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Exotic elliptic surfaces without 1-handles

Motoo Tange

TL;DR

The paper addresses whether exotic elliptic surfaces produced by knot-surgery $E(n)_K$ or double log-transformations $E(n)_{p,q}$ admit handle decompositions free of $1$-handles. It develops a bridge-number-based framework and analyzes the global monodromy to produce explicit handle-cancellation schemes, yielding a broad sufficient condition for eliminating all $1$-handles. Specifically, if the knot $K$ satisfies $b(K)\le 9n$, then $E(n)_K$ admits a $1$-handle-free decomposition; in particular, $E(1)_{p,q}$ has such a decomposition when $\min\{p,q\}\le 9$. Furthermore, for $\gcd(p,q)=1$ and $\min\{p,q\}\le 4$, the double log-transformations $E(n)_{p,q}$ admit a $1$-handle-free decomposition for any $n$. These results extend prior work on $1$-handle obstructions and provide systematic handle-calculus techniques for constructing exotic smooth structures with simplified handle decompositions.

Abstract

In this article, we consider a sufficient condition that a knot-surgery or log-transformation of $E(n)$ admits a handle decomposition without 1-handles. We show that if $K$ is a knot that the bridge number is $b(K)\le 9n$, then the knot-surgery $E(n)_K$ of the elliptic surface $E(n)$ admits a handle decomposition without 1-handles. This means that if $\gcd(p,q)=1$, and $\min\{p,q\}\le 9$, then $E(1)_{p,q}$ admits a handle decomposition without 1-handles. We also show that if $\gcd (p,q)=1$, $\min\{p,q\}\le 4$, then the double log-transformation $E(n)_{p,q}$ admits a handle decomposition without 1-handles for any positive integer $n$.

Exotic elliptic surfaces without 1-handles

TL;DR

The paper addresses whether exotic elliptic surfaces produced by knot-surgery or double log-transformations admit handle decompositions free of -handles. It develops a bridge-number-based framework and analyzes the global monodromy to produce explicit handle-cancellation schemes, yielding a broad sufficient condition for eliminating all -handles. Specifically, if the knot satisfies , then admits a -handle-free decomposition; in particular, has such a decomposition when . Furthermore, for and , the double log-transformations admit a -handle-free decomposition for any . These results extend prior work on -handle obstructions and provide systematic handle-calculus techniques for constructing exotic smooth structures with simplified handle decompositions.

Abstract

In this article, we consider a sufficient condition that a knot-surgery or log-transformation of admits a handle decomposition without 1-handles. We show that if is a knot that the bridge number is , then the knot-surgery of the elliptic surface admits a handle decomposition without 1-handles. This means that if , and , then admits a handle decomposition without 1-handles. We also show that if , , then the double log-transformation admits a handle decomposition without 1-handles for any positive integer .
Paper Structure (13 sections, 4 theorems, 10 equations, 11 figures)

This paper contains 13 sections, 4 theorems, 10 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Handle decomposition of a 4-manifold.
  3. Handle decomposition of $E(n)_{K}$ or $E(n)_{p,q}$
  4. Main results.
  5. Preliminaries
  6. Knot-surgery
  7. Bridge presentation of knot
  8. Handle decomposition of knot-surgery
  9. Log-transformation
  10. Proofs
  11. The global monodromy of $E(n)$
  12. Proof of Theorem \ref{['main']}.
  13. Proof of Theorem \ref{['main3']}.

Key Result

Theorem 1.2

Let $K$ be a knot in $S^3$ with $b(K)\le 9n$. Then $E(n)_K$ admits a handle decomposition without 1-handles.

Figures (11)

  • Figure 1: A normal form of an $n$-bridge knot. $B$ is a pure braid.
  • Figure 2: A generator element $\mathcal{T}_{i,j}$ of $PB_n$.
  • Figure 3: A handle decomposition of $T^2\times D^2$.
  • Figure 4: A deformation of the handle diagram of $T^2\times D^2$. All the components with no dots are 0-framed 2-handles.
  • Figure 5: A deformation of diagram for a generator element $\mathcal{T}_{2,5}$ in $PB_6$
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Theorem 1.4
  • Proposition 2.3
  • proof