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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 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] |
Catalan's Constant
In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is irrational number, irrational, let alone transcendental number, transcendental. has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven". Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865. Uses In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal polyhedron, ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link. It is 1/8 of the volume of the complement of the Borromean rings. In combinatorics and statistical mechanics, it arises in connection with counting domino tilings, spannin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Whitehead Double
Whitehead may refer to: * Whitehead, a blocked sweat/sebaceous duct of the skin known medically as a closed comedo. * Whitehead (bird), a small species of passerine bird, endemic to New Zealand. * Whitehead building, heritage listed residence of the principal of the University of Adelaide's Lincoln College. * Whitehead (patience), a patience game related to Klondike. * Whitehead (surname), a surname. * Whitehead torpedo, the first effective self-propelled torpedo, invented by Robert Whitehead in 1866. * Whiteheads, another name for the wheat disease take-all. * USS Whitehead, USS ''Whitehead'' (1861–1865), American Civil War, 136-ton screw steam gunboat. Places * Canada: ** The Rural Municipality of Whitehead, Manitoba ** Whitehead, Nova Scotia, on Tor Bay * Hong Kong ** Whitehead, Hong Kong, a cape (geography), cape at Wu Kai Sha * Northern Ireland ** Whitehead, County Antrim, a small town in Northern Ireland * United States: ** Cape Whitehead, Cumberland County, Maine 43.38 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solomon's Knot
Solomon's knot () is a traditional decorative motif used since ancient times, and found in many cultures. Despite the name, it is classified as a link, and is not a true knot according to the definitions of mathematical knot theory. Structure The Solomon's knot consists of two closed loops, which are doubly interlinked in an interlaced manner. If laid flat, the Solomon's knot is seen to have four crossings where the two loops interweave under and over each other. This contrasts with two crossings in the simpler Hopf link. In most artistic representations, the parts of the loops that alternately cross over and under each other become the sides of a central square, while four loopings extend outward in four directions. The four extending loopings may have oval, square, or triangular endings, or may terminate with free-form shapes such as leaves, lobes, blades, wings etc. Occurrences The Solomon's knot often occurs in ancient Roman mosaics, usually represented as two int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Manifold
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to \R^3. discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper where he incorrectly claimed that no such manifold exists. A contractible manifold is one that can continuously be shrunk to a point inside the manifold itself. For example, an open ball is a contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask whether ''all'' contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the Riemann mapping theorem. Dimension 3 presents the first counterexample: the Whitehead manifold. Construction Take a copy of S^3, the three-dimensional sphere. Now find a compact unknotted solid torus T_1 inside the sphere. (A solid torus is an ordinary three-dimensional doughnut, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by Henri Poincaré in 1904, the Grigori Perelman's theorem concerns spaces that locally look like ordinary Euclidean space, three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each path (topology), loop in the space can be continuously tightened to a point, then it is necessarily a 3-sphere, three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century. The Perelman's proof built upon Richard S. Hamilton's ideas of using the Ricci flow to solve the problem. By developing a number of breakthrough new techniques and results in the theory of Ricci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weeks Manifold
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… () and showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by as well as . Volume Since the Weeks manifold is an arithmetic hyperbolic 3-manifold, its volume can be computed using its arithmetic data and a formula due to Armand Borel: : V_w = \frac = 0.942707\dots where k is the number field generated by \theta satisfying \theta^3-\theta+1=0 and \zeta_k is the Dedekind zeta function of k. Alternatively, : V_w = \Im(\rm_2(\theta)+\ln, \theta, \ln(1-\theta)) = 0.942707\dots where \rm_n is the polylogarithm and , x, is the absolute value of the complex root \theta (with positive imaginary part) of the cubic. Related manifolds The cusped hyperboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure-eight Knot (mathematics)
Figure 8 (figure of 8 in British English) may refer to: * 8 (number), in Arabic numerals Entertainment * ''Figure 8'' (album), a 2000 album by Elliott Smith * "Figure of Eight" (song), a 1989 song by Paul McCartney * '' Figure Eight EP'', a 2008 EP by This Et Al * "Figure 8" (song), a 2012 song by Ellie Goulding from ''Halcyon'' * "Figure Eight", an episode and song from the children's educational series ''Schoolhouse Rock!'' * "Figure of Eight", song by Status Quo from ''In Search of the Fourth Chord'' * "Figure 8", a song by FKA Twigs from the EP ''M3LL155X'' Geography * Figure Eight Island, North Carolina, United States * Figure Eight Lake, Alberta, Canada * Figure-Eight Loops, feature of the Historic Columbia River Highway in Guy W. Talbot State Park Mathematics and sciences * Figure-eight knot (mathematics), in knot theory * ∞, symbol meaning infinity * Lemniscate, various types of mathematical curve that resembles a figure 8 * Figure 8, a two-lobed Lissajous c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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