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In this paper, 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. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we give a Khovanov homology theory for unoriented virtual links. The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented Jones polynomial for virtual links, which is a new invariant in the category of non-classical virtual links. The unoriented Jones polynomial continues to satisfy an important property of the usual one: for classical or even virtual links, the unoriented Jones polynomial evaluated at one is two to the power of the number of components of the link. As part of extending the main results of this paper to non-classical virtual links, a new framework for computing integral Khovanov homology based upon arc labeled diagrams is described. This framework can be efficiently and effectively implemented on a computer. We define an unoriented Lee homology theory for virtual links based upon the unoriented version of Khovanov homology. [ABSTRACT FROM AUTHOR]

In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood. [ABSTRACT FROM AUTHOR]

We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to ℓ > 2 , then the girth of the link is equal to ℓ. [ABSTRACT FROM AUTHOR]

This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by Melvin and Kirby that underlie the quantum group-based Reshetikhin–Turaev model for SU (2) Chern–Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented. [ABSTRACT FROM AUTHOR]

In this paper we prove that the family of colored Jones polynomials of a knot in S3 determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On the one hand, the first variable q can be evaluated at 2r roots of unity with r 2 N and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable A evaluated at A D qn gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is holonomic and is annihilated by the same polynomial as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum sl2 completion studied by Habiro in [J. Pure Appl. Algebra 211 (2007), 265-292], showing that it corresponds in fact to the two-variable colored Jones invariant defined by Habiro in [Invent. Math. 171 (2008), 1-81]. [ABSTRACT FROM AUTHOR]

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links. [ABSTRACT FROM AUTHOR]

It has been known that the coefficients of the series expansion of the Jones polynomial evaluated at ex are rational-valued Vassiliev invariants. In this paper, we calculate minimal multiplying factor, λ, needed for these rational-valued invariants to become integer-valued Vassiliev invariants. By doing that, we obtain a set of integer-valued Vassiliev invariants. [ABSTRACT FROM AUTHOR]

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 change the Jones polynomial and conjecture (based on experimental evidence from all rational knots with determinant <900) that these moves are sufficient to generate all Jones rational coincidences. In the process we give a new formula for the Jones polynomial of a rational knot based on a continued fraction expansion of the associated rational, which has significantly fewer terms than other formulae known to us. The paper is based on the second author’s Ph.D. thesis and gives an essentially self-contained account. [ABSTRACT FROM AUTHOR]

This essay is a remembrance of Vaughan Jones and a diagrammatic exposition of the remarkable breakthroughs in knot theory and low-dimensional topology that were catalyzed by his work. [ABSTRACT FROM AUTHOR]

We introduce an alternative stratification of knots: by the size of the lattice on which a knot can be first met. Using this classification, we find the fraction of unknots and knots with more than minimal crossings inside different lattices and answer the question of which knots can be realized inside and lattices. In accordance with previous research, the fraction of unknots decreases exponentially with the growth of the lattice size. Our computational results are consistent with theoretical estimates for the number of knots with a fixed crossing number inside lattices of a given size. [ABSTRACT FROM AUTHOR]

In this paper, we derive formulae for the determinant of weaving knots W (3 , n) and W (p , 2). We calculate the dimension of the first homology group with coefficients in Z 3 of the double cyclic cover of the 3-sphere S 3 branched over W (3 , n) and W (p , 2) respectively. As a consequence, we obtain a lower bound of the unknotting number of W (3 , n) for certain values of n. [ABSTRACT FROM AUTHOR]

Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript, we give a novel definition of the Jones polynomial that generalizes the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, coincides with that of the corresponding link. Using the framework introduced in (Panagiotou E, Kauffman L. 2020 Proc. R. Soc. A476, 20200124. ((doi:10.1098/rspa.2020.0124)), this enables us to define the Jones polynomial of collections of open and closed curves in 3-space. For collections of open curves in 3-space, the Jones polynomial has real coefficients and it is a continuous function of the curves' coordinates. As the endpoints of the curves tend to coincide, the Jones polynomial tends to that of the resultant link. We demonstrate with numerical examples that the novel Jones polynomial enables us to characterize the topological/geometrical complexity of collections of open curves in 3-space for the first time. [ABSTRACT FROM AUTHOR]

For classical knots, Δ -moves and crossing changes can both unknot any knot. But many virtual knots cannot be unknotted with these moves. Moreover, there are many virtual knots that can be unknotted by crossing changes but cannot be unknotted by Δ -moves. We show that the derivatives of the Jones polynomial can detect whether a virtual knot can be unknotted by Δ -moves. [ABSTRACT FROM AUTHOR]

The braid group on strands plays a central role in knot theory and low dimensional topology. 3-braids were classified, up to conjugacy, into normal forms. Basing on Burau's representation of the braid group, Birman introduced a simple way to calculate the Jones polynomial of closed 3-braids. We use Birman's formula to study the structure of the Jones polynomial of links of braid index 3. More precisely, we show that in many cases the normal form of the 3-braid is determined by the Jones polynomial and the signature of its closure. In particular we show that alternating pretzel links , which are known to have braid index 3, cannot be represented by alternating 3-braids. Also we give some applications to the study of symmetries of 3-braid links. [ABSTRACT FROM AUTHOR]

Let N = 2 n 2 - 1 or N = n 2 + n - 1 , for any n ≥ 2 . Let M = N - 1 2 . We construct families of prime knots with Jones polynomials (- 1) M ∑ k = - M M (- 1) k t k . Such polynomials have Mahler measure equal to 1. If N is prime, these are cyclotomic polynomials Φ 2 N (t) , up to some shift in the powers of t. Otherwise, they are products of such polynomials, including Φ 2 N (t) . In particular, all roots of unity ζ 2 N occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials. [ABSTRACT FROM AUTHOR]

We prove that the length of any gap in the differential grading of the Khovanov homology of any quasi-alternating link is one. As a consequence, we obtain that the length of any gap in the Jones polynomial of any such link is one. This establishes a weaker version of Conjecture 2.3 in (Topol Appl 264:1–11, 2019). Moreover, we obtain a lower bound for the determinant of any such link in terms of the breadth of its Jones polynomial. This establishes a weaker version of Conjecture 3.8 in (Algebr Geom Topol 15:1847–1862, 2015). The main tool in obtaining this result is establishing the Knight Move Conjecture [(Algebr Geom Topol 2:337-370, 2002), Conjecture 1] for the class of quasi-alternating links. [ABSTRACT FROM AUTHOR]

We classify closed 3-braids which are L-space knots. [ABSTRACT FROM AUTHOR]