Non-orientable genus of knots in punctured spin 4-manifolds
Non-orientable genus of knots in punctured spin 4-manifolds
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刺穿自旋 4 流形中的不可定向结属
DOI：
10.1016/j.topol.2015.02.008
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发表时间：
2015
影响因子：
0.6
通讯作者：
佐藤光樹
中科院分区：
文献类型：
--
作者：
Mitsuishi;Ikuyuki et al.;佐藤光樹
对于一个闭的4-流形X和在被穿孔X的边界上的一个纽结K，我们定义γX0(K)为被穿孔X中边界为K的不可定向零同调曲面的最小第一Betti数。请注意，γS 4 0等价于不可定向的4球亏格，因此γX0是不可定向4球亏格的推广。虽然很可能对于给定的X，γX 0没有上界，但很难证明它。事实上，即使是在γS 40的情况下，其无界性在2012年也首次在巴特森身上表现出来。本文证明了对于任意自旋4维流形X，γX0都没有上界。
For a closed 4-manifold X and a knot K in the boundary of punctured X, we define γ X 0 (K) to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured X with boundary K. Note that γ S 4 0 is equal to the non-orientable 4-ball genus and hence γ X 0 is a generalization of the non-orientable 4-ball genus. While it is very likely that for given X, γ X 0 has no upper bound, it is difficult to show it. In fact, even in the case of γ S 4 0, its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that for any Spin 4-manifold X, γ X 0 has no upper bound.