Knot polynomials

In summary, failure to detect a rigidly achiral presentation does not mean that such a presentation cannot be found among the infinitely many presentations of a knot failure to interconvert enantiomorphous presentations by ambient isotopy does not exclude the possibility that an interconversion pathway can be found among the infinitely many pathways that are available and a palindromic knot polynomial does not necessarily mean that the knot is amphicheiral. Consequently, it may be impossible in certain cases to determine with complete certainty whether a knot is topologically chiral or not. The fundamental task of the theory of knots was stated over a hundred years ago by its foremost pioneer Given the number of its double points, to find all the essentially different forms which a closed curve can assume. 15 Yet to find invariants that will definitively determine whether or not a knot is chiral remains an unsolved problem to this day.63a Vassiliev invariants have been conjectured to be such perfect invariants.63b... 

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 ... 

For any pD-model p 3), we can develop descriptors of variable dimensionality d. Examples of zero-dimensional descriptors are single numbers such as the radius of gyrationio (used for OD, ID, and 2D models) or the molecular volume 1 (used for 2D models). One-dimensional descriptors such as radial distribution functions or knot polynomials are used in OD and ID models, respectively. Two-dimensional descriptors include distance maps and Rama-chandran torsional-angle maps for some OD and ID models. Similarly, molecular graphs (2D descriptors) can be associated with ID models (contour lines), 2D models (molecular surfaces), or 3D models (e.g., the entire electron density function). Shape descriptors of higher dimensionality can also be constructed. 

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 ... 

The key notion in shape group analysis is evaluation of topological invariants not only for the original surface, but also for a family of surfaces derived by using geometrical properties of the initial surface. Conceptually, this is the same approach discussed in the preceding section for the derivation of a family of knot polynomials from a given molecular space curve. 

Kauffinan, L.H., Lomonaco, S. q-Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation, quant-ph/0606114 v2 Kauffman, L.H., Lomonaco, S.J. Braiding Operators are Universal QuantumGates. New Journal of Physics 6(134), 1-39 (2004)... 

Sphnes are functions that match given values at the points X, . . . , x t and have continuous derivatives up to some order at the knots, or the points X9,. . . , x vr-i-Cubic sphnes are most common see Ref. 38. The function is represented by a cubic polynomial within each interval Xj, X, +1) and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end points, which are commonly taken as zero this gives the natural cubic splines. 

Note that the theorem does not detect all topologically chiral knots and oriented links, because there are topologically chiral knots and oriented links whose P-polynomials are nonetheless symmetric with respect to / and l"1. For example, consider the knot which is illustrated in Figure 11. This knot is known by knot theorists as 942 because this is the forty second knot with 9 crossings listed in the standard knot tables (see the tables in Rolfsen s book [9]). Using a computer program we find that the P-polynomial of the knot 942 is P(942) = (-21 2 - 3-212) + m2 l 2 + 4 + l2) - m. Observe that this polynomial is symmetric with respect... 

The spline surface S(x,y) consists of a set of bicubic polynomials, one in each panel, joined together with continuity up to the second derivative across the panel boundaries. Because each B-spline only extends over four adjacent knot intervals, the functions B.(x)C.(y) are each non-zero only over a rectangle composed of 16 adjacent panels in a 4 x 4 arrangement. The amount of calculation required to evaluate the coefficients y may be reduced by making use of this property. As before, least-squares methods may be used if the number of data exceeds (h+4)(jJ+4), which is usually the case. 

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... 

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 ... 

Nevertheless, even though these polynomials are normally capable of detecting topological chirality, none of them is infallible In several cases, chiral knots yield palindromic polynomials.13 The cause for these anomalies is still unknown. 

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. 

We would like to describe in this section the very beautiful idea proposed by L.H. Kauffman for the analytical construction of powerful polynomial invariants of knots and links. 

The partition function of the system described above represents the polynomial in A, B and d values. Just this function, for some special choice of relations among weights A, B and d, is the topological invariant of regular isotopic knots. Let us prove this fact and derive the necessary relation among A, B and d values. 

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... 

To emphasize the broad region of applicability of the system described in this section, we would like to stress the following fact. Recently, in Refs. [48,49] during investigation of 3D-quantum field theory with Chem-Simon s action a strong connection was established between expectation values of Wilson lines with non-trivial topology and partition function determining the polynomial invariant of the knot or link. 

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. 

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