Finding
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 $t^n-t^{n-1}+t^{n-2}+\cdots$.
Paper
Seifert fibered and reducible surgeries on hyperbolic fibered knots
Citations: 1
Full textAbstract
Let $K\subset S^3$ be a hyperbolic fibered knot such that $S^3_{p/q}(K)$, the $\frac pq$--surgery on $K$, is either reducible or an atoroidal Seifert fibered space. 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 $K$ is an L-space knot. If the monodromy of $K$ is neither right-veering nor left-veering, then $|q|=1$ and the surgery is irreducible. 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 $t^n-t^{n-1}+t^{n-2}+\cdots$. 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.
Authors
Yi Ni
Journal
arXiv: Geometric Topology