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Skein Theory
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots. In general, the converse does not hold. Skein relations are often used to give a simple definition of knot polynomials. A skein relation gives a linear relation between the values of a knot polynomial on a collection of three links which differ from each other only in a small region. For some knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively. Definition A skein relationship requires three link diagrams that are identical except at one crossing. The three diagrams must exhibit the three possibilities that could occur for the two line seg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot (mathematics)
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's ances ... [...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] |
Alternating Knot
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram. Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions. The simplest non-alternating prime knots have 8 crossings (and there are three such: 819, 820, 821). It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly. Alternating links end up having an important role in knot theory and 3-manifold theory, due to their complements having useful and interesting geometric and topological properties. This led Ralph Fox to ask, "What is an alternating knot?" By this he was asking what non-diag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
