Immersed surfaces with knot group $\mathbb{Z}$

TL;DR

This paper classifies immersed surfaces with fundamental group Z in 4-manifolds, providing criteria for isotopy, enumeration of disks, and relations to knot crossing changes, with applications to understanding surface uniqueness and knot bounds.

Contribution

It introduces a classification of immersed surfaces with Z fundamental group using equivariant intersection forms and secondary invariants, and applies this to knot and disk problems in 4-manifolds.

Findings

Classification of immersed surfaces with fixed genus and double points.

Criteria for when an immersed Z-surface in S^4 is standard.

Enumeration of Z-disks in D^4 with a single double point, possibly infinite.

Abstract

This article is concerned with locally flatly immersed surfaces in simply-connected $4$-manifolds where the complement of the surface has fundamental group $\mathbb{Z}$. Once the genus and number of double points are fixed, we classify such immersed surfaces in terms of the equivariant intersection form of their exterior and a secondary invariant. Applications include criteria for deciding when an immersed $\mathbb{Z}$-surface in $S^4$ is isotopic to the standard immersed surface that is obtained from an unknotted surface by adding local double points. As another application, we enumerate $\mathbb{Z}$-disks in $D^4$ with a single double point and boundary a given knot; we prove that the number of such disks may be infinite. We also prove that a knot bounds a $\mathbb{Z}$-disk in $D^4$ with $c_+$ positive double points and $c_-$ negative double points if and only if it can be converted into an Alexander polynomial one knot via changing $c_+$ positive crossings and $c_-$ negative crossings. In $4$-manifolds other than $D^4$ and $S^4$, applications include measuring the extent to which immersed $\mathbb{Z}$-surfaces are determined by the equivariant intersection form of their exterior. Along the way, we prove that any two $\mathbb{Z}^2$-concordances between the Hopf link and an Alexander polynomial one link $L$ are homeomorphic rel. boundary.