On the Gassner Invariant of Braids and String links

In this thesis we delve into the computation of the Gassner invariant for string links, which are a more generalized form than braids, utilizing a (co)homological approach. We restrict this (co)homology invariant, denoted as $\mathcal{G}_h$, to pure braids, leading to the derivation of the Gassner representation. We introduce the concept of "flying cars," which assigns an invariant $\mathcal{C}(L)$ to an $(n+1)$-component string link $L$. This invariant, an $n \times n$ matrix, has entries in the field $\mathbb{Q}(t_0, t_1,\ldots, t_n)$. We establishes a connection between the invariant $\mathcal{C}(L)$ and the homology Gassner invariant $\mathcal{G}_{h}\left(L\right)$ of $L$ through the formula $\mathcal{G}_{h}\left(L\right) = \left(D_n \cdot \mathcal{C}\left(L\right) \cdot D_n^{-1}\right)//\rho_{col}//{m^t}$. Here, $D_n$ is a diagonal matrix, $m^t$ denotes matrix transpose, and $\rho_{col}$ represents column permutation. We prove that $\mathcal{C}(L)$ is indeed an invariant of string links under the Reidemeister moves, thereby directly verifying the invariance of the homology Gassner invariant. Moreover, we provide formulas for the intersection product $\mu:=\langle-, -\rangle: H_1(P;\mathcal{F}) \times H_1(P;\mathcal{F}) \rightarrow \mathcal{F}$, which is defined on the cycles of the homology group $H_1(P;\mathcal{F})$. In this context, $P$ is an $(n+1)$-punctured disk viewed as a subspace of the complement $X$ of an $n+1$ string link, and $\mathcal{F}$ is a local coefficient system on $X$ determined by the abelianization map $\epsilon: \pi_1(X,x_0) \rightarrow \langle t_0, t_1, \cdots, t_n \rangle$. This map takes values in the free abelian group $\langle t_0, t_1, \cdots, t_n \rangle$. We conclude by verifying that the homology Gassner invariant is unitary with respect to this intersection product.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.