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Knot surgered elliptic surfaces for a $(2,2h+1)$-torus knot

Naoyuki Monden, Reo Yabuguchi

TL;DR

This work proves that knot surgered elliptic surfaces $E(n)_{T(2,2h+1)}$ admit handle decompositions free of $1$- and $3$-handles for all $h\ge1$ and $n\ge1$, advancing the Kirby problem on eliminating such handles in simply connected 4-manifolds. The authors leverage Kirby diagrams derived from Lefschetz fibrations on $E(n)_K$, together with a detailed analysis of the monodromy and fiber-sum constructions, to translate the problem into explicit handle cancellations on a surface diagram. A key tool is the explicit global monodromy ${}_{\Phi_K}(W)^2 \cdot W^2$ for knot-surgered elliptic surfaces and its behavior under cyclic permutations and diffeomorphisms $\Phi_K$, especially for torus knots $T_{2,2h+1}$. The result generalizes prior work on genus-$1$ and higher-genus Lefschetz fibrations and contributes to a broader understanding of when 4-manifolds admit 1- and 3-handle-free decompositions, with potential implications for exotic smooth structures and Kirby's problem list.

Abstract

We show that for any positive integer $h$, a knot surgered elliptic surface $E(n)_{T(2,2h+1)}$ for a $(2,2h+1)$-torus knot $T(2,2h+1)$ and the elliptic surface $E(1)_{2,2h+1}$ admit handle decompositions without 1- and 3-handles using the Kirby diagrams derived from Lefschetz fibrations on them.

Knot surgered elliptic surfaces for a $(2,2h+1)$-torus knot

TL;DR

This work proves that knot surgered elliptic surfaces admit handle decompositions free of - and -handles for all and , advancing the Kirby problem on eliminating such handles in simply connected 4-manifolds. The authors leverage Kirby diagrams derived from Lefschetz fibrations on , together with a detailed analysis of the monodromy and fiber-sum constructions, to translate the problem into explicit handle cancellations on a surface diagram. A key tool is the explicit global monodromy for knot-surgered elliptic surfaces and its behavior under cyclic permutations and diffeomorphisms , especially for torus knots . The result generalizes prior work on genus- and higher-genus Lefschetz fibrations and contributes to a broader understanding of when 4-manifolds admit 1- and 3-handle-free decompositions, with potential implications for exotic smooth structures and Kirby's problem list.

Abstract

We show that for any positive integer , a knot surgered elliptic surface for a -torus knot and the elliptic surface admit handle decompositions without 1- and 3-handles using the Kirby diagrams derived from Lefschetz fibrations on them.

Paper Structure

This paper contains 7 sections, 5 theorems, 7 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Basics of Lefschetz fibrations
  3. Kirby diagrams
  4. A Kirby diagram derived from a Lefschetz fibration
  5. A Kirby diagram on a surface with a boundary component
  6. Knot surgered elliptic surfaces
  7. Proof of Theorem \ref{['thm:1']}

Key Result

Theorem 1.2

For any positive integer $h$ and $n\geq 1$, $E(n)_{T(2,2h+1)}$ admits a handle decomposition without 1- and 3-handles.

Figures (31)

  • Figure 1: A handle decomposition of $\Sigma_g$ and a Kirby diagram of $\Sigma_g \times \mathbb{D}^2$.
  • Figure 2: The vanishing cycles $x_1,x_2,x_3$ of the genus-$2$ Lefschetz fibration $f_0$ on $X_0$ and a Kirby diagram of $X_0$.
  • Figure 3: RIGHT: a Kirby diagram "on $\Sigma_g^1$" of $X_0$, LEFT: a standard Kirby diagram of $X_0$.
  • Figure 4: The generators $\alpha_i,\beta_i$ of $\pi_1(\Sigma_g^1,p)$, and the 1-handles $\beta_i^\ast,\alpha_i^\ast$ for $i=1,2,\ldots,g$.
  • Figure 5:
  • ...and 26 more figures

Theorems & Definitions (8)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 4.1: FS4,Yun2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof : Proof of Theorem \ref{['thm:1']}
  • proof : Proof of Lemma \ref{['lem:1001']}