Exotic elliptic surfaces without 1-handles

TL;DR

This paper establishes conditions under which certain elliptic surfaces obtained via knot-surgery or log-transformation can be decomposed without 1-handles, expanding understanding of their handlebody structures.

Contribution

It provides new sufficient conditions for elliptic surfaces to admit handle decompositions without 1-handles, based on knot bridge numbers and parameters of log-transformations.

Findings

Knot-surgeries with bridge number ≤ 9n admit no 1-handle decompositions.

Log-transformations with gcd(p,q)=1 and min(p,q) ≤ 4 admit no 1-handle decompositions.

Specific cases for E(1) and E(n) surfaces are explicitly characterized.

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 $n$.