The Jones Polynomial and its Limitations

Journal of Mathematical Chemistry, 2011

In this paper, we first recall some known architectures of polyhedral links . Motivated by these architectures we introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering, which include all polyhedral links in as special cases. The analysis of chirality of polyhedral links is very important in stereochemistry and the Jones polynomial is powerful in differentiating the chirality . Then we give a detailed account of a result on the computation of the Jones polynomial of polyhedral links based on edge covering developed by the present authors in and, at the same time, by using this method we obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study. We introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering. These new architectures include polyhedral links in [10-16] as special cases. We hope that these new architectures will become the potential synthetical objects of chemists and biologists. Then we review a general method on the computation of the Jones polynomial of polyhedral links based on the edge covering developed by the present authors in , at the same time, by using this method to obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of polyhedral links based on edge covering. Finally, we give some remarks and pose some open problems for further study.

Theoretical Computer Science, 2007

We give fast algorithms for computing Jones polynomials of 2-bridge links, closed 3-braid links and Montesinos links from a progressive expression. The algorithms run with $O(n)$ operations of polynomials of degree $O(n)$ , where $n$ is the number of the crossings of the link diagram. We also give linear time algorithms for computing a progressive expression from the Tait graph of a link diagram of 2-bridge links and closed 3-braid links.

Journal of Differential Geometry, 1993

In this paper, a representation of the category of tangles will be given, which has a natural meaning in terms of certain twisted homology groups. The representation is demonstrated explicitly in terms of a presentation of this category found by Turaev. The invariant of links obtained is identified with the one-variable Jones polynomial via the skein relation, and some remarks are made on how the procedure can be extended to give the two-variable Jones polynomial.

2010

This article contains general formulas for the Tutte and Jones polynomials of families of knots and links given in Conway notation and the corresponding plots of zeroes for the Jones polynomials.

2018

In this paper we introduce the notion of a knot, knot equivalence, and their related terms. In turn, we establish that these terms correspond to the much simpler knot diagram, Reidemeister moves, and their related terms. It is then shown that working with these latter terms, one can construct a number of invariants that can be used to distinguish inequivalent knots. Lastly, the use of these invariants is applied in setting of molecular biology to study DNA supercoiling and recombination.

Applied Mathematics and Computation, 2007

In this paper, we define a polynomial invariant of regular isotopy, G L , for oriented knot and link diagrams L. From G L by multiplying it by a normalizing factor, we obtain an ambient isotopy invariant, N L , for oriented knots and links. We compare the polynomial N L with the original Jones polynomial and with the normalized bracket polynomial. We show that the polynomial N L yields the Jones polynomial and the normalized bracket polynomial. As examples, we give the polynomial G L of some knot and link diagrams and compute the polynomial G L for torus links of type (2, n), and applying computer algebra (MAPLE) techniques, we calculate the polynomial G L of torus links of type (2, n). Furthermore we give its applications to alternating links.

Kodai Mathematical Journal, 2016

We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.

1986

A short proof is given, using linear skein theory, of the theorem of V.F.R. Jones that the one variable "Jones" polynomial associated to an oriented link is independent of the choice of strand orientations, up to a multiple of the variable.

Banach Center Publications, 1995

We describe, in this talk, 1 three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa.

Journal of Knot Theory and Its Ramifications

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this invariant is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distinguish more pairs of nonisotopic links than the original Jones polynomial, such as the Thistlethwaite link from the unlink with two components.