Ismar Volic
We study the cohomology of spaces of string links and braids in $\mathbb{R}^n$ for $n\geq 3$ using configuration space integrals. For $n>3$, these integrals give a chain map from certain diagram complexes to the deRham algebra of differential forms on these spaces. For $n=3$, they produce all finite type invariants of string links and braids.
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The Cohomology of the Mod 4 Braid Group
本文构造了分类模4辫群扩张的2-上循环，证明其为已知上循环的模2约化，并给出扩张表示及level 4同余子群的正规生成集。
For a tree $T$ and a positive integer $n$, let $B_nT$ denote the $n$-strand braid group on $T$. We use discrete Morse theory techniques to show that the cohomology ring $H^*(B_nT)$ is encoded by an explicit abstract simplicial complex $K_nT$ that measures $n$-local interactions among essential vertices of $T$. We show that, in many cases (for instance when $T$ is a binary tree), $H^*(B_nT)$ is the exterior face ring determined by $K_nT$.
If G is a finite graph and n is a natural number, then the n-strand braid group of G is the fundamental group of the configuration space of n points on G. This article gives a complete computation of the integral cohomology rings of the n-strand braid groups in case G is a tree. Some of the argument builds on earlier joint work with Lucas Sabalka.
The purpose of this article is to describe the integral cohomology of the braid group B_3 and SL_2(Z) with local coefficients in a classical geometric representation given by symmetric powers of the natural symplectic representation. These groups have a description in terms of the so called "divided polynomial algebra". The results show a strong relation between torsion part of the computed cohomology and fibrations related to loop spaces of spheres.
Cohomology of quasi-abelianized braid groups
本研究针对广义辫群模纯辫群换位子群的商群有理上同调问题，通过图族同构类给出组合描述，建立了庞加莱对偶性并证明了无穷反射群序列的稳定性质。
The Gysin braid for $S^3$-actions on manifolds
针对非半自由的 $\mathbb S^3$ 光滑作用，本文构造联系 $M$ 的上同调与轨道空间相交上同调的 Gysin 辫，并揭示其依赖于 $M^{\mathbb S^1}$ 相交上同调的奇异项。
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$-primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$-primary part of the Farrell cohomology of a few pure mapping class groups.
Let $Γ$ be a finite connected graph. The (unlabelled) configuration space $UC^n Γ$ of $n$ points on $Γ$ is the space of $n$-element subsets of $Γ$. The $n$-strand braid group of $Γ$, denoted $B_nΓ$, is the fundamental group of $UC^n Γ$. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" to get a partial description of the cohomology rings $H^*(B_n T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a large number of counterexamples to Ghrist's conjecture…
The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero, and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a "p-divided power algebra". The work is an extension of the work in arXiv:1204.5390v1 as well as extensions of a classica…
We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism between diffeomorphism groups.
The fundamental group of the real locus of the Deligne-Mumford compactification of the moduli space of rational curves with $n$ marked points, the pure cactus group, resembles the pure braid group in many ways. As it is the case for several "pure braid like" groups, it is known that its cohomology ring is generated by its first cohomology. In this note we survey what the $FI$-module theory developed by Church, Ellenberg and Farb can tell us about those examples. As a consequence we obtain uniform representation stability for the sequence of cohomology groups of the pure cactus group.
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long l…
We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $λ$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values…
Configuration Spaces and Braid Groups
本文通过配置空间、辫群与映射类群的联系分析某些K(π,1)空间，给出函数空间解释，并计算带标记点的亏格0、1、2曲面的上同调。
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic tables.
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads and length of this braid does not exceed n(4n-5)+2.
Integral cohomology of classical Artin groups: spherical types $A,C,D$ and affine types $\widetilde B,\widetilde C,\widetilde D$
本文利用代数Salvetti复形和Fuchs复形，描述了除仿射族$\widetilde A$外的无限经典族Artin群的整上同调代数加法结构。
In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups.
The Braid Indices of Pretzel Links: A Comprehensive Study, Part I
本文通过基于标准链环图 Seifert 圈分解的系统分类方法，研究并完全确定了所有 Type 1 和 Type 2 非交错 pretzel 链环的精确辫指数。
The Braid Indices of Pretzel Links: A Comprehensive Study, Part II
本文作为pretzel链辫子指标综合研究的第二部分，针对类型3 pretzel链给出了辫子指标的确定方法，除少数例外可精确确定，例外情形也能限定在两个连续整数内，并猜想实际指标为其中较大的整数。
The closure of a braid in a closed orientable surface $Σ$ is a link in $Σ\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.
Let $ϕ: S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $ϕ$ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of $ϕ$ has no critical points at all). We introduce new isotopy invariants for closed braids and almost closed braids in the solid torus V. These invariants refine finite type invariants. They are still calculable with polynomial complexity with respect to the number of crossings of K. Let the solid torus V be standardly embedded …
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strings) and their isotopies via ribbon surfaces and abstract link diagrams. We also give reduced presentations for the virtual braid group, the flat virtual braid group, the welded braid group and several other categories of braids. The paper includes a discussion of the topological …
In the present paper the representation of the virtual braid group $VB_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that these already known representations are not faithful for $n \geq 4$ is verified. Using representations of $VB_n$, the virtual link group is defined. Also representations of welded braid group $WB_n$ are constructed and the welded link group is defined.
The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of B…
Positive 3-braids, Khovanov homology and Garside theory
本文利用Garside理论对正3辫共轭类进行分类，并计算其闭包Khovanov同调表的前四列和最低三行，进一步通过共轭最大化下确界以在线性时间内确定更多同调单元。
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