Knot theory topology

In this short review, we have extended the topological considerations of Ranada and Trueba [1] to 0(3) electrodynamics [3] and therefore also linked these concepts to the Sachs theory reviewed elsewhere in this three-volume compilation [2]. In the same way that topology and knot theory applied to the Maxwell-Heaviside theory produce a rich structure, so does topology applied to the higher-symmetry forms of electrodynamics such as the Sachs theory and 0(3) electrodynamics. 

The knots in Figure 21 are all prime knots because they cannot be divided (factored) into smaller, nontrivial knots. Prime knots are the building blocks of composite knots and of links. Like prime numbers, which yield composite numbers upon multiplication, or like atoms in chemistry, which yield molecules upon combination, prime knots are the elementary units of knot theory. Composite knots are exemplified by the topologically achiral square knot and the topologically chiral granny knot (Figure 22). In each of these knots, a plane perpendicular to the... 

In summary, failure to detect a rigidly achiral presentation does not mean that such a presentation cannot be found among the infinitely many presentations of a knot failure to interconvert enantiomorphous presentations by ambient isotopy does not exclude the possibility that an interconversion pathway can be found among the infinitely many pathways that are available and a palindromic knot polynomial does not necessarily mean that the knot is amphicheiral. Consequently, it may be impossible in certain cases to determine with complete certainty whether a knot is topologically chiral or not. The fundamental task of the theory of knots was stated over a hundred years ago by its foremost pioneer Given the number of its double points, to find all the essentially different forms which a closed curve can assume. 15 Yet to find invariants that will definitively determine whether or not a knot is chiral remains an unsolved problem to this day.63a Vassiliev invariants have been conjectured to be such perfect invariants.63b... 

In Figure 3.5, regular projections of knots with crossing numbers less than seven are shown, together with their symbolic notations commonly used in knot theory. For each topologically chiral knot only one of the two topological enantiomers is shown. 

The most prominent shape features in these structures are described by topological invariants derived from knot theory. I3.i25.i26 Although the minimum number of overcrossings can be used as an invariant, i the more powerful and discriminating descriptors are the knot polynomials. These knot invariants are polynomials of a scalar t with rational coefficients. These functions translate in algebraic terms the basic topological features of space curves (or, more accurately, the space about them). They are related to the curve overcrossings, i ... 

Knot invariants are used in knot theory in order to characterize, distinguish, and classify topological properties of knots. A knot invariant is a function of a knot which takes the same value for all equivalent knots. There are numerical, matrix, polynomial, and finite-type invariants. In this section, the apphcation of some numerical and polynomial invariants to textiles will be... 

When the two end points r(0) and r(l) of the mathematical curve r(t) are formally joined, we obtain an object which is topologically a loop, possibly a knot [17,18]. The use of modem knot theory in chemical applications has an extensive literature [14,15,19,20]. In this work, we use the conventions and notations of ref. 20d for the knots, and the procedure discussed in ref. 14 to derive them from molecular space curves. In what follows, we shall assume that the coordinates specifying the protein backbones are available, for example, in the format of the Protein Data Bank (PDB) of X-ray structures. 

From the other side other many problems in low dimension topology and knot theory are incomputable problems and again, the alternative models and particularly the GTQC may be useful in the theory of computability. 

From these perspectives we think that the new advances in knot theory and topological quantum computation are very important in computer science and for hence all these new alternative models of computation deserve an intensive investigation. 

Topological quantum field theory has become a fascinating and fashionable subject in mathematical physics. At present, the main applications of topological field theory are in mathematics (topology of low-dimensional manifolds) rather than in physics. Its application to the issue of classification of knots and links is one of the most interesting. To approach this problem, one usually tries to somehow encode the topology of a knot or link. As was first noted by Witten... 

ZEEMAN, ERIK CHRISTOPHER (1925-). Zeeman was an English mathematician. His doctoral work was in pure mathematics and he received his Ph D. in 1954 for a thesis on knots and all the algebra you need to actually prove the existence of knots. He did research in topology, which is a type of geometry that examines the properties of shapes in many dimensions. His best known work was in catastrophe theory. His work has consequences for a broad range of fields from weather to psychiatry. Zeeman also made contributions in the development of the chaos theory. 

As a first step in constructing a topological model of the electromagnetic field, let us consider the set of electromagnetic knots defined by pairs of dual scalars (<)>, 0). If we try a theory based on these two scalars, the most natural election for the action integral is... 

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