Doubly slicing knots and embedding 3-manifolds in 4-manifolds

TL;DR

This paper introduces new invariants for knots related to their embeddings in 4-manifolds, extending classical notions and providing obstructions using $L^2$-signatures, with results showing these invariants can be arbitrarily large.

Contribution

The paper extends the concepts of double slice genus and superslice genus to arbitrary simply connected 4-manifolds and introduces the double stabilizing number, with new $L^2$-signature obstructions.

Findings

Existence of knots with arbitrarily large $X$-double slice genus.

Construction of knots with arbitrarily large generalized superslice genus.

Introduction of the double stabilizing number as a new invariant.

Abstract

For a knot $K$ in the 3-sphere and a simply connected closed 4-manifold $X$, we define the $X$-double slice genus of $K$, extending the notion from the case when $X$ is the 4-sphere. We show that for each integer $n$, there exists an algebraically doubly slice and ribbon knot $K$ whose $X$-double slice genus is greater than $n$. Our arguments use new $L^2$-signature obstructions to embedding closed 3-manifolds with infinite cyclic first homology into closed 4-manifolds with infinite cyclic fundamental group, in a way that preserves first homology. We also extend the concept of the superslice genus of a knot to simply connected 4-manifolds and show that there exist doubly slice knots whose generalized superslice genera are arbitrarily large. Furthermore, we define the double stabilizing number of a knot, extending the stabilizing number introduced by Conway and Nagel, and show that this invariant can also be arbitrarily large.