Ribbon knots and iterated cables of fibered knots

TL;DR

This paper introduces the concept of b3_0-sharp knots, linking their Seifert genus detection to ribbon properties, and explores implications for the structure of the smooth concordance group and the slice-ribbon conjecture.

Contribution

It establishes a characterization of ribbon knots as connected sums of b3_0-sharp fibered knots of the form K d7 -K, connecting knot invariants with concordance properties.

Findings

Connected sum of b3_0-sharp fibered knots is ribbon iff it is of the form K d7 -K.

Iterated cables of tight fibered knots are either linearly independent or the slice-ribbon conjecture fails.

The invariant b3_0$ detects the Seifert genus for b3_0-sharp knots.

Abstract

We define a knot to be $\gamma_0$-sharp if its Seifert genus is detected by the concordance invariant $\gamma_0$, which arises from the immersed curve formalism in bordered Heegaard Floer homology. We show that a connected sum of $\gamma_0$-sharp fibered knots is ribbon exactly when it is of the form $K \mathbin{\#} -K$. Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice--ribbon conjecture fails.