Let $M=\mathcal{W}{\cup }_{\mathcal{T}}\mathcal{V}$$M=\mathcal{W}{\cup }_{\mathcal{T}}\mathcal{V}$ be an amalgamation of two compact 3-manifolds along a torus, where $\mathcal{W}$$\mathcal{W}$ is the exterior of a knot in a homology sphere. Let $N$$N$ be the manifold obtained by replacing $\mathcal{W}$$\mathcal{W}$ with a solid torus such that the boundary of a Seifert surface in $\mathcal{W}$$\mathcal{W}$ is a meridian of the solid torus. This means that there is a degree-one map $f:M\to N$$f:M\to N$, pinching $\mathcal{W}$$\mathcal{W}$ into a solid torus while fixing $\mathcal{V}$$\mathcal{V}$. We prove that $g(M)⩾g(N)$$g(M)⩾g(N)$, where $g(M)$$g(M)$ denotes the Heegaard genus. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.

中文翻译：

---

Heegaard 属、一级图和 3 流形的合并

让$米=\mathcal{W}{\cup }_{\mathcal{吨}}\mathcal{五}$$M=\mathcal{W}{\cup }_{\mathcal{T}}\mathcal{V}$是沿环面的两个紧凑 3 流形的合并，其中$\mathcal{W}$$\mathcal{W}$是同调球中一个结的外部。让$ñ$$N$是通过替换得到的流形$\mathcal{W}$$\mathcal{W}$具有实心圆环，使得 Seifert 曲面的边界在$\mathcal{W}$$\mathcal{W}$是实心圆环的子午线。这意味着存在一阶映射$F：米\to ñ$$f:M\to N$, 捏$\mathcal{W}$$\mathcal{W}$固定时进入坚固的圆环$\mathcal{五}$$\mathcal{V}$. 我们证明$G(米)⩾G(ñ)$$g(M)⩾g(N)$， 在哪里$G(米)$$g(M)$表示 Heegaard 属。一个直接的推论是卫星结的隧道数至少与其模式结的隧道数一样大。