Related papers: Multiscale Jones Polynomial and Persistent Jones P…

Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applications to higher Chern-Simons theory. For a flat 2-connection, the holonomy of surface knots of arbitrary genus has been defined and its…

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…

This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave…

We construct two distinct yet related M-theory models that provide suitable frameworks for the study of knot invariants. We then focus on the four-dimensional gauge theory that follows from appropriately compactifying one of these M-theory…

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…

Two geometric spaces are in the same topological class if they are related by certain geometric deformations. We propose machine learning methods that automate learning of topological invariance and apply it in the context of knot theory,…

Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a…

We apply big data techniques, including exploratory and topological data analysis, to investigate quantum invariants. More precisely, our study explores the Jones polynomial's structural properties and contrasts its behavior under four…

Knot theory provides a powerful tool for the understanding of topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in the quantum spin system. Exactly solvable models with…

In this report, I will start by first giving a brief introduction on knots to build some intuition before beginning the more rigorous review in the Literature Review section. There, I will define knot equivalence, the Jones polynomial…

The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation,…

We extend the entanglement bootstrap approach to (3+1)-dimensions. We study knotted excitations of (3+1)-dimensional liquid topological orders and exotic fusion processes of loops. As in previous work in (2+1)-dimensions, we define a…

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 paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a…

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…

A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and…

We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses…

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…

We formulate the holographic principle for knots and links. For the "space" of all knots and links, torus knots T(2m+1,2) and torus links L(2m,2) play the role of the "boundary" of this space. Using the holographic principle, we find the…

We study continuity of the roots of nonmonic polynomials as a function of their coefficients using only the most elementary results from an introductory course in real analysis and the theory of single variable polynomials. Our approach…