Knot invariants

One approach to the problem of establishing a knot s chirality or achirality is through the use of knot invariants. The first invariant capable of distinguishing between enantiomorphs, a one-variable Laurent polynomial (a polynomial that has both positive and negative powers), was discovered only as recently as 1985, by Vaughan Jones. 00 More powerful two-variable polynomials have subsequently been developed by others.101 102... 

Statistical mechanics of knots. To solve the problem of statistical mechanics of knots, one needs, first of all, a knot invariant. Indeed a closed chain can be unknotted or can form knots of different types. The very beginning of the table of knots is shown in Fig.8. However, an analytical expression for the knot in variant is unknown. Therefore, we had to use a computer and analgebraic invariant elaborated in the topological theory of knots. We found that the most convenient in variant was the Alexander polynomial (reviewed by Frank-Kamenetsku and Vologodskh, 1981 [24] and Vologodskh and Frank-Kamenetskii, 1992 [81]). 

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 ... 

By formally closing a chain into a loop, i we can also use the foregoing approach for open chains. It has been used, as well, for the analysis of proteins and secondary motifs, In praaice, because the evaluation of knot invariants for generic curves is difficult and time-consuming, this shape characterization is hard to implement algorithmically. 

T. Deguchi and K. Tsurusaki, Phys. Lett. A, 174, 29 (1993). A New Algorithm for Numerical Calculation of Link Invariants. D. Bar-Natan, Topology, 34, 423 (1995). On the Vas-siliev Knot Invariants. 

Key words hbre, yarn, woven fabric, knitted fabric, non-wovens, design of experiment, hypothesis testing, analysis of variance (ANOVA), regression analysis, geometrical models, structural models, fibre migration, unit cell, knot invariants, textile mechanics, physical properties of textiles, homogenization, optimization. 

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... 

Chmutov S, Duzhin S and Mostovoy J (2010)), Introduction to Vassiliev Knot Invariants 1st Draft, http //www.math.osu.edu/ chmutov/preprints/. 

Grishanov S, Meshkov V and VassUiev V (2009c), Recognizing textile structures by finite type knot invariants , I Knot Theor Ramif, 18(2), 209-235. 

These measures are inspired by Vassiliev knot invariants.They form a natural progression of curve descriptors, much as moments of inertia and their correlations define solids. 

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