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Thurston–Bennequin Number
In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a Legendrian knot is defined as the writhe of the diagram minus the number of right cusps. It is named after William Thurston and Daniel Bennequin. The maximum Thurston–Bennequin number over all Legendrian representatives of a knot is a topological knot invariant 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 i .... References * * {{knottheory-stub Knot invariants ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
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] |
Legendrian Knot
In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into which is tangent to the standard contact structure on It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a contact manifold In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability' ... that is always tangent to the contact hyperplane. Two Legendrian knots are equivalent if they are isotopic through a family of Legendrian knots. There can be inequivalent Legendrian knots that are isotopic as topological knots. Many inequivalent Legendrian knots can be distinguished by considering their Thurston-Bennequin invariants and rotation number, which are together known as the "classical invariants" of Legendrian knots. More sophisticated invar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Writhe
In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link (knot theory), link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a knot (mathematics), mathematical knot (or any curve, closed simple curve) in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe. Writhe of link diagrams In knot theory, the writhe is a property of an oriented link (knot theory), link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings. A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while travel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
