Heegaard Reducing Spheres for the 3-Sphere
Heegaard Reducing Spheres for the 3-Sphere
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用于 3 球体的 Heegaard 还原球体
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
M. Scharlemann
中科院分区：
文献类型：
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作者：
M. Scharlemann
在文[4]中，对于具有解结隧道γ的纽结K，定义了Q/2Z中的一个不变量。证明不变量仅依赖于(K，γ)对的一个重要因素是，在某种意义上，由于Goeritz[1]的刻画，S 3的亏格两个Heegaard分裂的H1∪H2的约化球面。证明了H1∪H2的任何一对约化球面可以由一系列约化球面连接，使得序列中的任何连续对都非常简单地相交。这里，我们将用Powell的[3]推广的Goeritz结果来描述S3的任意亏格Heegaard分裂的约化球面。我们从扩展符号和定义开始。设H1∪F H2是S 3的亏格g>；1 Heegaard分裂。众所周知，H1∪F H2是标准亏格g分裂(见[6]或[5])，因此特别的H1∪F H2是可约的。
In [4] an invariant in Q/2Z was defined for a knot K with unknotting tunnel γ. An important ingredient in the proof that the invariant depended only on the pair (K,γ) was a characterization, in some sense due to Goeritz [1], of reducing spheres for the genus two Heegaard splitting H1 ∪H2 of S 3. It was shown there that any pair of reducing spheres for H1 ∪H2 can be connected by a sequence of reducing spheres, so that any successive pair in the sequence intersect very simply. Here we will use Powell’s [3] extension of Goeritz’ result similarly to describe reducing spheres for arbitrary genus Heegaard splittings of S3. We begin by extending notation and definitions. Let H1 ∪F H2 be a genus g > 1 Heegaard splitting of S 3. It is well-known that H1 ∪F H2 is the standard genus g splitting (see [6] or [5]) so in particular H1 ∪F H2 is reducible.