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knot theory | A Wisdom Archive on knot theory | | knot theory A selection of articles related to knot theory | |
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knot theory
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ARTICLES RELATED TO knot theory | |
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| | |  knot theory: Encyclopedia - 3-manifoldIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.
The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds. There are close connections to other fields, such as 4-manifolds, surfaces, knot theory, topological quantum field theory, and gauge theory. 3-manifold theory is a part o ...
Including:
Read more here: » 3-manifold: Encyclopedia - 3-manifold |
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| | | | knot theory: Encyclopedia II - Signed graph - Matroid theory There are two matroids associated with a signed graph, called the signed-graphic matroid (or the frame matroid or bias matroid) and the lift matroid, both of which generalize the cycle matroid of a graph. They are special cases of the same matroids of a biased graph.
The signed-graphic matroid M(G) (Zaslavsky, 1982) has for its ground set the edge set E. An edge set is independent if each component contains either no circles or just one circle, which is negative. (In matroid theo ...
See also:Signed graph, Signed graph - Examples, Signed graph - Adjacency matrix, Signed graph - Orientation, Signed graph - Incidence matrix, Signed graph - Switching, Signed graph - Fundamental theorem, Signed graph - Matroid theory, Signed graph - Other kinds of signed graph, Signed graph - Generalizations Read more here: » Signed graph: Encyclopedia II - Signed graph - Matroid theory |
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| | | | knot theory: Encyclopedia II - Polynomial - Elementary properties of polynomials All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication. Every polynomial in expanded form is a sum of terms (where subtraction is carried out by addition of negative numbers).
Polynomials are classified by their degree and number of variables. The degree of a term in a polynomial is the su ...
See also:Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables Read more here: » Polynomial: Encyclopedia II - Polynomial - Elementary properties of polynomials |
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| | | | knot theory: Encyclopedia II - Topology - History The root of topology was in the study of geometry in ancient cultures. Leibniz was the first to employ the term analysus situs, later employed in the 19th century to refer to what is now known as topology. Yet, this is only part of the argument in favor of his anticipation. As Benoit Mandelbrot, in his monumental The Fractal Geometry of Nature, wrote (taken from Leibniz's Cultural Pluralism And N ...
See also:Topology, Topology - History, Topology - Elementary introduction, Topology - Some theorems in general topology, Topology - Some useful notions from algebraic topology, Topology - Outline of the deeper theory, Topology - Generalizations Read more here: » Topology: Encyclopedia II - Topology - History |
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| | | | knot theory: Encyclopedia II - Polynomial - Elementary properties of polynomials All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication (where whole number powers are viewed as repeated multiplication). Every polynomial in expanded form is a sum of terms ...
See also:Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables Read more here: » Polynomial: Encyclopedia II - Polynomial - Elementary properties of polynomials |
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| | | | knot theory: Encyclopedia II - Homotopy - Homotopy-invariant properties and functions; homotopy category Homotopy equivalence is important because in algebraic topology many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence. For example, if X and Y are homotopy equivalent spaces, then:
if X is path-connected, then so is Y
if X is simply connected, then so is Y
the (singular) homology and cohomology groups of X and Y are isomorphic
if X and Y are path-connected, then the fundamental g ...
See also:Homotopy, Homotopy - Formal definition, Homotopy - Properties, Homotopy - Homotopy equivalence and null-homotopy, Homotopy - Homotopy-invariant properties and functions; homotopy category, Homotopy - Relative homotopy, Homotopy - Isotopy Read more here: » Homotopy: Encyclopedia II - Homotopy - Homotopy-invariant properties and functions; homotopy category |
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| | | | knot theory: Encyclopedia II - Von Neumann algebra - Definitions There are three common ways to define von Neumann algebras.
The first and most common way is to define them as weakly closed * algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by almost any other common topology other than the norm topology, in particular by the strong or ultrastrong topologies. (The * algebras of bounded operators that are closed in the norm topology are C* algebras, so in particular a ...
See also:Von Neumann algebra, Von Neumann algebra - Definitions, Von Neumann algebra - Commutative von Neumann algebras, Von Neumann algebra - Projections, Von Neumann algebra - Factors, Von Neumann algebra - Type I factors, Von Neumann algebra - Type II factors, Von Neumann algebra - Type III factors, Von Neumann algebra - Weights states and traces., Von Neumann algebra - Amenable von Neumann algebras, Von Neumann algebra - Examples, Von Neumann algebra - Applications Read more here: » Von Neumann algebra: Encyclopedia II - Von Neumann algebra - Definitions |
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