A refinement of the Ozsváth-Szabó large integer surgery formula and knot concordance

Abstract

We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot.

PL-Genus of surfaces in homology balls

“…The proof of Theorem 1.2 relies on showing that CFK(𝑌 𝑛 , 𝐾 𝑛 ) is sufficiently complicated so as to not admit local maps to and from CFK(𝑆 3 , 𝐽) of certain bigradings (see Section 2 for more details). For constructing our examples (𝑌 𝑛 , 𝐾 𝑛 ), we rely on recent work of the last author [14], which combines work of Hedden-Levine [3] and Truong [11] to give a description of the knot Floer complex for ( 𝑝, 1)-cables of the meridian in the image of surgery along a knot in 𝑆 3 . Preliminaries on this filtered mapping cone are given in Section 3 and the computation is carried out in Section 4.…”

“…For constructing our examples , we rely on recent work of the last author [14], which combines work of Hedden-Levine [3] and Truong [11] to give a description of the knot Floer complex for -cables of the meridian in the image of surgery along a knot in . Preliminaries on this filtered mapping cone are given in Section 3 and the computation is carried out in Section 4.…”

PL-genus of surfaces in homology balls

“…For constructing our examples (Y n , K n ), we rely on recent work of the last author [Zho22], which combines work of Hedden-Levine [HL19] and Truong [Tru21] to give a description of the knot Floer complex for (p, 1)-cables of the meridian in the image of surgery along a knot in S 3 . Preliminaries on this filtered mapping cone are given in Section 3 and the computation is carried out in Section 4.…”