论文标题
在4个manifolds中深而浅层
Deep and shallow slice knots in 4-manifolds
论文作者
论文摘要
我们考虑在平滑紧凑的4个manifold $ x^{4} $的边界中的打结片。如果有$ x $的平滑嵌入$ x $的平滑嵌入2盘，而边界$ k $，我们称为$ k \ subset \ subset \ partial x $ deep slice in $ x $，但$ k $与套装$ \ parter nekool $ \ partial $ \ partial x \ times x \ times x i $ y the Bargues of The Bridaige of The Bridaige n of Boundare $ k $。我们指出，这个概念如何与各种知名的猜想相关，并为这种深层结的不存在提供了一些标准。然后，我们使用墙壁自我交流不变和Rohlin的结果表明，每4个manifold仅由一个0- 0-和非零数的2个手柄组成，始终在边界中有一个深切的结。我们结束时考虑了4个manifolds，其中每个结都在边界界定内部的嵌入式磁盘。 Murasugi-Tristram不平等的概括是用来表明，没有球形边界的紧凑，定向的4-manifold $ V $，因此每个结$ k \ subset S^3 = \ partial v $均通过$ v $ slice slice s slice s slice s slice s s $ v $ s $ v $ s $ v $。
We consider slice disks for knots in the boundary of a smooth compact 4-manifold $X^{4}$. We call a knot $K \subset \partial X$ deep slice in $X$ if there is a smooth properly embedded 2-disk in $X$ with boundary $K$, but $K$ is not concordant to the unknot in a collar neighborhood $\partial X \times I$ of the boundary. We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of 2-handles always has a deep slice knot in the boundary. We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented 4-manifold $V$ with spherical boundary such that every knot $K \subset S^3 = \partial V$ is slice in $V$ via a null-homologous disk.
