Concordance of knots in S1×S2
Concordance of knots in S1×S2
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S1×S2 中结的一致性
DOI：
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发表时间：
2017
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通讯作者：
Arunima Ray
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作者：
C. Davis;M. Nagel;Junghwan Park;Arunima Ray
建立了S1×S2中纽结的光滑性和拓扑协调性的若干结果。定义S1×S2中纽结的缠绕数为H1（S1×S2;Z）<$Z中纽结的类。我们表明，有一个独特的光滑和谐类的纽结缠绕数为1。这改进了Friedl-Nagel-Orson-Powell在拓扑范畴中的相应结果.我们说S1×S2中的一个纽结是切片的（分别是拓扑切片的），如果它在D2×S2中界定一个光滑（分别是局部平坦的）圆盘。我们证明了有无穷多个非切片纽结的拓扑协调类，而且，对于±1以外的任何缠绕数，甚至在切片纽结的集合中也有无穷多个拓扑协调类。此外，我们通过构造成对拓扑但不光滑协调的切片结的无限族，以及拓扑切片且成对拓扑但不光滑协调的非切片结，来证明光滑和拓扑范畴之间的区别。
We establish a number of results about smooth and topological concordance of knots in S1×S2 . The winding number of a knot in S1×S2 is defined to be its class in H1(S1×S2;Z)≅Z . We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl–Nagel–Orson–Powell in the topological category. We say a knot in S1×S2 is slice (respectively, topologically slice) if it bounds a smooth (respectively, locally flat) disk in D2×S2 . We show that there are infinitely many topological concordance classes of non‐slice knots, and moreover, for any winding number other than ±1 , there are infinitely many topological concordance classes even within the collection of slice knots. Additionally, we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are pairwise topologically but not smoothly concordant, as well as non‐slice knots that are topologically slice and are pairwise topologically, but not smoothly, concordant.