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

Abstract

In this paper, we show that the lens space $L(s,1)$ for $s\ne 0$ is obtained by a distance one surgery along a knot in the lens space $L(n,1)$ with $n\ge 5$ odd only if $n$ and $s$ satisfy one of the following cases: (1) $n\ge 5$ 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 $s\ne 0$ is obtained by a band surgery from $T(2,n)$ with $n\ge 5$ 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.

arXiv commentWe have improved the techniques presented in this paper and extended the results to L(n,1) for all integers n, as detailed in the new paper arXiv:2504.02325. Additionally, we have corrected an error regarding the chirally cosmetic banding of the torus knot T(2,9) in the new paper

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• Lens space surgeries and L-space homology spheres
• Floer Simple Manifolds and L-Space Intervals
• A spanning tree model for the Heegaard Floer homology of a branched double-cover
• Some knots with surgeries yielding lens spaces
• Distance one lens space fillings and band surgery on the trefoil knot