Virtual Covers of Links II

TL;DR

This paper introduces semi-fibered concordance for links with a fibered component, using virtual covers to define invariants that obstruct certain slice and ribbon discs, and explores applications to satellite operators and knots in fibered 3-manifolds.

Contribution

It develops the concept of semi-fibered concordance for links, utilizing virtual covers to produce new invariants and applications in knot theory.

Findings

The virtual knot associated to a link is a semi-fibered concordance invariant.

Obstructions to slice and ribbon discs are derived from these invariants.

Applications include results on satellite operators and knots in fibered 3-manifolds.

Abstract

A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first component. Motivated by some examples of Gompf-Scharlemann-Thompson, we further limit our attention to those links $L$ where $K$ is "close to" a fiber of $J$. Such $L$ are studied with virtual covers, where a virtual knot $\upsilon$ is associated to $L$. We show that the concordance class of $\upsilon$ is a semi-fibered concordance invariant. This gives obstructions for certain slice and ribbon discs for the $K$ component. Further applications are to injectivity of satellite operators in semi-fibered concordance and to knots in fibered $3$-manifolds.