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A note on band surgery and the signature of a knot
Bulletin of the London Mathematical Society ( IF 0.8 ) Pub Date : 2020-07-21 , DOI: 10.1112/blms.12397
Allison H. Moore 1 , Mariel Vazquez 2
Affiliation  

Band surgery is an operation relating pairs of knots or links in the three‐sphere. We prove that if two quasi‐alternating knots K and K of the same square‐free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer d ‐invariants for pairs of L‐spaces that are related by distance one Dehn fillings and satisfy a certain condition in first homology. These results imply that T ( 2 , 5 ) is the only torus knot T ( 2 , m ) with m square‐free that admits a chirally cosmetic banding, that is, a band surgery operation to its mirror image. We conclude with a discussion on the scarcity of chirally cosmetic bandings.

中文翻译:

关于乐队手术和打结的笔记

带外科手术是指在三个球体中成对的结或链接。我们证明如果有两个准交替的结 ķ ķ 无平方行列式的一个定理通过带手术进行关联,则其签名差异的绝对值为0或8。这种障碍来自关于Heegaard Floer差异的更一般的定理 d 一对L空间的不变量,它们与一个Dehn填充物的距离相关并且在第一同源性中满足特定条件。这些结果表明 Ť 2 5 是唯一的圆环结 Ť 2 无平方的,允许手性装饰带,即对其镜像进行带状手术。最后,我们讨论了手性化妆品条带的稀缺性。
更新日期:2020-07-21
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