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Knot Invariants
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a ''knot invariant'' is a rule that assigns to any knot a quantity such that if and are equivalent then ."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification,Purcell, Jessica (2020). ''Hyperbolic Knot Theory'', p.7. American Mathematical Society. "A ''knot invariant'' is a function from the set of knots to some other set whose value depends only on the equiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Table
In 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, 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. 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 it 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 describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Khovanov Homology
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov. Overview To any link diagram D representing a link L, we assign the Khovanov bracket \left D \right/math>, a cochain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise \left D \right/math> by a series of degree shifts (in the graded vector spaces) and height shifts (in the cochain complex) to obtain a new cochain complex C(D). The cohomology of this cochain complex turns out to be an invariant of L, and its graded Euler characteristic is the Jones polynomial of L. Definition This definition follows the formalism given in Dror Bar-Natan's 2002 paper. Let denote the ''degree shift'' operation on graded vector spaces—that is, the homogeneous comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mostow Rigidity Theorem
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by and extended to finite volume manifolds by in 3 dimensions, and by in all dimensions at least 3. gave an alternate proof using the Gromov norm. gave the simplest available proof. While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic n-manifold (for n >2) is a point, for a hyperbolic surface of genus g>1 there is a moduli space of dimension 6g-6 that parameterizes all metrics of constant curvature (up to diffeomorphism), a fact essential for Teichmüller theory. There is also a rich theory of deformation spaces of hyperbolic structures on ''infinite'' volume manifolds in three dimensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peripheral Subgroup
In algebraic topology, a peripheral subgroup for a topological pair, space-subspace pair ''X'' ⊃ ''Y'' is a certain subgroup of the fundamental group of the complementary space, π1(''X'' − ''Y''). Its conjugacy class is an invariant of the pair (''X'',''Y''). That is, any homeomorphism (''X'', ''Y'') → (''X''′, ''Y''′) induces an isomorphism π1(''X'' − ''Y'') → π1(''X''′ − ''Y''′) taking peripheral subgroups to peripheral subgroups. A peripheral subgroup consists of Loop (topology), loops in ''X'' − ''Y'' which are peripheral to ''Y'', that is, which stay "close to" ''Y'' (except when passing to and from the Pointed space, basepoint). When an ordered generating set of a group, set of generators for a peripheral subgroup is specified, the subgroup and generators are collectively called a peripheral system for the pair (''X'', ''Y''). Peripheral systems are used in knot th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
