Related papers: Conformal invariance of the writhe of a knot

Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that…

The survey we are presenting is over 22 years old but it has still some ideas which where never published (except in Polish). This survey is the base of the third Chapter of my book: KNOTS: From combinatorics of knot diagrams to…

We show that (as conjectured by Lin and Wang) when a Vassiliev invariant of type $m$ is evaluated on a knot projection having $n$ crossings, the result is bounded by a constant times $n^m$. Thus the well known analogy between Vassiliev…

We show that the quantum covering group associated to osp(1|2n) has an associated colored quantum knot invariant \`a la Reshetikhin-Turaev, which specializes to a quantum knot invariant for osp(1|2n), and to the usual quantum knot invariant…

Let $p$ be a prime number and $m$ an integer coprime to $p$. In the spirit of arithmetic topology, we introduce the notions of the twisted Iwasawa invariants $\lambda, \mu, \nu$ of ${\rm GL}_N$-representations and…

In this paper we clarify an issue in the knot surgery construction of Fintushel and Stern. Using knot surgery, they construct an infinite number of smooth structures on 4-manifolds satisfying certain conditions, but they do not explicitly…

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We…

Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to $24$ crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, \gamma)$. The invariant $\rho$ has…

In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial in 1984. The focus of our account will be recent glimmerings of understanding of the topological…

Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p/q surgery on another knot K' in the 3-sphere, then K and K' are…

We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose…

We model the typical behavior of knots and links using grid diagrams. Links are ubiquitous in the sciences, and their "normal" or "typical" behavior is of significant importance in understanding situations such as the topological state of…

Using the Racah coefficients in our earlier paper arXiv:1107.3918, we explicitly write the Chern-Simons field theory invariants for many non-torus knot and links. Further, we have tabulated the reformulated invariants which agrees with the…

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant $t^{I\left( \mathcal{L} \right) }$ is constructed for a link $\mathcal{L}$, where $I$ is the abelian Chern-Simons…

In this paper we prove a mirror symmetry conjecture based on the work of Brini-Eynard-Mari\~no \cite{BEM} and Diaconescu-Shende-Vafa \cite{DSV}. This conjecture relates open Gromov-Witten invariants of the conifold transition of a torus…

We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian…

The ropelength of a knot or link is the minimal number of inches of 1-inch-thick rope that it takes to tie it. The relationship of this measurement to knot and link invariants has been studied by various authors. We give the first results…

Generalized knot groups $G_n(K)$ were introduced independently by Kelly (1991) and Wada (1992). We prove that $G_2(K)$ determines the unoriented knot type and sketch a proof of the same for $G_n(K)$ for $n>2$.

The first and last named authors have demonstrated the existence of knots for which every integral slope is non-characterizing. In this short note, we extend this result in two ways. There exists a knot that shares for every integer n the…