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Topic: Topological invariant

	PlanetMath: genus of topological surface
		The genus is a topological invariant of surfaces.
		It is one of the oldest known topological invariants and, in fact, much of topology has been created in order to generalize this notion to more general situations than the topology of surfaces.
		This is version 26 of genus of topological surface, born on 2002-08-15, modified 2006-09-11.
	planetmath.org /encyclopedia/GenusOfTopologicalSurface.html (242 words)

	TAU Quantum Group - Topological Effects in Quantum Mechanics (Site not responding. Last check: 2007-11-03)
		Topological and geometrical effects are among the most striking quantum phenomena discovered since the creation of quantum mechanics in 1926.
		An analogous topological effect is the Aharonov-Casher (AC) effect [7], in which particles that carry a magnetic moment acquire a nontrivial phase while encircling a charged wire.
		Topological quantum phases are crucial in explaining superconductivity, the quantum Hall effect, the Josephson junction, flux quantization and many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behavior.
	www.tau.ac.il /~quantum/publicat/topo-effects.html (1118 words)

	Topologically constrained relaxation method and apparatus for producing reversed-field pinch with inner divertor in ...
		This is a homotopic invariant, implying that it is insensitive to local change of topology of the magnetic surfaces, and that it may therefore be of comparable life-time to that of global helicity having a central role in modeling relaxation.
		To strictly ensure the conservation of the homotopic invariant, it is required to prevent the possibility of null point occurrence, in particular at the poloidal null, and on the axis (as present in spheromak).
		The location of the topological invariant according to the present invention is such as to exert a stabilizing influence on global instabilities preventing decay to lower energy Taylor states with unfavourable magnetic shear, or to total reconnection to open field-lines.
	www.freepatentsonline.com /5147596.html (7982 words)

	Topological property - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.
		A common problem in topology is to decide whether two topological spaces are homeomorphic or not.
		This is, strictly speaking, not a topological invariant of the space, but depends on how it is embedded as a subspace.
	en.wikipedia.org /wiki/Topological_property (1281 words)

	Kuratowski's Introduction to Topology
		For example, the property of a circle to separate the plane into two regions is a topological invariant; if we transform the circle into an ellipse or into the perimeter of a triangle, this property is retained.
		The generality of topological methods rests not only on the generality of the assumptions concerning the transformations considered but also on the generality of the sets considered to which these transformations are applied.
		This period was preceded by the transition from the investigation of subsets of Euclidean space in set-theoretic topology to the investigation of arbitrary topological spaces.
	www-groups.dcs.st-and.ac.uk /~history/Extras/Kuratowski_Topology.html (965 words)

	A Class of P,T-Invariant Topological Phases of Interacting Electrons
		As with other topologically ordered phases, the 2d FQHE ground states are non-degenerate when placed on surfaces of non-trivial topology, such as a torus.
		This appealingly accessible description of 2d topological quantum field theories is apparently familiar to topologists, but is not widely appreciated within the condensed matter theory community.
		next analyze a class of topological quantum field theories which have excitations with non-abelian braiding statistics, and illustrate how these highly non-trivial excitations can again be understood pictorially in terms of curves on surfaces with their particular surgery and fusion rules.
	www.bell-labs.com /jc-cond-mat/october/jccm_oct03_03.html (607 words)

	KNOTS (Site not responding. Last check: 2007-11-03)
		Topological information is information about a knot that does not depend upon the material from which it is made and is not changed by stretching or bending that material so long as it is not torn in the process.
		He was led to this invariant by a trail that began with the study of von Neumann algebras [JO1] (a branch of algebra directly related to quantum theory and to statistical mechanics) and ended in braids, knots and links.
		Topological invariance of the amplitude is a convenient and fundamental way to produce such independence.
	www.math.uic.edu /~kauffman/Tots/Knots.htm (16146 words)

	Springer Online Reference Works
		Later a great deal of importance was attached to topological invariants which are groups, and later to still other algebraic structures such as, for example, the Betti groups or homology groups of different dimensions (cf.
		In the case of polyhedra important topological invariants are often, indeed principally, defined as properties of a simplicial complex which is a triangulation of the given polyhedron.
		Such definitions required a proof of an invariance theorem, asserting that the corresponding property does not change on passing from one triangulation of a given polyhedron to another triangulation of the same polyhedron or of a homeomorphic polyhedron.
	eom.springer.de /T/t093080.htm (374 words)

	Topological Classification of Objects in Digital Images through Boundary Invariant (Site not responding. Last check: 2007-11-03)
		Topological features of object constitute the highest abstraction in object representation which is important for various pattern recognition, image analysis, medical image processing, and machine vision tasks.
		We propose a topological boundary invariant of 2 and 3-dimensional digital objects, called BIUP2 and BIUP3, that can be obtained through a special deformation: homogenous front propagation, BIUP standing for boundary/ surface invariant under propagation.
		Further investigation suggests that BIUP could be invariant in Z4, and that surfaces of digital objects in even and odd dimensional spaces may have different topological properties.
	www.fst.umac.mo /seminar/2004/sem20040930.html (534 words)

