Kevin P. Knudson
We study the Gassner representation of the pure braid group $P_n$ by considering its restriction to a free subgroup $F$. The kernel of the restriction is shown to lie in the subgroup $[Γ^3 F,Γ^2 F]$, sharpening a result of Lipschutz.
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We give a 3-page description of the Gassner invariant / representation of braids / pure braids, along with a description and a proof of its unitarity property.
We show how Conway's multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel-Turaev, of the Alexander-Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the sign ambiguity in the current formulas which relate the multivariable Alexander polynomial to the reduced Gassner representation. We also relate the distinct definitions of this representation which…
The Gassner representation of the pure braid group to $GL_n(Z[Z^n])$ can be extended to give a representation of the concordance group of $n$-strand string links to $GL_n(F)$, where $F$ is the field of quotients of $\zz[\zz^n]$, $ F = Q(t_1,...,t_n)$; this was first observed by Le Dimet. Here we present simple new perspectives on its definition, its computation, and its applications to obtain both general results and interesting family of examples.
Gassner and Burau representations over $\mathbb{Z}_p$-modules
本文研究Artin辫群的Gassner与Burau表示及其模p约化，运用拓扑方法证明Gassner表示对所有n均忠实且模p忠实，并给出Burau表示模p忠实性的新证明，展望高辫群应用。
We give an explicit isomorphism between the Gassner representation and the first weight level of a representation of quantum sl(2). Then we construct and provide matrices for colored versions of the BKL representation and higher Lawrence's representations.
Long and Moody gave a method of constructing representations of the braid group B_n. We discuss some ways to generalize their construction. One of these gives representations of subgroups of B_n, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke algebra.
Ihara initiated to study a certain Galois representation which may be seen as an arithmetic analogue of the Artin representation of a pure braid group. We pursue the analogies in Ihara theory further, following after some issues and their inter-relations in the theory of braids and links such as Milnor invariants, Johnson homomorphisms, Magnus-Gassner cocycles and Alexander invariants, and study relations with arithmetic in Ihara theory.
The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie algebraic condition which guarantees that a homomorphism out of the classical pure braid group is faithful. However, it is unclear whether the methods here can be applied to any open cases such as the Gassner representation.
Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the op…
Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$
本文通过 Fox 微积分建立 Burau 表示与 Knot Floer 同调的明确联系，定义范畴化行列式的不变量，并利用量子 $\mathfrak{gl}(1 \vert 1)$ 证明其恢复辫表示。
We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Dehornoy ordering of the braid groups.
On kernels of homological representations of mapping class groups
本文研究映射类群在构型空间扭曲同调上表示的核，借助扭曲相交配对证明忠实性判据，并分析 Heisenberg 局部系统表示 ρ_n 与 Lawrence 表示的关系。
Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of "Johnson-type" homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of Bur_n, we do this with the assistance of a computer calculation.
For $n$ an even number, we study representations of the mapping class group of the $n$-punctured sphere arising from $\mathrm{SU}(2)$-TQFT when all punctures are colored by the same integer $N \geq 1$. We prove that the conjecture of Andersen, Masbaum and Ueno holds for the $4$-punctured sphere for all $N \geq 2$. In the case $n \geq 6$ of punctures, we prove it for the pseudo-Anosovs satisfying a homological condition, namely they should act with a non trivial stretching factor on certain eigenspaces of homology of a $\frac{n}{2}$-fold branched cover considered by McMullen. The main idea is t…
We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.
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