1.
Torus
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In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a shape and is called a torus of revolution. Real-world examples of objects include inner tubes, swim rings, and the surface of a doughnut. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, a solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, in topology, a ring torus is homeomorphic to the Cartesian product of two circles, S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1, the ring torus is one way to embed this space into three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane. This produces an object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any space that is topologically equivalent to a torus. R is known as the radius and r is known as the minor radius. The ratio R divided by r is known as the aspect ratio, a doughnut has an aspect ratio of about 2 to 3. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is 2 + z 2 = r 2, or the solution of f =0, algebraically eliminating the square root gives a quartic equation,2 =4 R2. The three different classes of standard tori correspond to the three aspect ratios between R and r, When R > r, the surface will be the familiar ring torus. R = r corresponds to the torus, which in effect is a torus with no hole. R < r describes the self-intersecting spindle torus, when R =0, the torus degenerates to the sphere. When R ≥ r, the interior 2 + z 2 < r 2 of this torus is diffeomorphic to a product of an Euclidean open disc, the losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. In traditional spherical coordinates there are three measures, R, the distance from the center of the system, and θ and φ. As a torus has, effectively, two points, the centerpoints of the angles are moved, φ measures the same angle as it does in the spherical system. The center point of θ is moved to the center of r and these terms were first used in a discussion of the Earths magnetic field, where poloidal was used to denote the direction toward the poles
2.
Braid theory
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In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the operation is do the first braid on a set of strings. Such groups may be described by explicit presentations, as was shown by Emil Artin, for an elementary treatment along these lines, see the article on braid groups. Braid groups are also understood by a deeper mathematical interpretation, as the group of certain configuration spaces. To explain how to reduce a braid group in the sense of Artin to a fundamental group and that is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it. A path in the symmetric product is the abstract way of discussing n points of X, considered as an unordered n-tuple. Since we must require that the strings never pass through other, it is necessary that we pass to the subspace Y of the symmetric product. That is, we remove all the subspaces of Xn defined by conditions xi = xj and this is invariant under the symmetric group, and Y is the quotient by the symmetric group of the non-excluded n-tuples. Under the dimension condition Y will be connected, with this definition, then, we can call the braid group of X with n strings the fundamental group of Y. The case where X is the Euclidean plane is the one of Artin. In some cases it can be shown that the homotopy groups of Y are trivial. When X is the plane, the braid can be closed, i. e. corresponding ends can be connected in pairs, to form a link, i. e. a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, a theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the closure of a braid. Different braids can give rise to the link, just as different crossing diagrams can give rise to the same knot. Markov describes two moves on braid diagrams that yield equivalence in the corresponding closed braids, a single-move version of Markovs theorem, was published by Lambropoulou & Rourke. Vaughan Jones originally defined his polynomial as an invariant and then showed that it depended only on the class of the closed braid. The braid index is the least number of strings needed to make a closed braid representation of a link and it is equal to the least number of Seifert circles in any projection of a knot. Additionally, the length is the longest dimension of a braid
3.
Bridge number
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In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot. Given a knot or link, draw a diagram of the using the convention that a gap in the line denotes an undercrossing. Call an 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 number was 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 number of bridges whose projections onto the plane are straight lines. Equivalently the bridge number is the number of local maxima of the projection of the knot onto a vector. Every non-trivial knot has bridge number at least two, so the knots that minimize the number are the 2-bridge knots. It can be shown that every knot can be decomposed into two trivial n-tangles and hence 2-bridge knots are rational knots. If K is the sum of K1 and K2, then the bridge number of K is one less than the sum of the bridge numbers of K1. Crossing number Linking number Stick number Unknotting number Cromwell, Peter
4.
Crossing number (knot theory)
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In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. By way of example, the unknot has crossing number zero, tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant. The listing goes 31,41,51,52,61 and this order has not changed significantly since P. G. Tait published a tabulation of knots in 1877. There has been little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question if the crossing number is additive when taking knot sums. It is also expected that a satellite of a knot K should have larger crossing number than K, additivity of crossing number under knot sum has been proven for special cases, for example if the summands are alternating knots, or if the summands are torus knots. Marc Lackenby has also given a proof that there is a constant N >1 such that 1 N ≤ c r, but his method, there are connections between the crossing number of a knot and the physical behavior of DNA knots. For prime DNA knots, crossing number is a predictor of the relative velocity of the DNA knot in agarose gel electrophoresis. Basically, the higher the number, the faster the relative velocity. For composite knots, this not appear to be the case. There are related concepts of average crossing number and asymptotic crossing number, both of these quantities bound the standard crossing number. Asymptotic crossing number is conjectured to be equal to crossing number, other numerical knot invariants include the bridge number, linking number, stick number, and unknotting number
5.
