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Unknotting Problem
In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether the problem lies in the complexity class P (complexity), P. Computational complexity First steps toward determining the computational complexity were undertaken in proving that the problem is in larger complexity classes, which contain the class P. By using normal surfaces to describe the Seifert surfaces of a given knot, showed that the unknotting problem is in the complexity class NP (complexity), NP. claimed the weaker result that unknotting is in Arthur–Merlin protocol, AM ∩ co-AM; however, later they retracted this claim. In 2011, Greg Kuperberg proved that (assuming the generalized Riemann hypothesis) the unknotting problem is in co-NP, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undirected Graph
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing money is not necessarily reciprocated. Gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residually Finite
In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a finite group, such that :h(g) \neq 1.\,{{Cite journal , last=Magnus , first=Wilhelm , date=March 1969 , title=Residually finite groups , url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-75/issue-2/Residually-finite-groups/bams/1183530287.full , journal=Bulletin of the American Mathematical Society , volume=75 , issue=2 , pages=305–316 , doi=10.1090/S0002-9904-1969-12149-X , issn=0002-9904, doi-access=free There are a number of equivalent definitions: *A group is residually finite if for each non-identity element in the group, there is a normal subgroup of finite index not containing that element. *A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial. *A group is residually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a ''solid torus'' include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels. Topological properties The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to S^1 \times S^1, the ordinary torus. Since the disk D^2 is contractible, the solid torus has the homotopy type o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pachner Moves
In topology, a branch of mathematics, Pachner moves, named after Udo Pachner, are ways of replacing a triangulation of a piecewise linear manifold by a different triangulation of a homeomorphic manifold. Pachner moves are also called bistellar flips. Any two triangulations of a piecewise linear manifold are related by a finite sequence of Pachner moves. Definition Let \Delta_ be the (n+1)-simplex. \partial \Delta_ is a combinatorial ''n''-sphere with its triangulation as the boundary of the ''n+1''-simplex. Given a triangulated piecewise linear (PL) ''n''-manifold N, and a co-dimension ''0'' subcomplex C \subset N together with a simplicial isomorphism \phi : C \to C' \subset \partial \Delta_, the Pachner move on ''N'' associated to ''C'' is the triangulated manifold (N \setminus C) \cup_\phi (\partial \Delta_ \setminus C'). By design, this manifold is PL-isomorphic to N but the isomorphism does not preserve the triangulation. See also * Flip graph * Unknotting problem * Rei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the 3-sphere). Let ''N'' be a tubular neighborhood of ''K''; so ''N'' is a solid torus. The knot complement is then the complement of ''N'', :X_K = M - \mbox(N). The knot complement ''XK'' is a compact 3-manifold; the boundary of ''XK'' and the boundary of the neighborhood ''N'' are homeomorphic to a two-torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu .... Sometimes the ambient manifold ''M'' is understood to be the 3-sphere. Context is needed to determine the usage. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reidemeister Move
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. and, independently, , demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves. Each move operates on a small region of the diagram and is one of three types: No other part of the diagram is involved in the picture of a move, and a planar isotopy may distort the picture. The numbering for the types of moves corresponds to how many strands are involved, e.g. a type II move operates on two strands of the diagram. One important context in which the Reidemeister moves appear is in defining knot invariants. By demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves, an invariant is defined. Many important invariants can be defined in this way, including the Jones polynomial. The type I move is the only move that affects the writh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Foliation
A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Enumeration Problem
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities: :Ax \leq b where ''A'' is an ''m''×''n'' matrix, ''x'' is an ''n''×1 column vector of variables, and ''b'' is an ''m''×1 column vector of constants. The inverse (dual) problem of finding the bounding inequalities given the vertices is called '' facet enumeration'' (see convex hull algorithms). Computational complexity The computational complexity of the problem is a subject of research in computer science. For unbounded polyhedra, the problem is known to be NP-hard, more precisely, there is no algorithm that runs in polynomial time in the combined input-output size, unless P=NP. A 1992 art ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Cone
In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, C is a cone if x\in C implies sx\in C for every . This is a broad generalization of the standard cone in Euclidean space. A convex cone is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered field, although the field of real numbers is used most often. Definition A subset C of a vector space is a cone if x\in C implies sx\in C for every s>0. Here s>0 refers to (strict) positivity in the scalar field. Competing definitions Some other authors require ,\infty)C\subset C or even 0\in C. Some require a cone to be convex and/or satisfy C\cap-C\subset\. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
