A ribbon obstruction and derivatives of knots

TL;DR

This paper introduces a new obstruction for knots to be Z[Z]-homology ribbon, providing restrictions on derivative link invariants and revealing non-doubly slice knots, with implications for knot concordance and solvability filtrations.

Contribution

It defines a novel obstruction for Z[Z]-homology ribbon knots and explores its implications for derivative link invariants and knot sliceness properties.

Findings

Identifies new non-doubly slice knots.

Shows doubly algebraically slice ribbon knots are not necessarily doubly (1)-solvable.

Suggests potential limitations of the new obstruction on slice knots.

Abstract

We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots. In particular this gives new information on the doubly solvable filtration of Taehee Kim: doubly algebraically slice ribbon knots need not be doubly (1)-solvable, and doubly algebraically slice knots need not be (0.5,1)-solvable. We also discuss potential connections to unsolved conjectures in knot concordance, such as generalised versions of Kauffman's conjecture. Moreover it is possible that our obstruction could fail to vanish on a slice knot.