Jones polynomials

There are several link polynomials which are convenient to use, notably the Jones polynomial [3], the Kauffman polynomial [4], and the 2-variable HOMFLY polynomial [5]. These polynomials are all easy to explain and somewhat similar... 

L. Kauffman, State models and the Jones polynomial. Topology, 1987, 26, 395-407. 

As illustrated by the example of the Jones polynomial for the figure-eight knot, the polynomial of an amphicheiral knot must be palindromic with respect to the coefficients633 ... 

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. 

The state model and bracket polynomials introduced by L.H. Kauffman seem to be very special because they explore only the peculiar geometrical rules such as summation over all possible knot (link) splittings with simple defined weights. But L.H. Kauffman also showed that bracket polynomials are strongly connected with the Jones polynomials [38]. The substitution A —1 1/4 converts... 

One of the most interesting and useful knot polynomials is the Jones polynomial Vk(0 discovered in 1985 [244]. This is a polynomial of a rather general type it can contain both positive and negative fractional powers of the variable t. The Jones polynomial V (t) has the following intriguing property the polynomial VK(t) of a knot K and the polynomial VkoW of the mirror image of the knot K are related in a simple way ... 

The recursive rules for the construction of the Jones polynomial VK(t) for an arbitrary knot K are shown in Figure 3.6. By definition, the Jones polynomial Vjj(t) of the unknot U is 1,... 

Figure 3.7 If two knots K] and K2 are cut anywhere and then are interconnected orientations matching, then the Jones polynomial Vj, 3 t) of the resulting new knot K3 is the product of the Jones polynomials VK (t) and of the two original knots Kj and K2, respectively.

Consider now three knots, denoted by K, K , and K+. These three knots are identical under the cover Q, shown in Figure 3.6, and they differ only in their exposed parts, where they have a left-handed crossing, an avoided crossing, and a right-handed crossing, respectively. For any three such knots their Jones polynomials VK (t), VKo(t), and VK+(t) are interrelated by the equation... 

On level b the task is to characterize the projection, without direct reference to the actual space curve K. By selecting one or several of the knots Kb that generate the same 2D projection (with crossing information supressed), and by using their Jones polynomials VKb(t)> nonvisual, algorithmic characterization of the projection is obtained. 

In order to exploit the full characterization power of the Jones polynomials, it is of some interest to find those knots K% of family Kb that cannot have simpler 2D projections than the actual 2D projection of knot Ka These are the knots K"b of family Kb that have crossing numbers equal to n. If no such knot (or link) exists, then one may take K b as a knot which has a crossing number that differs the least from the number of crossings in the projection. 

Frequently, certain crossings of the projection cannot contribute to knottedness. These crossings can be eliminated from the knot model. We shall take n as the number of crossings obtained after eliminating those crossings that cannot contribute to knottedness. The actual Jones polynomials of these knots are in most... 

KNOT REPRESENTATIONS AND JONES POLYNOMIALS V(t) OF MYOGLOBIN TERTIARY STRUCTURE... 

Equations [23] illustrate a general property the Jones polynomials of chiral loop isomers are related by the transformation t t. ... 

In the next sections we present a brief discussion of the derivation of knots from the molecular space curves representing the protein backbone, using projections to a sphere. The knots are characterized by topological invariants. As in ref. 14, we use the Jones polynomials [15]. The occurrence of basic structural patterns can be recognized in terms of the knots we discuss briefly some of the results derived in ref. 16. Finally, we comment on the application of this procedure to study conformational motions in proteins, and to recognize the occurrence of essential changes in the shape or folding patterns. 

The above operations on the protein backbone produce a loop in most cases. The simple loop (called the "unknot") is the trivial knot, and it has V(Ko)=l as Jones polynomial. Nevertheless, one can derive nontrivial knots when constructing a family of loops from the original curve, by introducing a sequence of formal switches in the original overcrossing pattern. 

The Jones polynomials of the knots Kn, obtained by the switches specified by vectors (vn), are in general different from the polynomial of the actual, original knot Kq. Consequently, they provide a more detailed characterization of the projection. We shall use the complete set of knots (Kn) as a shape descriptor, following a formal vector notation knot vector K) ... 

The symbols in (6) are a short-hand notation for the knots found [20d]. The notations 0i,3i,3i, and 4j identify the unknot, left-handed trefoil knot, right-handed trefoil knot, and the figure-eight knot, respectively. The corresponding Jones polynomials are as follows [14] ... 

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