Michael H. Freedman, Michael J. Larsen, Zhenghan Wang
Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations
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A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.
Using the Huynh and Le quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that for a knot which is the closure of a positive braid, the first N coefficients of the N-th colored Jones polynomial are trivial.
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots. We give a counterexample to, and a corrected version of a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot…
We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the problem by Kawamuro.
In the 1920's Artin defined the braid group in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the…
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative results.
We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general infinite braids are also considered.
A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.
We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.
Twisted links are obtained from a base link by starting with a $n$-braid representation, choosing several ($m$) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a certain form. The Jones polynomial of full $m$-twisted links have some interesting properties. It is known that when sufficiently many full $m$-twists are added that the coefficients break up into disjoint blocks which are independent of the number of full twists. These blocks are …
We show that the family of colored Jones polynomials of the closure of a braid compute weighted sums of abelianized Lefschetz numbers associated with the action of the braid on configuration spaces. The sum is over the number of configuration points. Then we interpret this sum in terms of Poincare--Lefschetz duality intersection pairing between homology classes.
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial. We also discuss an extension of the Bennequin inequality, showing that t…
For a closed n-braid L with a full positive twist and with k negative crossings, 0\leq k \leq n, we determine the first n-k+1 terms of the Jones polynomial V_L(t). We show that V_L(t) satisfies a braid index constraint, which is a gap of length at least n-k between the first two nonzero coefficients of (1-t^2)V_L(t). For a closed positive n-braid with a full positive twist, we extend our results to the colored Jones polynomials. For N>n-1, we determine the first n(N-1)+1 terms of the normalized N-th colored Jones polynomial.
On unifying Jones representations of braid groups and Thompson's group
本文引入有向二元辫群，统一了辫群与定向Thompson群的Jones表示，并构造了扩展的酉表示及嵌入，使辫闭包恢复定向链环构造。
We prove the $K$ and $L$ theoretic versions of the Fibered Isomorphism Conjecture of F. T. Farrell and L. E. Jones for braid groups on a surface.
We establish a relationship between the Jones polynomial of generalized weaving knots of type $W(3,n,m)$ and the Chebyshev polynomial of the first kind. Consequently, we prove that the coefficients of the Jones polynomial of weaving knots are basically the Whitney numbers of Lucas lattices. Furthermore, we give an explicit formula for the Alexander polynomial of weaving knots $W(3,n)$ and we prove that it satisfies Fox's trapezoidal conjecture.
A determinant formula of the Jones polynomial for a family of braids
本文扩展了Cohen等人的方法，将Jones多项式恢复为辫群某些子族加权邻接矩阵的行列式，并首次用平衡叠加Tait图在多项式时间内计算了无法由HOMFLYPT多项式导出的(2,q)环面结的Kauffman多项式。
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 representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof o…
We prove a generalization of the Jones-Kawamuro conjecture that relates the self-linking number and the braid index of closed braids, for planar open books with certain additional conditions and modifications. We show that our result is optimal in some sense by giving several counter examples for naive generalizations of the Jones-Kawamuro conjecture.
Given a knot, we develop methods for finding the braid representative that minimizes the number of simple walks. Such braids lead to an efficient method for computing the colored Jones polynomial of $K$, following an approach developed by Armond and implemented by Hajij and Levitt. We use this method to compute the colored Jones polynomial in closed form for the knots $5_2, 6_1,$ and $7_2$. The set of simple walks can change under reflection, rotation, and cyclic permutation of the braid, and we prove an invariance property which relates the simple walks of a braid to those of its reflection u…
In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group $\mathbb{B}_6$ into a group of $5\times 5$ matrices. We also can calculate the Jones polynomial of the $2n$-plat presentations of knots by generalizing the method for the 6-plat presentations of knots.
In a remark in his seminal 1987 paper, Jones describes a way to define the Burau matrix of a positive braid using a metaphor of bowling a ball down a bowling alley with braided lanes. We extend this definition to allow multiple bowling balls to be bowled simultaneously. We obtain the Iwahori-Hecke algebra and a cabled version of the Temperley-Lieb representation.
Closed 4-braids and the Jones unknot conjecture
针对Jones纽结猜想中4股辫子情形的未解问题，研究发现一族以拧数为参数、可能对应非平凡纽结的双变量HOMFLY-PT多项式（其Jones多项式平凡），探究其性质并提出检验方法与多股推广思路。
Braiding structures on categorical multi-Interval Jones-Wassermann subfactor
研究在与酉模融合范畴相关的多区间琼斯 - 瓦塞尔曼子因子平面代数上构建编织结构，借此给出子因子自对偶性新证，证明编织诱导映射类群的射影酉表示，还推导出广义Verlinde公式。
We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the case of n=6, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4.
