Modelling the structure of textiles

The topological approach has been used in many areas of physics and chemistry, such as the dynamics of DNA supercoiUng and recombination 

Basic definitions in a simplified form can be given and illustrated in Fig. 1.6 (Prasolov and Sosinskii, 1997 Cromwell, 2004 Chmutov et ai,2010). A knot is a smooth closed curve embedded in three-dimensional space without selfintersections a knot which is equivalent to a circle is called an unknot . A link is a collection of knots chained together where each individual knot is a link s component. A braid is a set of ascending simple non-intersecting strings which connect points Aj, A2. A on a line with points Sj, 

B on a parallel line. A tangle is a generalization of knots, links, and braids which can contain an arbitrary collection of closed and opened strings with their ends fixed in space. 

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 application of some numerical and polynomial invariants to textiles will be 

6 Examples of basic topological objects (a) knot (b) link (c) braid (d) tangle. 

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