	Introduction
		A physicist's ``dynamical quest'' consists of first dissecting the topological form of a strange set, and second, ``dressing'' this topological form with its metric structure.
		Topological signatures and ergodic measures usually present different aspects of the same dynamical system, though there are some unifying principles between the two approaches, which can often be found via symbolic dynamics [11].
		Topological invariants, on the other hand, can be stable under parameter changes and therefore are useful in identifying the same dynamical system at different parameter values.
	cnls.lanl.gov /People/nbt/Book/node136.html (1149 words)

	PlanetMath: topological invariant
		Properties of a space depending on an extra structure such as a metric (i.e.
		volume, curvature, symplectic invariants) typically are not topological invariants, though sometimes there are useful interpretations of topological invariants which seem to depend on extra information like a metric (for example, the Gauss-Bonnet
		This is version 2 of topological invariant, born on 2003-06-18, modified 2003-06-24.
	planetmath.org /encyclopedia/TopologicalInvariant.html (119 words)

	Balents Group Home Page
		topological invariant of time-reversal invariant insulators in two dimensions, showed that the nontrivial ``topological insulator'' phase has an intrinsic spin Hall effect distinct from earlier proposals, and argued that graphene is a viable system in which to observe this effect.
		A direct and experimentally relevant characterization of the invariant was given in terms of edge states at the boundary of a 2D insulator: the topological insulator has an odd number of Kramers pairs of edge modes, while the ordinary insulator has an even number.
		invariant in two dimensions, and this paper by Kane and Mele which (two weeks after our paper) not only gives the same classification of 3d topological insulators as our own, but explores much more fully the physical consequences and also presents an explicit microscopic toy model for the effect.
	www.physics.ucsb.edu /~balents (2134 words)

	Igor et Grichka Bogdanoff - Publications / Published papers (théorie instant zéro)
		Interestingly, this invariant also corresponds to the invariant topological current yield by the hyperfinite II von Neumann algebra describing the zero scale of spacetime.
		In such a context, the instantaneous propagation and the infinite range of the inertial interaction might be explained in terms of the topological amplitude connected with the singular zero size gravitational instanton corresponding to the Initial Singularity of space-time.
		This means that within the limits of the KMS strip the Lorentzian and the Euclidean metric are in a “quantum superposition state” (or coupled), thus entailing a “unification” (or coupling) between the topological (Euclidean) and the physical (Lorentzian) states of spacetime.
	users.skynet.be /catherinev/publications.htm (934 words)

	Non-Integer Topological Invariant for Thin-Walled Primitives (Site not responding. Last check: 2007-11-03)
		Although such parts are usually non-manifold objects, the paper establishes a general topological invariant f = s + b + e + w - v - gnm + m regarding the number of facets, components, bends, free edges, welds, vertices holes and volumes, respectively.
		With this invariant, it is possible to reason about manufacturing processes, such as number of components and arrangement of bend lines and weld lines, using only a single qualitative model of the product.
		A corresponding topological invariant v-e+f=s+m-gnm is also proposed for general sheet models and thin walled objects.
	mecadserv1.technion.ac.il /public_html/Research/hod/smtopo.htm (155 words)

	Topological field theory of the initial singularity of spacetime
		Connected with some unexpected topological data corresponding to the zero scale of spacetime, the initial singularity is thus not considered in terms of divergences of physical fields but can be resolved within the framework of topological field theory.
		Then we suggest that the (pre-)spacetime is in thermodynamical equilibrium at the Planck-scale and is therefore subject to the KMS condition.
		Then we conjecture that the transition from the topological phase of the spacetime (around the zero scale) to the physical phase observed beyond the Planck scale should be deeply connected to the supersymmetry breaking of the N = 2 supergravity.
	stacks.iop.org /0264-9381/18/4341 (416 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		In regard to (ii), we show that the topological information in a spatial database can be precisely summarized by a finite structure which can be viewed as a topological annotation to the raw spatial data.
		The languages considered are first-order on the spatial database side, and fixpoint and first-order on the topological invariant side.
		This suggests that topological invariants are particularly well-behaved with respect to descriptive complexity.
	math.ucsd.edu /~asl99/abstracts/vianu.txt (303 words)

	[No title]
		A sphere is topologically equivalent to an ellipsoid, for similar reasons.
		The study of EDGES as a topological invariant originated with Euler founding topology by solving the famous "Könisberg problem".
		Any topological structure can be described as a sphere with zero or one or more holes, zero or one or more "handles" (cylinders), and zero or one or more "crosscaps" (möbius construction in 4-D).
	members.fortunecity.com /jonhays/topadult.htm (780 words)

	DNA Structure and Topology
		Topology is the branch of mathematics that studies properties of objects that are invariant to continuous deformations of the object.
		Since the linking number is a topological constant, that means that no matter how we deform the object, provided it is a continuous deformation (no "bonds" are broken) then the linking number remains the same.
		If you had made a clockwise turn in the ribbon before joining the ends and cutting, the topological relationship would be opposite to that which would occur had you first twisted it counterclockwise.
	www.rpi.edu /dept/bcbp/molbiochem/BiochSci/sbello/new_page_1.htm (5090 words)

