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List Of Mathematical Knots And Links
This article contains a list of mathematical knots and links. See also list of knots, list of geometric topology topics. Knots Prime knots *01 knot/Unknot - a simple un-knotted closed loop *31 knot/Trefoil knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together *41 knot/Figure-eight knot (mathematics) - a prime knot with a crossing number four *51 knot/ Cinquefoil knot, (5,2)-torus knot, Solomon's seal knot, pentafoil knot - a prime knot with crossing number five which can be arranged as a star polygon (pentagram) *52 knot/ Three-twist knot - the twist knot with three-half twists *61 knot/Stevedore knot (mathematics) - a prime knot with crossing number six, it can also be described as a twist knot with four twists * 62 knot - a prime knot with crossing number six * 63 knot - a prime knot with crossing number six * 71 knot, septafoil knot, (7,2)-torus knot - a prime knot with crossing number seven, which can be arranged as a star polygon (heptagr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Table
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] |
Carrick Mat
The carrick mat is a flat woven decorative knot which can be used as a mat or pad. Its name is based on the mat's decorative-type Carrick bend#Decorative uses, carrick bend with the ends connected together, forming an endless knot. A larger form, called the prolong knot, is made by expanding the basic carrick mat by extending, twisting, and overlapping its outer bight (knot), bights, then weaving the free ends through them. This process may be repeated to produce an arbitrarily long mat. In its basic form it is the same as a 3-lead, 4-bight Turk's head knot. The basic carrick mat, made with two passes of rope, also forms the central motif in the logo of the International Guild of Knot Tyers. When tied to form a cylinder around the central opening, instead of lying flat, it can be used as a woggle. See also * List of knots References External links * Alternating knots and links Fibered knots and links Fully amphichiral knots and links Hyperbolic knots and links {{D ... [...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 tangles made of two intertwined circular helices. The tangles are connected cyclicly, the first component of the first tangle is connected to the second component of the second tangle, etc., with the first component of the last tangle connected to the second component of the first. A pretzel link which is also a knot (i.e. 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_1 left-handed crossings in the first , tangle, p_2 in the second, and, in general, p_n in the nth. A pretzel link can also be described as a Montesinos link with integer tangles. Some basic results The (p_1,p_2,\dots,p_n) pretzel link is a knot iff both n and all the p_i ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L10a140 Link
In the knot theory, mathematical theory of knots, L10a140 is the name in thThistlethwaite link tableof a link (knot theory), link of three loops, which has ten crossings between the loops when presented in its simplest visual form. It is of interest because it is presumably the simplest link which possesses the Brunnian link, Brunnian property — a link of connected components that, when one component is removed, becomes entirely unconnected — other than the six-crossing Borromean rings. In other words, no two loops are Hopf link, directly linked with each other, but all three are collectively interlinked, so removing any loop frees the other two. In the image in the infobox at right, the red loop is not interlinked with either the blue or the yellow loops, and if the red loop is removed, then the blue and yellow loops can also be disentangled from each other without cutting either one. According to work by Slavik V. Jablan, the L10a140 link can be seen as the second in an inf ... [...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] |
Brunnian Link
In knot theory, a branch of topology, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops (so that no two loops can be directly linked). The name ''Brunnian'' is after Hermann Brunn. Brunn's 1892 article ''Über Verkettung'' included examples of such links. Examples The best-known and simplest possible Brunnian link is the Borromean rings, a link of three unknots. However for every number three or above, there are an infinite number of links with the Brunnian property containing that number of loops. Here are some relatively simple three-component Brunnian links which are not the same as the Borromean rings: Image:Brunnian-3-not-Borromean.svg, 12-crossing link. Image:Three-triang-18crossings-Brunnian.svg, 18-crossing link. Image:Three-interlaced-squares-Brunnian-24crossings.svg, 24-crossing link. The simplest Brunnian link other than the 6-cros ... [...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] |
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] |
Hopf Link
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. Geometric realization A concrete model consists of two unit circles in perpendicular planes, each passing through the center of the other.. See in particulap. 77 This model minimizes the ropelength of the link and until 2002 the Hopf link was the only link whose ropelength was known. The convex hull of these two circles forms a shape called an oloid. Properties Depending on the relative orientations of the two components the linking number of the Hopf link is ±1. The Hopf link is a (2,2)-torus link with the braid word :\sigma_1^2.\, The knot complement of the Hopf link is R × ''S''1 × ''S''1, the cylinder over a torus. This space has a locally Euclidean geometry, so the Hopf link is not a hyperbolic link. The knot group of the Hopf link (the fund ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unlink
In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane. Properties * An ''n''-component link ''L'' ⊂ S3 is an unlink if and only if there exists ''n'' disjointly embedded discs ''D''''i'' ⊂ S3 such that ''L'' = ∪''i''∂''D''''i''. * A link with one component is an unlink if and only if it is the unknot. * The link group of an ''n''-component unlink is the free group on ''n'' generators, and is used in classifying Brunnian links. Examples * The Hopf link is a simple example of a link with two components that is not an unlink. * The Borromean rings form a link with three components that is not an unlink; however, any two of the rings considered on their own do form a two-component unlink. * Taizo Kanenobu has shown that for all ''n'' > 1 there exists a hyperbolic link of ''n'' components such that any proper sublink is an unlink ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Granny Knot (mathematics)
In knot theory, the granny knot is a composite knot obtained by taking the connected sum of two identical trefoil knots. It is closely related to the square knot, which can also be described as a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the granny knot and the square knot are the simplest of all composite knots. The granny knot is the mathematical version of the common granny knot. Construction The granny knot can be constructed from two identical trefoil knots, which must either be both left-handed or both right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the granny knot. It is important that the original trefoil knots be identical to each another. If mirror-image trefoil knots are used instead, the result is a square knot. Properties The crossing number of a granny knot is six, which is the smallest possible crossing number for a composite kn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Knot (mathematics)
In knot theory, the square knot is a composite knot obtained by taking the connected sum of a trefoil knot with its reflection. It is closely related to the granny knot, which is also a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the square knot and the granny knot are the simplest of all composite knots. The square knot is the mathematical version of the common reef knot. Construction The square knot can be constructed from two trefoil knots, one of which must be left-handed and the other right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the square knot. It is important that the original trefoil knots be mirror images of one another. If two identical trefoil knots are used instead, the result is a granny knot. Properties The square knot is amphichiral, meaning that it is indistinguishable from its own mirror image. The crossing number of a squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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