In this paper we provide a counterexample to a 22-year-old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody. We find more relations than previously predicted for the Kauffman bracket skein module of the connected sum of handlebodies, when one of them is not a solid torus. Additionally, we speculate on the structure of the Kauffman bracket skein module of the connected sum of two solid tori.

中文翻译：

---

连接的车体总和的Kauffman支架绞线模块：一个反例

在本文中，我们提供了一个22岁的定理的反例，该定理关于两个车体的连接总和的Kauffman支架绞链模块的结构。我们通过分析手柄主体中压缩盘上的手柄滑动来实现这一点。当其中一个不是坚固的圆环时，我们发现的关系比以前对连接的和体的Kauffman支架绞线模块的预测要多。另外，我们推测了两个实心花托的连接总和的Kauffman支架绞线模块的结构。