# Studies of distance one surgeries on the lens space $L(p,1)$

## TL;DR

This paper classifies which lens spaces can be obtained from specific lens spaces via distance one surgeries, providing explicit conditions for $L(5,1)$ and $L(7,1)$ cases, and explores band surgeries between certain torus knots.

## Contribution

It offers a complete classification of lens spaces reachable by distance one surgeries from $L(5,1)$ and $L(7,1)$, and analyzes band surgeries between $T(2,p)$ and $T(2,n)$.

## Key findings

- $L(n,1)$ is obtained from $L(5,1)$ by a distance one surgery only for specific $n$ values.
- $L(n,1)$ is obtained from $L(7,1)$ by a distance one surgery only for specific $n$ values.
- Characterization of band surgeries from $T(2,p)$ to $T(2,n)$.

## Abstract

In this paper, we study distance one surgeries between lens spaces $L(p,1)$ with $p \geq 5$ prime and lens spaces $L(n,1)$ for $n \in \mathbb{Z}$ and band surgeries from $T(2,p)$ to $T(2,n)$. In particular, we prove that $L(n,1)$ is obtained by a distance one surgery from $L(5,1)$ only if $n=\pm 1$, $4$, $\pm 5$, $6$ or $\pm 9$, and $L(n,1)$ is obtained by a distance one surgery from $L(7,1)$ if and only if $n=\pm 1$, $3$, $6$, $7$, $8$ or $11$.