Alexander polynomial

Alexander Polynomials as a Tool for Numerical Investigations of Polymers with Topological Constraints... 

The so-called Jones polynomials [38] are even more strongly invariant than the Alexander ones. However, their calculation requires far more computer capacity calculation of an Alexander polynomial takes in the order 0(/3) operations, where / is the number of selfintersections of contour projection on the plane on the other hand, the calculation of a Jones polynomial takes in the order 0(e ) operations. This is why the existing attempts to use Jones polynomials in computer experiments with ring polymers have not been successful as yet Nevertheless, the construction of algebraic polynomial invariants of knots and links seems to be of great importance in principle, and we shall consider it in the next section. 

Let us return to Fig. 8, where the knot formation probability p is plotted as a function of the swelling ratio, a, in the globular region (a < 1). It can be seen that in the compression region, especially for a < 0.6, data of numerical simulation are absent. It is difficult to obtain such data because of the restricted capacity of computers. Really, it is necessary to calculate the Alexander polynomial for each generated closed contour. As mentioned above, it takes in the order of 0(l3) operations. This value is too large for the dense chain state because the denser the system is, the more selfcrossings l should be in the projection. 

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]). 

Kamenetskii and Vologodskii, 1981 [24] Vologodskii and Frank-Kamenetskii, 1992 [81]). Using these methods and teaching the computer to calculate the Alexander polynomials and therefore to distinguish the knots of different type, we could calculate the knotting probabihty. 

The simplest of such functions is the Alexander polynomial A(f).i3 For a closed, unknotted loop (the unknot U, homeomorphically equivalent to a circle) the polynomial is trivial = 1. The simplest knot is the trefoil knot... 

T shown in Figure 6. It has been observed in nucleic acidsi > 3 and recently traced in some proteins. Its corresponding Alexander polynomial... 

And what is t here Nothing It has no physical relevance whatsoever. Alexander A(t) is just an abstract algebraic object... This is the type of mathematics to which most physicists are usually deaf. But science teaches us time after time that prejudice of any sort is a bad advisor, that a scientist, to deserve the name, should keep his or her eyes open... In our story, the researchers in Prank Kamenetskii group realized that their computer could work out Alexander polynomial for several values of t for every loop generated and, for instance, if the result was A(—1) = 3, then the loop is most likely the trefoil. (Most likely and not for sure because some other rather complex knots also have the same Alexander polynomial as the trefoil also, Alexander polynomial does not distinguish left from right this was not a significant problem, so let s skip it here.)... 

Table 1.10 Multi-variable Alexander polynomials for textile structures ...

The Alexander polynomial p P of a finite-dimensional Z-acyclic E [z,)-module chain complex C is the generator (unique up to unit) of the maximal principal ideal contained in the order ideal (sez(z,z l sH j(C) = 0)4E(z,z. ... 

The Alexander polynomial, encountered in the algebraic topology, can be used to distinguish different topological states particularly in computer simulations. Interesting quantities like the probability that two rings are entangled as a function of distance between their centers of mass can readily be computed. However, the modification on the Rouse relaxation time is unknown. 

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