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Crossing Number (knot Theory)
In the mathematics, mathematical area of knot theory, the crossing number of a knot (mathematics), knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant. Examples By way of example, the unknot has crossing number 0 (number), zero, the trefoil knot three and the figure-eight knot (mathematics), figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases. Tabulation 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 (this sub-ordering is not based on anything in particular, except that torus knots then twist knots are listed first). The listing goes 31 (the trefoil knot), 41 (the figure-eight knot), 51, 52, 61, etc. This order has not changed significa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Crossing Numbers Trefoil
Crossing may refer to: * Crossing (2008 film), ''Crossing'' (2008 film), a South Korean film * Crossing (2024 film), ''Crossing'' (2024 film), an internationally co-produced film * Crossing (album), ''Crossing'' (album), a 1985 album by world music/jazz group Oregon * The Crossing (Joker Xue album), ''The Crossing'' (Joker Xue album), a 2017 album by Chinese artist Joker Xue * Crossing (architecture), the junction of the four arms of a cruciform church * Crossing (knot theory), a visualization of intersections in mathematical knots * Crossing (physics), the relation between particle and antiparticle scattering * Crossing (plant), deliberate interbreeding of plants * Crossing (opera), an opera composed by Matthew Aucoin * Crossing oneself, a ritual hand motion made by some Christians * William Crossing (1847–1928), English writer Transportation * Intersection (road), also known as a crossing * Level crossing or grade crossing, a railway crossing a street or path at the same level ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Satellite Knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite ''link'' is one that orbits a companion knot ''K'' in the sense that it lies inside a regular neighborhood of the companion. A satellite knot K can be picturesquely described as follows: start by taking a nontrivial knot K' lying inside an unknotted solid torus V. Here "nontrivial" means that the knot K' is not allowed to sit inside of a 3-ball in V and K' is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot. This means there is a non-trivial embedding f\colon V \to S^3 and K = f\left(K'\right). The central core curve of the solid torus V is sent to a knot H, which is called the "companion knot" a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stick Number
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 stick number of K, denoted by \operatorname(K), is the smallest number of edges of a polygonal path equivalent Known values 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 stick number of a (p,q)-torus knot T(p,q) in case the parameters p and q are not too far from each other: The same result was found independently around the same time by a research group around Colin Adams, but for a smaller range of parameters. Bounds The stick number of a knot sum can be upper bounded by the stick numbers of the summands: \text(K_1\#K_2)\le \text(K_1)+ \text(K_2)-3 \, Related invariants The stick number of a knot K is related to its crossing number c(K) by the following ineq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linking Number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds, where linking numbers can also be fractions or just not exist at all). The linking number was introduced by Gauss in the form of the linking integral. It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications in mathematics and science, including quantum mechanics, electromagnetism, and the study of DNA supercoiling. Definition Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of the following standard positions. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bridge Number
In the mathematical field of knot theory, the bridge number, also called the bridge index, is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot. Definition Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an unbroken 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 numbers were 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 finite number of bridges whose projections onto the plane are straight lines. Equivalently, the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Asymptotic Crossing Number
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Average Crossing Number
In the mathematical subject of knot theory, the average crossing number of a knot is the result of averaging over all directions the number of crossings in a knot diagram of the knot obtained by projection onto the plane orthogonal to the direction. The average crossing number is often seen in the context of physical knot theory. Definition More precisely, if ''K'' is a smooth knot, then for almost every unit vector ''v'' giving the direction, orthogonal projection onto the plane perpendicular to ''v'' gives a knot diagram, and we can compute the crossing number, denoted ''n''(''v''). The average crossing number is then defined as the integral over the unit sphere: : \frac\int_ n(v) \, dA where ''dA'' is the area form on the 2-sphere. The integral makes sense because the set of directions where projection doesn't give a knot diagram is a set of measure zero and ''n''(''v'') is locally constant when defined. Alternative formulation A less intuitive but computationally useful def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Composite Knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non- trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links. It can be a nontrivial problem to determine whether a given knot is prime or not. A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus ''p'' times in one direction and ''q'' times in the other, where ''p'' and ''q'' are coprime integers. Knots are characterized by their crossing numbers. The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3)-torus knot. The figure-eight knot, with four crossings, is the simplest non-torus knot. For any positive integer ''n'', there are a finite number of prime knots with ''n'' crossings. The first few values for exclusively prime knots and for prime ''or'' composite kn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gel Electrophoresis
Gel electrophoresis is an electrophoresis method for separation and analysis of biomacromolecules (DNA, RNA, proteins, etc.) and their fragments, based on their size and charge through a gel. It is used in clinical chemistry to separate proteins by charge or size (IEF agarose, essentially size independent) and in biochemistry and molecular biology to separate a mixed population of DNA and RNA fragments by length, to estimate the size of DNA and RNA fragments, or to separate proteins by charge. Nucleic acid molecules are separated by applying an electric field to move the negatively charged molecules through a gel matrix of agarose, polyacrylamide, or other substances. Shorter molecules move faster and migrate farther than longer ones because shorter molecules migrate more easily through the pores of the gel. This phenomenon is called sieving. Proteins are separated by the charge in agarose because the pores of the gel are too large to sieve proteins. Gel electrophoresi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Normal Surface
In mathematics, a normal surface is a Surface (topology), surface inside a triangulated 3-manifold that intersects each tetrahedron in several components called normal disks. Each normal disk is either a ''triangle'' which cuts off a vertex of the tetrahedron, or a ''quadrilateral'' which separates pairs of vertices. In a given tetrahedron there cannot be two quadrilaterals separating different pairs of vertices, since such quadrilaterals would intersect in a line, causing the surface to be self-intersecting. Dually, a normal surface can be considered as a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner, similar to the above. The concept of a normal surface can be generalized to arbitrary polyhedra. There are also related notions of almost normal surfaces and spun normal surfaces. In an almost normal surface, one tetrahedron in the triangulation has a single exceptional piece. This is either an ''octagon'' that separates p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Marc Lackenby
Marc Lackenby is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory. Lackenby studied mathematics at the University of Cambridge beginning in 1990, and earned his Ph.D. in 1997, with a dissertation on ''Dehn Surgery and Unknotting Operations'' supervised by W. B. R. Lickorish. After positions as Miller Research Fellow at the University of California, Berkeley and as Research Fellow at Cambridge, he joined Oxford as a Lecturer and Fellow of St Catherine's in 1999. He was promoted to Professor at Oxford in 2006. Lackenby's research contributions include a proof of a strengthened version of the 2 theorem on sufficient conditions for Dehn surgery to produce a hyperbolic manifold, a bound on the hyperbolic volume of a knot complement of an alternating knot, and a proof that every diagram of the unknot can be transformed into a diagram without crossings by only a polynomial number of Reidemeister move ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
