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arXiv 2007.11774math.GT
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双曲纤维结的特殊手术

Exceptional surgeries on hyperbolic fibered knots

Yi Ni

备注：
12 pages, 2 figures. v2: Changed the title, the main result is now stated for exceptional surgeries, thanks to a result of Ying-Qing Wu. Revised a wrong citation of a theorem of Gabai, and the conclusion in one case became $|q|\le 3$ instead of $|q|\le 2$
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摘要：

设$K\subset S^3$为双曲纤维结，使得$S^3_{p/q}(K)$、$\frac pq$ --$K$上的手术是非双曲的。证明了如果$K$的单值是右转向的，则$0\le\frac pq\le 4g(K)$ .如果$S^3_{p/q}(K)$是一个小的Seifert纤维L-空间，则不能得到$4g(K)$的上界.如果$K$的单值性既不是右转向也不是左转向，则为$|q|\le3$。作为推论，对于任何给定的正环面结$T$，如果$p/q\ge4g(T)+4$，则$p/q$是特征斜率。这改进了Ni-Zhang和McCoy的上界.我们还证明了一些有限/循环斜率的特征。更确切地说，$14$表征$T_{4,3}$，$17$表征$T_{5,3}$，$4n+1$表征$T_{2n+1,2}$，除了$n=5$。根据Tange最近的定理，这表明$T_{2n+1,2}$是$S^3$中允许晶状体间隙手术的唯一结，而Alexander多项式具有$t^n-t^{n-1}+t^{n-2}+\text{lower order terms}$的形式。在附录中，我们证明了如果一个纤维纽结的纽结Floer同调的第二项的秩是$1$，则单值是右转向的或左转向的。

英文摘要：

Let $K\subset S^3$ be a hyperbolic fibered knot such that $S^3_{p/q}(K)$, the $\frac pq$--surgery on $K$, is non-hyperbolic. We prove that if the monodromy of $K$ is right-veering, then $0\le\frac pq\le 4g(K)$. The upper bound $4g(K)$ cannot be attained if $S^3_{p/q}(K)$ is a small Seifert fibered L-space. If the monodromy of $K$ is neither right-veering nor left-veering, then $|q|\le3$. As a corollary, for any given positive torus knot $T$, if $p/q\ge4g(T)+4$, then $p/q$ is a characterizing slope. This improves earlier bounds of Ni--Zhang and McCoy. We also prove that some finite/cyclic slopes are characterizing. More precisely, $14$ is characterizing for $T_{4,3}$, $17$ is characterizing for $T_{5,3}$, and $4n+1$ is characterizing for $T_{2n+1,2}$ except when $n=5$. By a recent theorem of Tange, this shows that $T_{2n+1,2}$ is the only knot in $S^3$ admitting a lens space surgery while the Alexander polynomial has the form $t^n-t^{n-1}+t^{n-2}+\text{lower order terms}$. In the appendix, we prove that if the rank of the second term of the knot Floer homology of a fibered knot is $1$, then the monodromy is either right-veering or left-veering.

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