Related papers: Jones rational coincidences
This expository essay is aimed at introducing the Jones polynomial. We will see the encapsulation of the Jones polynomial, which will involve topics in functional analysis and geometrical topology; making this essay an interdisciplinary…
Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this paper, we use the Jones polynomial as a general topological invariant to capture the global knot…
We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form $T((p,q),(2,s))$ where $p$ and $q$ are coprime and $s$ is nonzero. When $s = 2n$, these links are the twisted torus knots…
Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and…
In this article the discovery of the Jones Polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of Mathematics, each with its own language, but initially without a dictionary.
It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight…
In previous joint work with Frohman and Lofaro a noncommutative generalization of the A-polynomial of a knot was introduced, consisting of a finitely generated ideal of polynomials (the noncommutative A-ideal) in the quantum plane. The…
This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we…
Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and…
In previous work of the first and third authors, we proposed a conjecture that the Kauffman bracket skein module of any knot in $S^3$ carries a natural action of the rank 1 double affine Hecke algebra $SH_{q,t_1, t_2}$ depending on 3…
The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by…
We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques…
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.
The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of SL(2). It was recently shown by TTQ…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…
In this note, we study solutions of the equation $J_K(t)=1$ for the Jones polynomial of knots and links. For the family $K_n$ of double-twist knots, we show that every root of unity (except $-1$) satisfies $J_{K_n}(\zeta)=1$ for some $n$.…
We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a…
The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope…
The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume…
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…