On the geography and botany of knot Floer homology
On the geography and botany of knot Floer homology
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论结弗洛尔同源物的地理学和植物学
DOI：
10.1007/s00029-017-0351-5
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Watson, Liam
中科院分区：
文献类型：
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作者：
Hedden, Matthew;Watson, Liam
本文探讨了两个问题：（1）在三球面中，哪些双阶群作为纽结的纽结Floer同调而出现？(2)给定一个纽结，有多少个不同的纽结共享它的Floer同调？关于第一个，我们发现存在满足所有以前已知的约束结Floer同源性不出现作为一个结的不变量的双阶群。这导致了一个新的约束，承认透镜空间手术的结，以及证明结Floer同源的秩检测三叶结。对于第二，我们表明，任何非平凡的两个unknots带和产生一个无限的家庭不同的结同构结弗洛尔同源。我们还证明了具有单位单值性的纤维纽结可被其纽结Floer同调强检测到，这意味着Floer同调解决了具有非空边界的曲面的映射类群的字问题.最后，我们总结了一些经验和问题，并在此基础上提出了一些新的经验和问题。
This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.