论文标题
结的嵌入结节和子
Embedding calculus and grope cobordism of knots
论文作者
论文摘要
我们表明，嵌入计算不变的$ ev_n $对于任意$ 3 $ - manifold的长结是溢出的。这解决了一些剩余的开放式案例 - klein-我们的连通性估计，同时证实了一半的猜想，即对于古典结$ ev_n $是整数上的通用添加剂vassiliev不变性。此外，我们为这个猜想提供了足够的条件，可以将系数组保持在一个系数中，这是Boavida de Brito和Horel的最新结果，为理性和$ p $ - adiC整数在一个范围内实现。因此，嵌入积分不变性比Kontsevich积分更强大。 此外，我们的工作表明它们也更具计算。也就是说，主定理计算一个结的可能不变的$ ev_n $，它与灰色结的灰分是灰色的，以完全等于相关图中灰grope的基础装饰树的等效类别。实际上，我们的技术适用于尺寸$ 3 $，提供了在任何维度的多种层中嵌入微积分的层的描述，并提出某些广义莎程序实现了相应的图形复杂类。
We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary $3$-manifold. This solves some remaining open cases of Goodwillie--Klein--Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the $p$-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral. Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariant $ev_n$ of a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension $3$, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes.
