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Tricoloring
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non- isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial. Rules of tricolorability In these rules a strand in a knot diagram will be a piece of the string that goes from one undercrossing to the next. A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:Weisstein, Eric W. (2010). ''CRC Concise Encyclopedia of Mathematics'', Second Edition, p.3045. . quoted at Accessed: May 5, 2013. :1. At least two colors must be used, and :2. At each crossing, the three incident strands are either all the same color or all different colors. Some references state instead that all three colors must be used.Gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricoloring
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non- isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial. Rules of tricolorability In these rules a strand in a knot diagram will be a piece of the string that goes from one undercrossing to the next. A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:Weisstein, Eric W. (2010). ''CRC Concise Encyclopedia of Mathematics'', Second Edition, p.3045. . quoted at Accessed: May 5, 2013. :1. At least two colors must be used, and :2. At each crossing, the three incident strands are either all the same color or all different colors. Some references state instead that all three colors must be used.Gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricolorable Knots And Links
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non- isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial. Rules of tricolorability In these rules a strand in a knot diagram will be a piece of the string that goes from one undercrossing to the next. A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:Weisstein, Eric W. (2010). ''CRC Concise Encyclopedia of Mathematics'', Second Edition, p.3045. . quoted at Accessed: May 5, 2013. :1. At least two colors must be used, and :2. At each crossing, the three incident strands are either all the same color or all different colors. Some references state instead that all three colors must be used.Gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trefoil Knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory. The trefoil knot is named after the three-leaf clover (or trefoil) plant. Descriptions The trefoil knot can be defined as the curve obtained from the following parametric equations: :\begin x &= \sin t + 2 \sin 2t \\ y &= \cos t - 2 \cos 2t \\ z &= -\sin 3t \end The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus (r-2)^2+z^2 = 1: :\begin x &= (2+\cos 3t) \cos 2t \\ y &= (2+\cos 3t )\sin 2t \\ z &= \sin 3t \end Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Failed Tricoloring Of Figure 8
Failure is the state or condition of not meeting a desirable or intended objective, and may be viewed as the opposite of success. The criteria for failure depends on context, and may be relative to a particular observer or belief system. One person might consider a failure what another person considers a success, particularly in cases of direct competition or a zero-sum game. Similarly, the degree of success or failure in a situation may be differently viewed by distinct observers or participants, such that a situation that one considers to be a failure, another might consider to be a success, a qualified success or a neutral situation. It may also be difficult or impossible to ascertain whether a situation meets criteria for failure or success due to ambiguous or ill-defined definition of those criteria. Finding useful and effective criteria, or heuristics, to judge the success or failure of a situation may itself be a significant task. In American history Cultural histori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fox N-coloring
In the mathematics, mathematical field of knot theory, Fox ''n''-coloring is a method of specifying a representation of a knot group or a group of a link (not to be confused with a link group) onto the dihedral group of order ''n'' where ''n'' is an odd integer by coloring arcs in a link diagram (the representation itself is also often called a Fox ''n''-coloring). Ralph Fox discovered this method (and the special case of tricolorability) "in an effort to make the subject accessible to everyone" when he was explaining knot theory to undergraduate students at Haverford College in 1956. Fox ''n''-coloring is an example of a conjugation Racks and quandles, quandle. Definition Let ''L'' be a Link (knot theory), link, and let \pi be the fundamental group of its complement. A representation \rho of \pi onto D_ the dihedral group of order ''2n'' is called a Fox ''n''-coloring (or simply an ''n''-coloring) of ''L''. A link ''L'' which admits such a representation is said to be ''n''-colorab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers ''p'' and ''q''. A torus link arises if ''p'' and ''q'' are not coprime (in which case the number of components is gcd(''p, q'')). A torus knot is trivial (equivalent to the unknot) if and only if either ''p'' or ''q'' is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot. Geometrical representation A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following. The (''p'',''q'')-torus knot winds ''q'' times around a circle in the interior of the torus, and ''p'' times around its axis of rotational symmetry.. If ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricolor Invariance On Reidemeister III
A tricolour () or tricolor () is a type of flag or banner design with a triband design which originated in the 16th century as a symbol of republicanism, liberty, or revolution. The flags of France, Italy, Romania, Mexico, and Ireland were all first adopted with the formation of an independent republic in the period of the French Revolution to the Revolutions of 1848, with the exception of the Irish tricolour, which dates from 1848 but was not popularised until the Easter Rising in 1916 and adopted in 1919. History The first association of the tricolour with republicanism is the orange-white-blue design of the Prince's Flag (''Prinsenvlag'', predecessor of the flags of the Netherlands), used from 1579 by William I of Orange-Nassau in the Eighty Years' War, establishing the independence of the Dutch Republic from the Spanish Empire. The flag of the Netherlands inspired both the French and Russian flags, which in turn inspired many further tricolour flags in other countries. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricolor Invariance On Reidemeister II
A tricolour () or tricolor () is a type of flag or banner design with a triband design which originated in the 16th century as a symbol of republicanism, liberty, or revolution. The flags of France, Italy, Romania, Mexico, and Ireland were all first adopted with the formation of an independent republic in the period of the French Revolution to the Revolutions of 1848, with the exception of the Irish tricolour, which dates from 1848 but was not popularised until the Easter Rising in 1916 and adopted in 1919. History The first association of the tricolour with republicanism is the orange-white-blue design of the Prince's Flag (''Prinsenvlag'', predecessor of the flags of the Netherlands), used from 1579 by William I of Orange-Nassau in the Eighty Years' War, establishing the independence of the Dutch Republic from the Spanish Empire. The flag of the Netherlands inspired both the French and Russian flags, which in turn inspired many further tricolour flags in other countries. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricolor Invariance On Reidemeister I
A tricolour () or tricolor () is a type of flag or banner design with a triband design which originated in the 16th century as a symbol of republicanism, liberty, or revolution. The flags of France, Italy, Romania, Mexico, and Ireland were all first adopted with the formation of an independent republic in the period of the French Revolution to the Revolutions of 1848, with the exception of the Irish tricolour, which dates from 1848 but was not popularised until the Easter Rising in 1916 and adopted in 1919. History The first association of the tricolour with republicanism is the orange-white-blue design of the Prince's Flag (''Prinsenvlag'', predecessor of the flags of the Netherlands), used from 1579 by William I of Orange-Nassau in the Eighty Years' War, establishing the independence of the Dutch Republic from the Spanish Empire. The flag of the Netherlands inspired both the French and Russian flags, which in turn inspired many further tricolour flags in other countries. ... [...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] |
Ambient Isotopy
In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let N and M be manifolds and g and h be embeddings of N in M. A continuous map :F:M \times ,1\rightarrow M is defined to be an ambient isotopy taking g to h if F_0 is the identity map, each map F_t is a homeomorphism from M to itself, and F_1 \circ g = h. This implies that the orientation must be preserved by ambient isotopies. For example, two knots that are mirror images of each other are, in general, not equivalent. See also * Isotopy * Regular homotopy *Regular isotopy References *M. A. Armstrong, ''Basic Topology'', Springer-Verlag Springer Science+Business Med ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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