Related papers: The Jones Polynomial and its Limitations
In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of…
Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. I will present my multivariate version of Bigelow's calculation. The advantage to my…
Motivated by the result that an `approximate' evaluation of the Jones polynomial of a braid at a $5^{th}$ root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP…
Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable ``interlace polynomial'' for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and…
The colored Jones polynomial is a series of one variable Laurent polynomials J(K,n) associated with a knot K in 3-space. We will show that for an alternating knot K the absolute values of the first and the last three leading coefficients of…
The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are $q$-holonomic, that is, they satisfy linear $q$-difference equations with coefficients Laurent…
We establish a relationship between the Jones polynomial of generalized weaving knots of type $W(3,n,m)$ and the Chebyshev polynomial of the first kind. Consequently, we prove that the coefficients of the Jones polynomial of weaving knots…
For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…
The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than…
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
The 2-bridge knots are a family of knots with bridge number 2. In this paper, we compute the Kauffman polynomials of 2-bridge knots using the Kauffman skein theory and linear algebra techniques. Our calculation can be easily carried out…
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…
The Jones polynomial $V_{L}(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which…
We study factorizations of HOMFLY polynomials of certain knots and oriented links. We begin with a computer analysis of knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for…
We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find…
A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only…
We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via…
The Kauffman-Vogel polynomials are three variable polynomial invariants of $4$-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented $4$-valent rigid vertex graphs was given by using the…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
It was discovered recently by Griffin, Ono, Rolen and Zagier that the Jensen polynomials associated to many sequences have Hermite polynomial limits. We develop this theory in detail, based on the log-polynomial property which is a…