Variations of 4-dimensional twists obtained by an infinite order plug

TL;DR

This paper introduces new infinite families of exotic 4-manifolds derived from a specific infinite order plug, explores their properties, and constructs twists that produce knot surgeries and potential exotic structures.

Contribution

It constructs two infinite families of exotic 4-manifolds with specific boundary properties and develops new plug twists that realize knot and link surgeries, including mutant knots.

Findings

Two infinite families of exotic 4-manifolds with $b_2=3,4$

A plug twist producing 2-bridge knot or link surgery

A twist between knot surgeries of mutant knots as a candidate for exotic $ ext{ extasciitilde}^2S^2 imes S^2$

Abstract

In the previous paper the author defined an infinite order plug $(P,\varphi)$ which gives rise to infinite Fintushel-Stern's knot-surgeries. Here, we give two 4-dimensional infinitely many exotic families $Y_n$, $Z_n$ of exotic enlargements of the plug. The families $Y_n$, $Z_n$ have $b_2=3$, $4$ and the boundaries are 3-manifolds with $b_1=1$, $0$ respectively. We give a plug (or g-cork) twist $(P,\varphi_{p,q})$ producing the 2-bridge knot or link surgery by combining the plug $(P,\varphi)$. As a further example, we describe a 4-dimensional twist $(M,\mu)$ between knot-surgeries for two mutant knots. The twisted double concerning $(M,\mu)$ gives a candidate of exotic $\#^2S^2\times S^2$.