论文标题
LMO不变的剪接公式
A splicing formula for the LMO invariant
论文作者
论文摘要
我们证明了LMO不变的“剪接公式”，这是理性同源性$ 3 $ -SPHERES的通用有限型不变性。具体而言，如果有理同源$ 3 $ -sphere $ m $是通过粘合两个带框结的外部$ k_1 \ k_1 \ subset m_1 $和$ k_2 \ g_2 \ subset m_2 $理性同源性$ 3 $ -SPHERES，我们的公式表达了lmo nmo $ m $ k $ m $在kontssessessse的情况下， $（M_1，K_1）$和$（M_2，K_2）$。该证明使用了Bar-Natan和Lawrence开发的技术来为LMO不变型获得合理的手术公式。在低度中，我们恢复了Casson-Walker不变的藤田公式，并且我们观察到Ohtsuki系列的第二项不是“标准”剪接下的添加剂。当每个$ m_i $都带有链接$ l_i $之外，剪接公式还可以工作，除了结$ k_i $，因此我们将获得Kontsevich-lmo不变的“卫星公式”。
We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's formula for the Casson-Walker invariant and we observe that the second term of the Ohtsuki series is not additive under "standard" splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$, hence we get a "satellite formula" for the Kontsevich-LMO invariant.
