adrian * ''Knots'' (Crash of Rhinos album), a 2013 album by Crash of Rhinos * ''Knots'' (EP), a 2018 extended play by Moira Dela Torre and Nieman Gatus * "Knots", a song by Gentle Giant Other uses in arts, entertainment, and media * KNOT, a radio station in Prescott, Arizona, United States * ''Knots'', a 1970 book of poetry by R. D. Laing Biology * Red knot, a wading bird (simply called "knot" in Europe) * Great knot, a wading bird * Trigger point or knot, a small, hard, tender spot in a muscle *''Bulbus glandis'' or knot, an erectile swell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
