Knot theory

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. 

M. Ochiai, Knot Theory by Computer, version 3.60, info available by anonymous FTP at ttftp.ics.nara-wu.ac.jp). 

E. Flapan, Rigidity of graph symmetries in the 3-sphere. J. Knot Theory and its Ramifications 1995, 4, 373-388. 

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... 

Crowell, R. H. Fox, R. H. Introduction to Knot Theory, Springer-Verlag New York, 1963. Livingston, C. Knot Theory Mathematical Association of America Washington, DC, 1993. 

Crowell RH, Fox RH (1963) Introduction to knot theory, Ginn, Boston... 

Kauffman LH (1989) Braid Group, Knot theory and statistical mechanics, World Scientific, Singapore... 

An alternate approach to the analysis of the network geometry from the viewpoint of multiplicity and stability is due to Beretta and his coworkers (1979, 1981) the latter approach is based on knot theory. The Schlosser-Feinberg theory reproduces some of the Beretta-type theory results concerning stability from a somewhat different viewpoint. 

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. 

Deruchi, T., Tsurusaki, K., 1994. A statistical study of random knotting using the Vassiliev invariants. J. Knot Theory and Its Ramifications 3 321-353. 

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 ... 

Menasco, W., and Thistlefhwaite, M., Eds., 2005. Handbook of Knot Theory, Amsterdam Elsevier. 

Murasugi K (1996) Knot theory and its applications. Birkhauser, Boston-Bazylea-Berlin... 

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... 

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