It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unit…
We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial and the invariant of singular knots is an extension of an invariant of singular knots previously defined by S. Lambropoulou and the second author.
In this paper we represent the classical braids in the Yokonuma--Hecke and the adelic Yokonuma--Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma--Hecke algebras, in analogy to the $p$--adic framed braids and the $p$--adic Yokonuma--Hecke algebras introduced in \cite{jula,jula2}. We further construct an adelic Markov trace, analogous to the $p$--adic Markov trace constructed in \cite{jula2}, and using the traces in \cite{ju} and the adelic Markov trace we define topological invariants of classical knots and links, upon impos…
Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate infinite families of links with the same Khovanov width from link diagrams satisfying certain conditions. Consequently, we compute the Khovanov width for all closed 3-braids.
We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum representations.
This article was submitted to a volume under preparation, with Benson Farb as the editor, on the topic of open problems in surface mapping class groups. The braid group B_n is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra H_n is a quotient of the braid group algebra of B_n by a quadratic relation in the standard generators. We discuss how to use H_n to define the Jones polynomial of a knot or link. We also summarize the classification of the irreducible representations of H_n. We conclude with some directions for future research that would apply mapping class …
Discrete Real Specializations of Sesquilinear Representations of the Braid Groups
本文提出一种利用Salem数寻找辫群半双线性表示离散实特化的方法，并将其应用于Jones和BMW表示，探讨了目标群的可公度性。
The Burau representation of the braid group is faithful for n = 4
利用前人想法证明辫群的Burau表示在n = 4时是忠实的这一定理，其推论是辫群的Jones表示在n = 4时也忠实。
We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at $q = -1$, are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno by extending their original argument for the sphere with four marked points to our more general case.
In this paper we survey some work on representations of $B_n$ given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all $n$. We will outline the methods used, applying them to a closely related representation for which the proof is slightly easier. The main tool is the Blanchfield pairing, a sesquilinear pairing between elements of relative homology. We discuss two other applications of the Blanchfield pairing, namely a proof that the Burau representation is not …
In this paper we will present a homological model for Coloured Jones Polynomials. For each colour $N \in \mathbb {N}$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homology classes in a covering of the configuration space on the punctured disk. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups.
Using the flow property of the R-matrix defining the colored Jones polynomial, we establish a natural bijection between the set of states on the part arc-graph of a link projection and the set of states on a corresponding bichromatic digraph, called arc-graph, as defined by Garoufalidis and Loebl. We use this to give a new and essentially elementary proof for a knot state-sum formula of Garoufalidis and Loebl. We will show that the state-sum contributions of states on the part arc-graph defined by the universal R-matrix of $U_q(sl(2,\mathbb{C}))$ correspond, under our bijection of sets of stat…
We consider two Laurent polynomials in two variables associated to a braid, given by {\em graded intersections} between {\em fixed Lagrangians in configuration spaces}. In order to get link invariants, we notice that we have to quotient by a quadratic relation. Then we prove by topological tools that this relation is sufficient and the first graded intersection gives an invariant which is the Jones polynomial. This shows a {\em topological model for the Jones polynomial} and a direct {\em topological proof}\hspace{0.4mm} that it is a well-defined invariant. The other intersection model in the …
We define and study the category of symmetric $\mathfrak{sl}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum $\mathfrak{sl}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric $\mathfrak{sl}_2$-spider is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and the colored Jones polynomials.
We investigate a family of (reducible) representations of Artin's braid groups corresponding to a specific solution to the Yang-Baxter equation. The images of the braid groups under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of the braid groups factoring over Temperley-Lieb algebras and the corresponding link invariants.
Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In this paper, we provide an algorithm to compute the trace of the Hecke algebra representation of any braid. We simplify this algorithm and write a Mathematica program to compute the invariants such as Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial and Khovanov homology…
In this paper, I give a method to calculate the HOMFLY polynomials of knots by using a representation of the braid group B4 into a group of 3 ? 3 matrices. Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same Jone polynomial, but different HOMFLY polynomials.