	Third-order topological invariant
		It is a topological invariant and therefore an invariant of ideal magnetohydrodynamics.
		The integral generalizes existing expressions for third-order invariants which are obtained from the Massey triple product, where the three fields are restricted to isolated flux tubes.
		The derivation and interpretation of the invariant show a close relationship with the well-known magnetic helicity, which is a second-order topological invariant.
	stacks.iop.org /0305-4470/35/3945 (299 words)

	B
		Second, it is easily seen that the procedure of removing the crossings one at a time by attaching a handle for every single crossing is nonunique and leads in general to a number of handles greater than the minimum number.
		The lack of uniqueness of the number of handles is bothersome, for indeed the number of handles is supposed to be a topological invariant.
		Question (ii) is whether the archetypical topological construction of "going to higher dimensions" has resulted in some decrease of the degree of singularity.
	www.usc.edu /dept/LAS/CAMS/EE/Edmond/mine2.htm (2976 words)

	Non-Integer Topological Invariant for Thin-Walled Primitives (Site not responding. Last check: 2007-11-03)
		The authors suggest a topological invariant supporting both manifold and thin-walled non-manifold objects based on this primitive.
		The validity of the proposed invariant constitutes a necessary condition for the validity of a geometrical representation of thin-walled products from a topological point of view.
		It specifically proposes the use of non-integer values in the standard Euler-Poincaré formula for representing non-manifold components, thus permitting the use of thin-walled primitives with topological coherence to traditional solid geometry schemes.
	www.mit.edu /~hlipson/papers/top_prop.htm (160 words)

	Polyakov's Model
		It's a nice introduction to solitons, topological quantum field theories, and Witten's explanation of the new knot polynomials in terms of topological quantum field theories.
		Since a small variation in the field configuation doesn't change the linking number (which after all is a topological invariant), the Euler-Lagrange equations (which come from differentiating the Lagrangian) don't notice this term at all.
		Physically, the contribution to the Hopf invariant due to ribbon twisting is interpreted as due to the rotation of individual anyons.
	math.ucr.edu /home/baez/braids/node3.html (1568 words)

	Re: Topological invariants and TFT (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-03)
		In article , Benjamin Gutierrez Garcia wrote: >I would like to ask you about references on the web or printed, about: >-What a topological invariant is There are many kinds of topological invariants and you'll need to focus in on certain particular ones before you'll be able to make much progress.
		These are all homotopy invariants: invariant not only under homeomorphism but also under a weaker relation called homotopy equivalence.
		After understanding the "classical" invariants you'll be in a much better position to understand the new-fangled "quantum" invariants, i.e.: >-How this relates to quantum field theory and topological field theory For this, start with: Charles Nash and Siddhartha Sen, Topology and Geometry for Physicists, Academic Press, 1983.
	www.lns.cornell.edu.cob-web.org:8888 /spr/2001-11/msg0036583.html (384 words)

	Homeomorphism
		The reader who has had abstract algebra will note that homeomorphism is the analogy in the setting of topological spaces and continuous functions to the notion of isomorphism in the setting of groups (or rings) and homomorphisms, and to that of linear isomorphism in the context of vector spaces and linear maps.
		A property of topological spaces which when possessed by a space is also possessed by every space homeomorphic to it is called a topological invariant.
		It is often easier to show that two spaces are not homeomorphic: simply exhibit an invariant which is possessed by one space and not the other.
	at.yorku.ca /course/atlas1/node17.html (255 words)

	Invariants and Linking Numbers
		A topological invariant of a knot or link is a quantity that does not change under continuous deformations of the strings.
		The calculation of topological invariants allows us to bypass directly showing the geometric equivalence of two knots, since distinct knots must be different if they disagree in at least one topological invariant.
		However, mathematicians have been successful in developing some very fine topological invariants capable of distinguishing large classes of knots [23].
	cnls.lanl.gov /~nbt/Book/node144.html (303 words)

	Binghamton University, Mathematical Sciences, Research Interests
		In many cases, topological invariants of such complexes provide a surprising amount of information regarding certain combinatorial parameters of the underlying graph, for instance, the chromatic number.
		The underlying theme of my research is the investigation of topological, geometric, and spectral invariants of (singular) Riemannian manifolds using techniques from partial differential equations.
		For example, the Euler characteristic of a surface is a topological invariant based its usual definition in terms of a triangulation of the surface.
	www.math.binghamton.edu /dept/server/research.html (1629 words)

	BIUP{3}: Boundary Topological Invariant of 3D Objects Through Front Propagation at a Constant Speed (Site not responding. Last check: 2007-11-03)
		Topological features constitute the highest abstraction in object representation.
		Euler characteristic is one of the most widely used topological invariants.
		In this paper, we show that a new topological surface invariant of 3D digital objects, called BIUP{3}, can be obtained through a special homeomorphic transform: front propagation at a constant speed.
	csdl.computer.org /comp/proceedings/gmp/2004/2078/00/20780369abs.htm (218 words)

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