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On surgeries from lens space $L(p,1)$ to $L(q,2)$

Boning Wang

TL;DR

The paper addresses the problem of classifying distance-one Dehn surgeries from a lens space $L(p,1)$ to another lens space $L(q,2)$ along a homologically essential knot, with $p,q$ odd. It employs the Wu–Yang $d$-invariant surgery formula together with Casson–Walker invariants and Seifert-fibered space plumbing techniques to obstruct most candidate surgeries and to identify a finite set of possible $(p,q)$ pairs, providing constructions for several cases. The main result enumerates viable pairs (and their constructions) while proving strong obstructions for many others through detailed $d$-invariant comparisons. Overall, the work illustrates how Heegaard Floer correction terms constrain distance-one surgeries between lens spaces and connects these invariants to classical 3-manifold invariants to refine the classification.

Abstract

We mainly use the d-invariant surgery formula established by Wu and Yang \cite{wu2025surgerieslensspacestype} to study the distance one surgeries along a homologically essential knot between lens spaces of the form $L(p,1)$ and $L(q,2)$ where $p,q$ are odd integers.

On surgeries from lens space $L(p,1)$ to $L(q,2)$

TL;DR

The paper addresses the problem of classifying distance-one Dehn surgeries from a lens space to another lens space along a homologically essential knot, with odd. It employs the Wu–Yang -invariant surgery formula together with Casson–Walker invariants and Seifert-fibered space plumbing techniques to obstruct most candidate surgeries and to identify a finite set of possible pairs, providing constructions for several cases. The main result enumerates viable pairs (and their constructions) while proving strong obstructions for many others through detailed -invariant comparisons. Overall, the work illustrates how Heegaard Floer correction terms constrain distance-one surgeries between lens spaces and connects these invariants to classical 3-manifold invariants to refine the classification.

Abstract

We mainly use the d-invariant surgery formula established by Wu and Yang \cite{wu2025surgerieslensspacestype} to study the distance one surgeries along a homologically essential knot between lens spaces of the form and where are odd integers.
Paper Structure (14 sections, 15 theorems, 117 equations)

This paper contains 14 sections, 15 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preparation
  3. Casson Walker invariant
  4. Seifert fibered space
  5. d-invariants
  6. d-invariant calculation
  7. surgery formula
  8. Casson Walker invariant
  9. negative framing
  10. positive framing
  11. m-k=-1
  12. calculation
  13. Exception
  14. Main proof of theorem \ref{['results']}

Key Result

Theorem 1.1

Suppose there exists a distance one surgery from lens spaces $L(p,1)$ along a homologically essential knot $K$ to $L(q,2)$, with $p>1$ is odd and $q\neq0,$$|q|>7$, then the only possible pairs $(p,q)$ are: $(1)$$(p,q)=(p,2p-1),$$(2)$$(p,q)=(p,2p+1),$$(3)$$(p,q)=(p,2p-9),$$(4)$$|q|=|p-k^2|,$$(5)$$(p,

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • proof : Proof of theorem\ref{['construction']}
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • Theorem 3.1
  • ...and 16 more