Distance one surgeries on the lens space $L(n,1)$

Abstract

In this paper, we show that the lens space $L(s,1)$ for $s \neq 0 $ is obtained by a distance one surgery along a knot in the lens space $L(n,1)$ with $n \geq 5$ odd only if $n$ and $s$ satisfy one of the following cases: (1) $n \geq 5$ is any odd integer and $s=\pm 1, n, n \pm 1$ or $n \pm 4$; (2) $n=5$ and $s=-5$; (3) $n=5$ and $s=-9$; (4) $n=9$ and $s=-5$. As a corollary, we prove that the torus link $T(2,s)$ for $s \neq 0 $ is obtained by a band surgery from $T(2,n)$ with $n \geq 5$ odd only if $n$ and $s$ are as listed above. Combined with the result of Lidman, Moore and Vazquez, it immediately follows that the only nontrivial torus knot $T(2,n)$ admitting chirally cosmetic banding is $T(2,5)$. The key ingredient of our proof is the Heegaard Floer mapping cone formula.