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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] |
Tricoloring
In the mathematics, mathematical field of knot theory, the tricolorability of a knot (mathematics), knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an Knot invariant, isotopy invariant, and hence can be used to distinguish between two different (non-Regular isotopy, 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. Som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mladen Bestvina
Mladen Bestvina (born 1959) is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah. Life and career Mladen Bestvina is a three-time medalist at the International Mathematical Olympiad (two silver medals in 1976 and 1978 and a bronze medal in 1977). He received a B. Sc. in 1982 from the University of Zagreb. He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh. He was a visiting scholar at the Institute for Advanced Study in 1987-88 and again in 1990–91. Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993.Mladen Bestvina: Distinguished Professor ''After ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the Vertex (graph theory), 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 (graph theory), edges 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 (graph theory), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fox N-coloring
In the 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 quandle. Definition Let ''L'' be a 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''-colorable, and \rho is called an ''n''-coloring of ''L' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Knot
In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), 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 greatest common divisor, gcd(''p, q'')). A torus knot is unknot, 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricolor Invariance On Reidemeister III
A triband is a vexillological style which consists of three stripes arranged to form a flag. These stripes may be two or three colours, and may be charged with an emblem in the middle stripe. Not all tribands are tricolour flags, which requires three unique colours. Design Outside of the name, which requires three bands of colour, there are no other requirements for what a triband must look like, so there are many flags that look very different from each other but are all considered tribands. Some triband flags (e.g. those of Germany, Russia and the Netherlands) have their stripes positioned horizontally, while others (e.g. that of Italy) position the stripes vertically. Often the stripes on a triband are of equal length and width, though this is not always the case, as can be seen in the flags of Colombia and Canada. Symbols on tribands may be seals, such as on the Belizean flag, or any manner of emblems of significance to the area the flag represents, such as in the flags o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
