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Trivial Knot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot. The unknot is the only knot that is the boundary of an embedded disk, which gives the characterization that only unknots have Seifert genus 0. Similarly, the unknot is the identity element with respect to the knot sum operation. Unknotting problem Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP. It is known that knot Floer homology and Khov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heegaard Floer Homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the sympl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3, :\pi_1(\mathbb^3 \setminus K). Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in S^3. Properties Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between certain pairs of inequivalent knots. This is because an equivalence between two knots is a self-homeomorphism of \mathbb^3 that is isotopic to the identity and sends the first knot onto the second. Such a homeomorphism restricts onto a homeomorphism of the complements of the knots, and this restricted homeomorphism induces an isomorphism of fundamental groups. However, it is possible for two inequivalent knots to have isomorphic knot groups (see below for an example). The ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Knot
In mathematics, in particular in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway. It is related by mutation to the Kinoshita–Terasaka knot, with which it shares the same Jones polynomial. Both knots also have the curious property of having the same Alexander polynomial and Conway polynomial as the unknot. The issue of the sliceness of the Conway knot was resolved in 2020 by Lisa Piccirillo, 50 years after John Horton Conway first proposed the knot. Her proof made use of Rasmussen's s-invariant, and showed that the knot is not a smoothly slice knot, though it is topologically slice (the Kinoshita–Terasaka knot is both). References External links Conway knoton The Knot Atlas ''The Knot Atlas'' is a website, an encyclopedia rather than atlas, dedicated to knot theory. It and its predecessor were created by mathematician Dror Bar-Natan, who maintains the current site with Scott Morrison. Acco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinoshita–Terasaka Knot
In knot theory, the Kinoshita–Terasaka knot is a particular prime knot. It has 11 crossings. The Kinoshita–Terasaka knot has a variety of interesting mathematical properties. It is related by mutation to the Conway knot, with which it shares a Jones polynomial. It has the same Alexander polynomial as the unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe .... References External links K11n42 at Knot Atlas {{DEFAULTSORT:Kinoshita-Terasaka knot Prime knots and links ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jones Polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^ with integer coefficients. Definition by the bracket Suppose we have an oriented link L, given as a knot diagram. We will define the Jones polynomial, V(L), using Louis Kauffman's bracket polynomial, which we denote by \langle~\rangle. Here the bracket polynomial is a Laurent polynomial in the variable A with integer coefficients. First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) :X(L) = (-A^3)^\langle L \rangle, where w(L) denotes the writhe of L in its given diagram. The writhe of a diagram is the number of positive crossings (L_ in the figure below) minus the number of negative crossings (L_). The writhe is not a knot invariant. X(L) is a knot invariant since it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander-Conway Polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial. Definition Let ''K'' be a knot in the 3-sphere. Let ''X'' be the infinite cyclic cover of the knot complement of ''K''. This covering can be obtained by cutting the knot complement along a Seifert surface of ''K'' and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a covering transformation ''t'' acting on ''X''. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stuck Unknot
In mathematics, a stuck unknot is a closed polygonal chain that is topologically equal to the unknot but cannot be deformed to a simple polygon by rigid motions of the segments.G. T. Toussaint, "A new class of stuck unknots in Pol-6," ''Contributions to Algebra and Geometry'', Vol. 42, No. 2, 2001, pp. 301–306. Similarly a stuck open chain is an open polygonal chain such that the segments may not be aligned by moving rigidly its segments. Topologically such a chain can be unknotted, but the limitation of using only rigid motions of the segments can create nontrivial knots in such a chain. Consideration of such "stuck" configurations arises in the study of molecular chains in biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and .... References Knots (knot theory) { ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Linkage (mechanical)
A mechanical linkage is an assembly of systems connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for example, and are called joints. A linkage modeled as a network of rigid links and ideal joints is called a kinematic chain. Linkages may be constructed from open chains, closed chains, or a combination of open and closed chains. Each link in a chain is connected by a joint to one or more other links. Thus, a kinematic chain can be modeled as a graph in which the links are paths and the joints are vertices, which is called a linkage graph. The movement of an ideal joint is generally associated with a subgroup of the group of Euclidean displacements. The number of parameters in the subgroup is called the degrees of freedom (DOF) of the joint. Mechanical linkages are usually designed to tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bight (knot)
In knot tying, a bight is a curved section or slack part between the two ends of a rope, string, or yarn.. "Any slack part of a rope between the two ends, particularly when curved or looped." A knot that can be tied using only the bight of a rope, without access to the ends, is described as in the bight. The term "bight" is also used in a more specific way when describing Turk's head knots, indicating how many repetitions of braiding are made in the circuit of a given knot. Bight vs. open loop Sources differ on whether an open loop or U-shaped curve in a rope qualifies as a bight. treats bights and loops as distinct, stating that a curve "no narrower than a semicircle" is a bight, while an open loop is a curve "narrower than a bight but with separated ends". However, ''The Illustrated Encyclopedia of Knots'' (2002) states: "Any section of line that is bent into a U-shape is a bight." Slipped knot In order to make a slipped knot (also slipped loop and quick release knot), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morwen Thistlethwaite
Morwen Bernard Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory. Biography Morwen Thistlethwaite received his BA from the University of Cambridge in 1967, his MSc from the University of London in 1968, and his PhD from the University of Manchester in 1972 where his advisor was Michael Barratt. He studied piano with Tanya Polunin, James Gibb and Balint Vazsonyi, giving concerts in London before deciding to pursue a career in mathematics in 1975. He taught at the North London Polytechnic from 1975 to 1978 and the Polytechnic of the South Bank, London from 1978 to 1987. He served as a visiting professor at the University of California, Santa Barbara for a year before going to the University of Tennessee, where he currently is a professor. His wife, Stella Thistlethwaite, also teaches at the University of Tennessee-Knoxville. Thistlet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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