Heegaard genus, degree-one maps, and amalgamation of 3-manifolds

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Topology Pub Date : 2022-07-07 DOI:10.1112/topo.12253

Tao Li

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引用次数: 0

Abstract

Let $M = W \cup_{T} V$ be an amalgamation of two compact 3-manifolds along a torus, where $W$ is the exterior of a knot in a homology sphere. Let $N$ be the manifold obtained by replacing $W$ with a solid torus such that the boundary of a Seifert surface in $W$ is a meridian of the solid torus. This means that there is a degree-one map $f : M \to N$ , pinching $W$ into a solid torus while fixing $V$ . We prove that $g (M) ⩾ g (N)$ , where $g (M)$ 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.

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高格属，度一映射，和3流形的合并

设M = W∪T V $M=\mathcal {W}\cup _\mathcal {T} \mathcal {V}$是沿环面两个紧致3流形的合并，其中W $\mathcal {W}$是同调球中一个结的外部。设N $N$为用实体环面代替W $\mathcal {W}$得到的流形，使得W $\mathcal {W}$中的Seifert曲面的边界是实体环面的子午线。这意味着存在一个一级映射f: M→N $f\colon M\rightarrow N$，将W $\mathcal {W}$捏成一个实体环面，同时固定V $\mathcal {V}$。我们证明g (M)小于g (N) $g(M)\geqslant g(N)$，其中g (M) $g(M)$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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Topology 数学-数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal. The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.