The Thurston norm \(x\colon H_2(Y,\partial Y;\mathbb{R})\to \mathbb{R}\) [W. P. Thurston, Mem. Am. Math. Soc. 339, 99–130 (1986; Zbl 0585.57006)] is a measure of complexity for \(3\)-manifolds \(Y\) with boundary. Connections between the Thurston norm and Heegaard-Floer homology have been known for some time. Notably, for a link \(L\subset S^3\) it is known that the link Floer homology determines the Thurston norm of the link complement [P. Ozsváth and Z. Szabó, J. Am. Math. Soc. 21, No. 3, 671–709 (2008; Zbl 1235.53090)]. In this article, the authors investigate a further connection, this time considering a certain type of concordance class of links in \(S^3\), and using the twisted Heegaard-Floer correction terms defined by the second author together with S. Behrens [Quantum Topol. 9, No. 1, 1–37 (2018; Zbl 1388.57026)].
Fix a 2-component link \(L=L_0\cup L_1\subset S^3\) with vanishing linking number, then perform 0-surgery on \(L_0\) and 1-surgery on \(L_1\) to obtain a closed \(3\)-manifold \(Y\). Let \(L'=L_0'\cup L_1'\) be concordant to \(L\) and write \([\Sigma]\in H_2(S^3,L;\mathbb{Z})\) for the class that is Poincaré-Lefschetz dual to the meridian of \(L_0'\) (the component concordant to \(L_0\)). The main result of the paper is the inequality \[ \left\lceil \frac{x([\Sigma])+1}{4}\right\rceil\geq \frac{\underline{d}(Y)+\underline{d}(-Y)+1}{2}. \] Here \(\underline{d}\) denotes the fully twisted Heegaard-Floer correction term of Behrens-Golla. (Recall that the original Heegaard-Floer correction terms were defined for rational homology \(3\)-spheres and the twisted versions are a generalisation that in particular are defined for more general \(3\)-manifolds.)
In practice, the authors apply the inequality among links where the \(0\)-surgery component is the unknot. Performing \(0\)-surgery on the unknot component yields \(S^1\times S^2\) and the second component determines a nullhomologous knot in \(S^1\times S^2\) (nullhomologous due to the linking number 0 assumption). In this case the left-hand side of the inequality can be interpreted as one quarter of the geometric winding number of the nullhomologous knot; that is the minimal number of geometric intersections between the knot and the \(2\)-sphere \(\{\text{pt}\}\times S^2\), within the isotopy class of that knot. Thus the inequality leads to a lower bound for the geometric winding number of the concordance class of a nullhomologous knot in \(S^1\times S^2\).
This bound is used to provide examples in \(S^1\times S^2\) of the following: knots with vanishing Schneiderman invariant (see [R. Schneiderman, Algebr. Geom. Topol. 3, 921–968 (2003; Zbl 1039.57005)]) but nonvanishing geometric winding number; knots that are not smoothly concordant to local knots; topologically but not smoothly slice knots in \(S^1\times D^3\); and knots with arbitrarily high geometric winding number. Although some of these example types have been previously obtained by other authors, these serve as a good showcase of the effectiveness of the authors’ method.