Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z\colon \mathcal{B}\to \hat{A}$, where $\mathcal{B}$ is the category of bottom tangles in handlebodies and $\hat{A}$ is the degree-completion of the category $\mathbf{A}$ of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, $\mathbf{A}$ is the linear PROP governing “Casimir Hopf algebras”, which are cocommutative Hopf algebras equipped with a primitive invariant symmetric $2$-tensor. The functor $Z$ induces a canonical isomorphism $\operatorname{gr}\mathcal{B}\cong\mathbf{A}$, where $\operatorname{gr}\mathcal{B}$ is the associated graded of the Vassiliev–Goussarov filtration on $\mathcal{B}$. To each Drinfeld associator $\varphi$ we associate a ribbon quasi-Hopf algebra $H_\varphi$ in $\hat{A}$, and we prove that the braided Hopf algebra resulting from $H_\varphi$ by “transmutation” is precisely the image by $Z$ of a canonical Hopf algebra in the braided category $\mathcal{B}$. Finally, we explain how $Z$ refines the LMO functor, which is a TQFT-like functor extending the Le–Murakami–Ohtsuki invariant.

中文翻译：

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车把底部缠结的 Kontsevich 积分

使用类似于 Andersen、Mattes 和 Reshetikhin 给出的构造的 Kontsevich 积分对把手中缠结的扩展，我们构造了一个函子 $Z\colon \mathcal{B}\to \hat{A}$，其中 $\mathcal{ B}$ 是把手中底部缠结的类别，$\hat{A}$ 是把手中雅可比图的类别 $\mathbf{A}$ 的完成度。作为对称幺半群线性范畴，$\mathbf{A}$ 是控制“Casimir Hopf 代数”的线性 PROP，它是配备原始不变对称 $2$-张量的共交换 Hopf 代数。函子 $Z$ 引入了规范同构 $\operatorname{gr}\mathcal{B}\cong\mathbf{A}$，其中 $\operatorname{gr}\mathcal{B}$ 是 Vassiliev 的相关分级 - $\mathcal{B}$ 上的 Goussarov 过滤。对于每个 Drinfeld 关联子 $\varphi$，我们在 $\hat{A}$ 中关联了一个带状拟 Hopf 代数 $H_\varphi$，并且我们证明了由 $H_\varphi$ 通过“嬗变”产生的编织 Hopf 代数是正是编织类别 $\mathcal{B}$ 中规范 Hopf 代数 $Z$ 的图像。最后，我们解释了 $Z$ 如何改进 LMO 函子，这是一个类似 TQFT 的函子，扩展了 Le-Murakami-Ohtsuki 不变量。