This paper aims to determine the images of the braid group under representations afforded by the Yang Baxter equation when the solution is a nontrivial $4 \times 4$ matrix. Making the assumption that all the eigenvalues of the Yang Baxter solution are roots of unity, leads to the conclusion that all the images are finite. Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as par…
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 meaning of the new invariants. A second theme will be the central role that braid theory has played in the subject. A third will be the unifying principles provided by representations of simple Lie algebras and their universal enveloping algebras. These choices in emphasis are our own. They represent, at best, particular aspects of the far-reaching ramifications …
The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in the literature that any braided 3-belt can be written in a pure twist form, specified by a vector of three multiples of half integers [a,b,c], a topological invariant. This paper identifies the set of braided 3-belts that can be written in a braid only form in which all twisting…
In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma--Hecke algebras ${\rm Y}_{d,n}(u)$ and the theory of singular braids. The Yokonuma--Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid $SB_n$ into the algebra ${\rm Y}_{d,n}(u)$. Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relat…
Orbifold braid groups
本文基于先前对 orbifold 辫群的研究，证明了 orbifold 的一类一般子 orbifold 包含映射诱导的单射同态，并得出大多数 orbifold 辫群的中心是平凡的结论。
Brauer and Jones tied monoids
本文围绕 Brauer 型幺半群构造分枝幺半群，给出 Brauer 情形的表示，并研究两类非分枝但可视为 Jones 与 Brauer 结合版本的新幺半群。
Virtual Thompson's group
本文受虚拟辫群推广自辫群且可生成虚拟链环、Thompson群$F$可构造有向链环的研究启发，定义了Thompson群$F$的虚拟版本并证明其可构造任意虚拟链环。
We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus 2 constructed by VFR Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori-Hecke algebra representations of Artin's braid group of 6 strings, and is defined over integral Laurent polynomials Z[t, t^{-1}]. We substitute the parameter t with -e^{h}, and then expand the powers e^h in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of th…
We construct a representation of the braid groups in a cluster C*-algebra coming from a triangulation of the Riemann surface S with one or two cusps. It is shown that the Laurent polynomials attached to the K-theory of such an algebra are topological invariants of the closure of braids. In particular, the Jones and HOMFLY polynomials of a knot correspond to the case S being a sphere with two cusps and a torus with one cusp, respectively.
Learning Topological Invariance
提出自动化学习拓扑不变性的机器学习方法，将其应用于纽结理论，通过对比与生成技术映射同一纽结类表示，还开发师生框架解释学习的不变量，并探索解决琼斯无结猜想的方法。
Multivariable knot polynomials from braided Hopf algebras with automorphisms
本文通过带自同构的辫子 Hopf 代数上的 Yetter--Drinfel'd 模求解 Yang--Baxter 方程，构造出包含 ADO、着色 Jones 及新 2 变量多项式在内的多元纽结不变量，新多项式可检测手性与突变并给出紧致亏格界。
Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular, for the Artin full braid groups. As a consequence we explicitly compute the surgery groups of the Artin pure braid groups. This is obtained as a corollary to a computation of the surgery groups of a more general class of groups, namely for the fundamental group of the complement of any fiber-type hyperplane arrangement in the complex n-space.
We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R=[2,2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6x6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate eigenvalues and is evaluated by the highest weight method. Together with the much harder calculation for R=[3,1] in arXiv:1605.02313 and with the new method to extract exclusive matrices S and \bar S from the inclusive ones, this completes the story of Racah…
We consider the classification problem for compact Lie groups $G\subset U(n)$ which are generated by a single conjugacy class with a fixed number $N$ of distinct eigenvalues. We give an explicit classification when N=3, and apply this to extract information about Galois representations and braid group representations.
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 algorithm to randomly generate $N$-crossing braids and their knot closures and discussing the induced prior on the distribution of knots, we apply binary classification to the UNKNOT decision problem. We find that the Reformer and shared-QK Transformer network architectures outperform fully-connected networks, though all perform well. Perhaps surprisingly, we f…
Configuration Lie groupoids and orbifold braid groups
本文提出配置李群胚的两种定义，证明一类李群胚的Fadell-Neuwirth型纤维化定理，并利用纯轨道折叠辫群的多重几乎自由结构，推导出相应Artin群及轨道折叠辫群的Farrell-Jones猜想。
We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel in arXiv:math/0006056. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action o…
The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted V_A(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4r-th root of unity (r=k+2). In each of these representations (as well as the general G case), Dehn twists act as transformations of finite order, so none represents the mapping class group M(Y) faithfully. However, taken together, the quantum SU(2) representations …
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