Related papers: The Jones Polynomial from a Goeritz Matrix
The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of…
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the…
This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation…
The Goeritz matrix of a link is obtained from the Jacobian matrix of a modified Dehn presentation associated to a diagram using Fox's free differential calculus. When the diagram is special the Seifert matrix can also be determined from the…
This paper will be an exposition of the Kauffman bracket polynomial model of the Jones polynomial, tangle methods for computing the Jones polynomial, and the use of these methods to produce non-trivial links that cannot be detected by the…
A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
In 2012, Cohen, Dasbach, and Russell presented an algorithm to construct a weighted adjacency matrix for a given knot diagram. In the case of pretzel knots, it is shown that after evaluation, the determinant of the matrix recovers the Jones…
Let $G$ be a signed graph. Let $\hat{G}$ be the graph obtained from $G$ by replacing each edge $e$ by a chain or a sheaf. We first establish a relation between the $Q$-polynomial of $\hat{G}$[6] and the $W$-polynomial of $G$ [9]. Two…
It is a challenging problem to construct an efficient quantum algorithm which can compute the Jones' polynomial for any knot or link obtained from platting or capping of a $2n$-strand braid. We recapitulate the construction of braid-group…
In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e., looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily…
Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using…
Given a 4-tangle shadow, we concatenate it with itself n times and form a knot by applying a closure operation that connects each top endpoint to the corresponding bottom endpoint on the same side without introducing any crossings. We then…
We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram. Our generalization is an epimorphism between skein modules of tangles in compact…
Given any diagram of a link, we define on the cube of Kauffman's states a "2-complex" whose homology is an invariant of the associated framed links, and such that the graded Euler characteristic reproduces the unnormalized Kauffman bracket.…
The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler…
We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a $q$-state Potts model defined as a planar…
It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs…
We explain how to compute the Jones polynomial of a link from one of its grid diagrams and we observe a connection between Bigelow's homological definition of the Jones polynomial and Kauffman's definition of the Jones polynomial.…
We utilize the trip matrix method of calculating the Jones Polynomial to give an alternative proof that the Jones Polynomial is multiplicative under connect sums.