Related papers: New points of view in knot theory
In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their…
We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call `Jones rational coincidences'. We provide moves on the continued fraction expansion of the associated rational which we prove do not…
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…
Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms,…
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…
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl…
This work is dedicated to the consideration of the construction of a representation of braid group generators from vertex models with $N$-states, which provides a great way to study the knot invariant. An algebraic formula is proposed for…
We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be…
Gram determinants earned traction among knot theorists after E. Witten's presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by N. Reshetikhin and V.…
We discuss the theory of knots, and describe how knot invariants arise naturally in gravitational physics. The focus of this review is to delineate the relationship between knot theory and the loop representation of non-perturbative…
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded…
In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field…
A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation…
The leading coefficient of the Alexander polynomial of a knot is the most informative element in this invariant, and the growth of orders of the first homology of cyclic branched covering spaces is also a familiar subject. Accordingly,…
This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying…
We recover the Jones polynomials of knots and links from the K-theory of a cluster C*-algebra of the sphere with two cusps. In particular, an interplay between the Chebyshev and Jones polynomials is studied.
We review a constructions of knots from elements of the Thompson groups due to Vaughan Jones, which comes in two flavours: oriented and unoriented.
We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an…
Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations…
An extension of the Artin Braid Group with new operators that generate double and triple intersections is considered. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple…