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Milnor Invariants
In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link complement. Definition The link group of an ''n''-component link is essentially the set of (''n'' + 1)-component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through immersions), knotting or unknotting itself, but is not allowed to move through other components. This is a weaker condition than isotopy: for example, the Whitehead link has linking number 0, and thus is link homotopic to the unlink, but it is not isotopic to the unlink. The link group is not the fundamental group of the link complement, since the components of the link are allowed to move through themselves, though not each other, but thus is a quotient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 Khova ... [...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] |
Brunnian Link
In knot theory, a branch of topology, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops (so that no two loops can be directly linked). The name ''Brunnian'' is after Hermann Brunn. Brunn's 1892 article ''Über Verkettung'' included examples of such links. Examples The best-known and simplest possible Brunnian link is the Borromean rings, a link of three unknots. However for every number three or above, there are an infinite number of links with the Brunnian property containing that number of loops. Here are some relatively simple three-component Brunnian links which are not the same as the Borromean rings: Image:Brunnian-3-not-Borromean.svg, 12-crossing link. Image:Three-triang-18crossings-Brunnian.svg, 18-crossing link. Image:Three-interlaced-squares-Brunnian-24crossings.svg, 24-crossing link. The simplest Brunnian link other than the 6-cros ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted . Definition For any positive integer , define as the sum of the primitive th roots of unity. It has values in depending on the factorization of into prime factors: * if is a square-free positive integer with an even number of prime factors. * if is a square-free positive integer with an odd number of prime factors. * if has a squared prime factor. The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where is the Kronecker delta, is the Liouville function, is the number of dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Lie Algebra
In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity. Definition The definition of the free Lie algebra generated by a set ''X'' is as follows: : Let ''X'' be a set and i\colon X \to L a morphism of sets (function) from ''X'' into a Lie algebra ''L''. The Lie algebra ''L'' is called free on ''X'' if i is the universal morphism; that is, if for any Lie algebra ''A'' with a morphism of sets f\colon X \to A, there is a unique Lie algebra morphism g\colon L\to A such that f = g \circ i. Given a set ''X'', one can show that there exists a unique free Lie algebra L(X) generated by ''X''. In the language of category theory, the functor sending a set ''X'' to the Lie algebra generated by ''X'' is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Type Invariant
In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can be extended (in a precise manner to be described) to an invariant of certain singular knots that vanishes on singular knots with ''m'' + 1 singularities and does not vanish on some singular knot with 'm' singularities. It is then said to be of type or order m. We give the combinatorial definition of finite type invariant due to Goussarov, and (independently) Joan Birman and Xiao-Song Lin. Let ''V'' be a knot invariant. Define ''V''1 to be defined on a knot with one transverse singularity. Consider a knot ''K'' to be a smooth embedding of a circle into \R^3. Let ''K be a smooth immersion of a circle into \mathbb R^3 with one transverse double point. Then : V^1(K') = V(K_+) - V(K_-), where K_+ is obtained from ''K'' by resolving the double point by pushing up one strand above the other, and ''K_-'' is obta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Links
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a ''trivial'' reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link. For example, a co-dimension 2 link in 3-dimensional space is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components are homeomorphic to circles. The simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked. The Borromean rings thus form a Brunnian link and in fact constitute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seifert Surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated 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 a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface ''S'' embedded in 3-space whose boundary is ''L'' such that the orientation on ''L'' is just the induced orientation from ''S''. 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. It is possible t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borromean Rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing. The Borromean rings are named after the Italian House of Borromeo, who used the circular form of these rings as a coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan. They have been used in Christian symbolism as a sign of the Trinity, and in modern commerce as the logo of Ballantine beer, giving them the alternative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massey Product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist. Massey triple product Let a,b,c be elements of the cohomology algebra H^*(\Gamma) of a differential graded algebra \Gamma. If ab=bc=0, the Massey product \langle a,b,c\rangle is a subset of H^n(\Gamma), where n=\deg(a)+\deg(b)+\deg(c)-1. The Massey product is defined algebraically, by lifting the elements a,b,c to equivalence classes of elements u,v,w of \Gamma, taking the Massey products of these, and then pushing down to cohomology. This may result in a well-defined cohomology class, or may result in indeterminacy. Define \bar u to be (-1)^u. The cohomology class of an element u of \Gamma will be denoted by /math>. The Massey triple product of three cohomology classes is defined by : \langle rangle = \. The Massey product of three cohomology classes is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Invariants
In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link complement. Definition The link group of an ''n''-component link is essentially the set of (''n'' + 1)-component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through immersions), knotting or unknotting itself, but is not allowed to move through other components. This is a weaker condition than isotopy: for example, the Whitehead link has linking number 0, and thus is link homotopic to the unlink, but it is not isotopic to the unlink. The link group is not the fundamental group of the link complement, since the components of the link are allowed to move through themselves, though not each other, but thus is a quotient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
