Bott-Cattaneo-Rossi invariants for long knots in asymptotic homology $\mathbb R^3$

Generalized Bott-Cattaneo-Rossi invariants of high-dimensional long knots

Bott, Cattaneo and Rossi defined invariants of long knots $\mathbb R^n \hookrightarrow \mathbb R^{n+2}$ as combinations of configuration space integrals for $n$ odd $\geq 3$. Here, we give a more flexible definition of these invariants. Our definition allows us to interpret these invariants as counts of diagrams. It extends to long knots inside more general $(n+2)$-manifolds, called asymptotic homology $\mathbb R^{n+2}$, and provides invariants of these knots.

Configuration space integral for long n-knots, the Alexander polynomial and knot space cohomology

There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n-knots and characterize the Bott-Cattaneo-Rossi invariant as a finite type invariant of long ribbon n-knots in [HKS]. As a consequence, we obtain a …

Generalized Bott-Cattaneo-Rossi invariants in terms of Alexander polynomials

The Bott-Cattaneo-Rossi invariant $(Z_k)_{k\in \mathbb N\setminus\{0,1\}}$ is an invariant of long knots $\mathbb R^n\hookrightarrow\mathbb R^{n+2}$ for odd $n$, which reads as a combination of integrals over configuration spaces. In this article, we compute such integrals and prove explicit formulas for (generalized) $Z_k$ in terms of Alexander polynomials, or in terms of linking numbers of some cycles of a hypersurface bounded by the knot. Our formulas, which hold for all null-homologous long knots in homology $\mathbb R^{n+2}$ at least when $n\equiv 1\mod 4$, conversely express the Reidemei…

长嵌入空间的余圈与多于一个环的BCR图

Cocycles of the space of long embeddings and BCR graphs with more than one loop

本文通过配置空间积分构造长嵌入空间的非平凡余圈，并利用BCR图与弦图的配对，为特定维数提供了非平凡平凡长嵌入族。

到球面三维流形的 $\mathbb{Z}\pi$-同调等价的 Theta 不变量

Theta-invariants of $\mathbb{Z}π$-homology equivalences to spherical 3-manifolds

本文利用有限群表示理论，计算到球面三维流形的 $\mathbb{Z}\pi$-同调等价上 Bott–Cattaneo $Θ$-不变量及 Garoufalidis–Levine 2 型有限型不变量所张成空间的维数上界。

与可能带自环的三价图相关的带框架3-流形的积分不变量

Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loops

本文研究带框架3-流形上由非循环局部系定义的积分不变量，证明哑铃图对应积分为零，并建立图复形与余闭链理论，使高圈不变量可用无自环和带自环两种等价方法定义，最终证明 Chern–Simons 微扰级数给出仅依赖无自环三价图的拓扑不变量。