Ribbon knots and iterated cables of fibered knots

Abstract We define a knot to be $$\gamma _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> - sharp if its Seifert genus is detected by the concordance invariant $$\gamma _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , which arises from the immersed curve formalism in bordered Heegaard Floer homology. We show that a connected sum of $$\gamma _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -sharp fibered knots is ribbon exactly when it is of the form $$K \mathbin {\#} -K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>#</mml:mo> <mml:mo>-</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> . Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice–ribbon conjecture fails.