Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On definite lattices bounded by integer surgeries along knots with slice genus at most 2

Marco Golla, Christopher Scaduto

Abstract

We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most two. The proofs use input from Yang--Mills instanton gauge theory and Heegaard Floer correction terms.

On definite lattices bounded by integer surgeries along knots with slice genus at most 2

Abstract

We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most two. The proofs use input from Yang--Mills instanton gauge theory and Heegaard Floer correction terms.

Paper Structure

This paper contains 13 sections, 24 theorems, 25 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Distinguished lattices
  3. Obstructions
  4. Algebraic constraints from surgery cobordisms
  5. Heegaard Floer correction terms
  6. Input from instanton theory
  7. Proofs of parts (i) of Theorems \ref{['thm:trefoil']} and \ref{['thm:2,5']}
  8. Constructions
  9. Seifert spaces and binary polyhedral spaces
  10. PL spheres and rational complex curves
  11. Fillings constructed from PL spheres
  12. Further examples
  13. Final remarks

Key Result

Theorem 1.2

(i) Let $K\subset S^3$ be a knot of slice genus at most $1$, and $n\in \mathbb{Z}_{> 0}$. If a positive definite lattice fills $S_n^3(K)$, then it is isomorphic to one of the following, for some $k\geqslant 0$: (ii) Furthermore, when the knot is the right-handed trefoil, all of these possibilities are realized, except for the trivial lattice.

Figures (6)

  • Figure 1: The trefoil lattices $\mathscr{T}_n$.
  • Figure 2: The cinquefoil lattices $\mathscr{C}_n$.
  • Figure 3: The surgery cobordism $W_n$ is obtained by attaching a $+1$-framed meridian to $n$-surgery on $K$ (left), and the outgoing boundary of $W_n$ is homeomorphic to $(n-1)$-surgery (right). Here we use angular braces for relative handlebodies GS.
  • Figure 4:
  • Figure 5: A rational homology ball filling $S^3_9(T_{3,4})$. Adding a $-1$-framed meridian to the dotted circle gives an embedding of $-X_9(T_{3,4})$ in $\overline{\mathbb{C}\mathbb{P}^2}$.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 36 more