Nonorientable slice genus can be arbitrarily large
Nonorientable slice genus can be arbitrarily large
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不可定向切片属可以任意大
DOI：
10.4310/mrl.2014.v21.n3.a1
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发表时间：
2014
影响因子：
1
通讯作者：
Joshua D. Batson
中科院分区：
文献类型：
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作者：
Joshua D. Batson
光滑嵌入到四个空间中的曲面的一般横截面是结或环。每个结都可以被实现为某个曲面的横截面，但如果要求该曲面是可定向的，那么Fox、Milnor和Murasugi的工作表明，它的亏格可能需要相当大。例如，(2，n)环结不是亏格小于n−1的可定向曲面的横截面，而是克莱因瓶的横截面。根据K的签名和−1-运算给出的整数同调球面的Heegaard-Floerd-不变量，我们给出了具有一个纽结的截面为K的曲面的第一Betti数的一个下界。特别地，我们证明了四维空间中具有(2k，2k−1)环纽结的光滑曲面的第一Betti数至少为2k−2。
A generic cross-section of a surface smoothly embedded in four-space is a knot or a link. Every knot can be realized as the cross-section of some surface, but if that surface is required to be orientable, then work of Fox, Milnor, and Murasugi show that its genus may need to be quite large. For example, the (2, n) torus knot is not a cross-section of any orientable surface with genus less than n−1, but is the cross-section of a Klein bottle. We give a lower bound on the first Betti number of a surface with cross-section a knot K in terms of the signature of K and the Heegaard–Floer d-invariant of the integer homology sphere given by −1-surgery on K. In particular, we show that any smooth surface in four-space with cross-section the (2k, 2k − 1) torus knot has first Betti number at least 2k − 2.