Seifert surface
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In mathematics, a Seifert surface is a surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the knot or link. For example, many knot invariants are most easily calculated using a Seifert surface, Seifert surfaces are also interesting in their own right, and the subject of considerable research. Specifically, let L be an oriented knot or link in Euclidean 3-space. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link, a single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented and it is possible to associate surfaces to knots which are not oriented nor orientable, as well. The standard Möbius strip has the unknot for a boundary but is not considered to be a Seifert surface for the unknot because it is not orientable, the checkerboard coloring of the usual minimal crossing projection of the trefoil knot gives a Mobius strip with three half twists. As with the example, this is not a Seifert surface as it is not orientable. Applying Seiferts algorithm to this diagram, as expected, does produce a Seifert surface, in case, it is a punctured torus of genus g=1. It is a theorem that any link always has an associated Seifert surface and this theorem was first published by Frankl and Pontrjagin in 1930. A different proof was published in 1934 by Herbert Seifert and relies on what is now called the Seifert algorithm, the algorithm produces a Seifert surface S, given a projection of the knot or link in question. Suppose that link has m components, the diagram has d crossing points, then the surface S is constructed from f disjoint disks by attaching d bands. The homology group H1 is free abelian on 2g generators, the intersection form Q on H1 is skew-symmetric, and there is a basis of 2g cycles a1, a2. a2g with Q= the direct sum of g copies of. The 2g × 2g integer Seifert matrix V= has v the linking number in Euclidean 3-space of ai, every integer 2g × 2g matrix V with V − V * = Q arises as the Seifert matrix of a knot with genus g Seifert surface. The Alexander polynomial is computed from the Seifert matrix by A = d e t, the Alexander polynomial is independent of the choice of Seifert surface S, and is an invariant of the knot or link. The signature of a knot is the signature of the symmetric Seifert matrix V + V ⊤ and it is again an invariant of the knot or link. The genus of a knot K is the knot invariant defined by the genus g of a Seifert surface for K. For instance, An unknot—which is, by definition, the boundary of a genus zero
6.
Linking number
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In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves, the linking number was introduced by Gauss in the form of the linking integral. Any two closed curves in space, if allowed to pass through themselves but not each other and this determines the linking number, Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as regular homotopy, which requires that each curve be an immersion. However, this condition does not change the definition of linking number. This fact is most easily proven by placing one circle in standard position, in detail, A single curve is regular homotopic to a standard circle. The complement of a circle is homeomorphic to a solid torus with a point removed. The fundamental group of 3-space minus a circle is the integers and this can be seen via the Seifert–Van Kampen theorem. Thus homotopy classes of a curve in 3-space minus a circle are determined by linking number and it is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument. There is an algorithm to compute the number of two curves from a link diagram. Label each crossing as positive or negative, according to the following rule and that is, linking number = n 1 + n 2 − n 3 − n 42 where n1, n2, n3, n4 represent the number of crossings of each of the four types. The two sums n 1 + n 3 and n 2 + n 4 are always equal, note that n 1 − n 4 involves only the undercrossings of the blue curve by the red, while n 2 − n 3 involves only the overcrossings. Any two unlinked curves have linking number zero, however, two curves with linking number zero may still be linked. Reversing the orientation of either of the curves negates the linking number, the linking number is chiral, taking the mirror image of link negates the linking number. The convention for positive linking number is based on a right-hand rule, the winding number of an oriented curve in the x-y plane is equal to its linking number with the z-axis. More generally, if either of the curves is simple, then the first homology group of its complement is isomorphic to Z, in this case, the linking number is determined by the homology class of the other curve. In physics, the number is an example of a topological quantum number
7.
Stick number
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In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight sticks stuck end to end needed to form a knot. Specifically, given any knot K, the number of K. Six is the lowest stick number for any nontrivial knot, there are few knots whose stick number can be determined exactly. Gyo Taek Jin determined the number of a -torus knot T in case the parameters p and q are not too far from each other, stick =2 q. The same result was found independently around the time by a research group around Colin Adams. Why knot, knots, molecules and stick numbers, Plus Magazine, an accessible introduction into the topic, also for readers with little mathematical background. The Knot Book, An elementary introduction to the theory of knots, Providence, RI, American Mathematical Society. Brennan, Bevin M. Greilsheimer, Deborah L. Woo, stick numbers and composition of knots and links, Journal of Knot Theory and its Ramifications,6, 149–161, doi,10. 1142/S0218216597000121, MR1452436. Calvo, Jorge Alberto, Geometric knot spaces and polygonal isotopy, Journal of Knot Theory and its Ramifications,10, 245–267, doi,10. 1142/S0218216501000834, MR1822491. Jin, Gyo Taek, Polygon indices and superbridge indices of torus knots and links, Journal of Knot Theory and its Ramifications,6, 281–289, doi,10. 1142/S0218216597000170, MR1452441. Negami, Seiya, Ramsey theorems for knots, links and spatial graphs, Transactions of the American Mathematical Society,324, 527–541, doi,10. 2307/2001731, MR1069741. Huh, Youngsik, Oh, Seungsang, An upper bound on stick number of knots, Journal of Knot Theory and its Ramifications,20, 741–747, doi,10. 1142/S0218216511008966, stick numbers for minimal stick knots, KnotPlot Research and Development Site
8.
Unknotting number
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In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number n, then there exists a diagram of the knot which can be changed to unknot by switching n crossings, the unknotting number of a knot is always less than half of its crossing number. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the numbers for the first few knots, In general. Known cases include, The unknotting number of a nontrivial twist knot is equal to one. The unknotting number of a knot is equal to /2. The unknotting numbers of knots with nine or fewer crossings have all been determined. Crossing number Bridge number Linking number Stick number Unknotting problem Three_Dimensional_Invariants#Unknotting_Number, The Knot Atlas
