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Fibered Knot
In knot theory, a branch of mathematics, a knot or link K in the 3-dimensional sphere S^3 is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t then the intersection of F_s and F_t is exactly K. Examples Knots that are fibered For example: * The unknot, trefoil knot, and figure-eight knot are fibered knots. * The Hopf link is a fibered link. Knots that are not fibered The Alexander polynomial of a fibered knot is monic, i.e. the coefficients of the highest and lowest powers of ''t'' are plus or minus 1. Examples of knots with nonmonic Alexander polynomials abound, for example the twist knots have Alexander polynomials qt-(2q+1)+qt^, where ''q'' is the number of half-twists. In particular the stevedore knot is not fibered. Related constructions Fibered knots and links ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blue Figure-Eight Knot
Blue is one of the three primary colours in the RYB colour model (traditional colour theory), as well as in the RGB (additive) colour model. It lies between violet and cyan on the spectrum of visible light. The eye perceives blue when observing light with a dominant wavelength between approximately 450 and 495 nanometres. Most blues contain a slight mixture of other colours; azure contains some green, while ultramarine contains some violet. The clear daytime sky and the deep sea appear blue because of an optical effect known as Rayleigh scattering. An optical effect called Tyndall effect explains blue eyes. Distant objects appear more blue because of another optical effect called aerial perspective. Blue has been an important colour in art and decoration since ancient times. The semi-precious stone lapis lazuli was used in ancient Egypt for jewellery and ornament and later, in the Renaissance, to make the pigment ultramarine, the most expensive of all pigments. In the ei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twist Knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots. Construction A twist knot is obtained by linking together the two ends of a twisted loop. Any number of half-twists may be introduced into the loop before linking, resulting in an infinite family of possibilities. The following figures show the first few twist knots: Image:One-Twist Trefoil.png, One half-twist (trefoil knot, 31) Image:Blue Figure-Eight Knot.png, Two half-twists (figure-eight knot, 41) Image:Blue Three-Twist Knot.png, Three half-twists ( 52 knot) Image:Blue Stevedore Knot.png, Four half-twists (stevedore knot, 61) Image:Blue 7_2 Knot.png, Five half-twists (72 knot) Image:Blue 8_1 Knot.png, Six half-twists (81 knot) Properties A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology (journal)
''Topology'' was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of ''Topology'' appeared in 2009. Pricing dispute On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published. In 2007 the former editors of ''Topology'' announced the launch of the ''Journal of Topology'', published by Oxford University Press ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(−2,3,7) Pretzel Knot
In geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery constructions. Mathematical properties The (−2, 3, 7) pretzel knot has 7 ''exceptional'' slopes, Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight knot The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ..., which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes. References Further reading * Kirby, R., ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Book Decomposition
In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold ''M'' into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a famous theorem of Emmanuel Giroux (given below) that shows that contact geometry can be studied from an entirely topological viewpoint. Definition and construction Definition. An ''open book decomposition'' of a 3-dimensional manifold ''M'' is a pair (''B'', π) where :*''B'' is an oriented link in ''M'', called the binding of the open book; :*π: ''M'' \ ''B'' → ''S''1 is a fibration of the complement of ''B'' such that for each θ ∈ ''S''1, π−1(θ) is the interior of a compact surface Σ ⊂ ''M'' whose boundary is ''B''. The surface Σ is called the page of the open book. This is the special case ''m'' = 3 of an open book decomposition of an ''m''-dimensional manifold, for any ''m''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Fibration
In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book ''Singular Points of Complex Hypersurfaces'' (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space. Definition Let f(z_0,\dots,z_n) be a non-constant polynomial function of n+1 complex variables z_0,\dots,z_n where the vanishing locus of :f(z)\ \text\ \frac(z) is only at the origin, meaning the associated variety X = V(f) is not smooth at the origin. Then, for K = X \cap S^_ (a sphere inside \mathbb^ of radius \varepsilon > 0) the Milnor fibrationpg 68 associated to f is defined as the map :\phi\colon (S_\varepsilon^\setminus K) \to S^1\ \text \ x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Point Of A Curve
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curves in the plane may be defined as the set of points satisfying an equation of the form f(x,y) = 0, where is a polynomial function If is expanded as f = a_0 + b_0 x + b_1 y + c_0 x^2 + 2c_1 xy + c_2 y^2 + \cdots If the origin is on the curve then . If then the implicit function theorem guarantees there is a smooth function so that the curve has the form near the origin. Similarly, if then there is a smooth function so that the curve has the form near the origin. In either case, there is a smooth map from to the plane which defines the curve in the neighborhood of the origin. Note that at the origin b_0 = \frac, \; b_1 = \frac, so the curve is non-singular or ''regular'' at the origin if at least one of the partial derivatives o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cusp (singularity)
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation :\begin x &= f(t)\\ y &= g(t), \end a cusp is a point where both derivatives of and are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope \lim (g'(t)/f'(t))). Cusps are ''local singularities'' in the sense that they involve only one value of the parameter , in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by an implicit equation :F(x,y) = 0, which is smooth, cusps are points where the terms of lowest degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone (topology)
In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching ''X'' into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by CX or by \operatorname(X). Definitions Formally, the cone of ''X'' is defined as: :CX = (X \times ,1\cup_p v\ =\ \varinjlim \bigl( (X \times ,1 \hookleftarrow (X\times \) \xrightarrow v\bigr), where v is a point (called the vertex of the cone) and p is the projection to that point. In other words, it is the result of attaching the cylinder X \times ,1/math> by its face X\times\ to a point v along the projection p: \bigl( X\times\ \bigr)\to v. If X is a non-empty compact subspace of Euclidean space, the cone on X is homeomorphic to the union of segments from X to any fixed point v \not\in X such that these segments intersect only by v itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane Curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. Symbolic representation A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f(x,y)=0 for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x'' – that is, rewritten as y=g(x) or x=h(y) for specific function ''g'' or ''h'' – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form (x,y)=(x(t), y(t)) for specific functions x(t) and y(t). Plane curves can sometimes also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Algebraic Geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas. Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties through the minimal model program and the construction of moduli spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
