Complements of Minimal Spanning Surfaces of Knots are Not Unique
Complements of Minimal Spanning Surfaces of Knots are Not Unique
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结的最小跨度曲面的补集不是唯一的
DOI：
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发表时间：
1970
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通讯作者：
W. Alford
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作者：
W. Alford
S 3中的每一条多面体简单闭曲线都有一个可定向曲面的边界[2];也就是说，如果k是一个纽结，则存在一个可定向曲面S使得D（S）= k。所有这些生成曲面的最小亏格是纽结的亏格，最小生成曲面是具有最小亏格的生成曲面。对于Neumerth-Stallings [4，5]纽结，已知这样的极小曲面是唯一的，因为存在将一个极小曲面抛向任何其他极小曲面的空间同胚。这里的重要性质是纽结群的换位子群是n-生成的，因此是自由的[4]。Stallings然后表明，S 3 k纤维在S'与纤维的最小生成表面的k。这样的纤维化可以用来证明最小生成曲面的唯一性。黑尔特罗特在给C. D. Feustel指出，有一个例子，一个结有两个不同的最小曲面。他的例子很大程度上依赖于不可逆纽结的存在[6]。Trotter的例子具有S3 S1同胚于S3 S2的性质。本文的目的是用两个极小生成曲面S1和S2构造一个纽结k，使得S3和S1不是同胚的。证明实际上表明Z1（S3 Si）不是同构的。作者感谢C教授。B。Schaufele，C博士。D. Feustel和A先生。C.康纳的有益的谈话。
Every polyhedral simple closed curve in S3 bounds an orientable surface [2]; that is, if k is a knot, then there exists an orientable surface S such that D(S) = k. The minimal genus of all such spanning surfaces is the genus of the knot and a minimal spanning surface is a spanning surface with minimal genus. For the Neuwirth-Stallings [4, 5] knots, it is known that such a minimal surface is unique in the sense that there is a space homeomorphism throwing one minimal surface to any other. The important property here is that the commutator subgroup of the knot group is finitely generated and, hence, free [4]. Stallings then shows that S3 k fibers over S' with fiber a minimal spanning surface of k. Such a fibration can then be used to show the uniqueness of the minimal spanning surface. Hale Trotter, in a letter to C. D. Feustel, indicates that there is an example of a knot with two different minimal surfaces. His example depends heavily on the existence of non-invertible knots [6]. Trotter's example has the property that S3 S1 is homeomorphic to S3 S2. The purpose of this note is to construct a knot k with two minimal spanning surfaces, SI and S2, such that S3 Si are not homeomorphic. The proof actually shows that Z1(S3 Si) are not isomorphic. The author is indebted to Professor C. B. Schaufele, Dr. C. D. Feustel, and Mr. A. C. Connor for helpful conversations.