On the Stein framing number of a knot (Q2306967)

According to a celebrated result of \textit{Y. Eliashberg} [Lond. Math. Soc. Lect. Note Ser. 151, 45--72 (1990; Zbl 0731.53036)], a smooth (real) \(4\)-manifold admits a Stein structure if and only if it admits a handle decomposition with only \(0\), \(1\) and \(2\)-handles, where each \(2\)-handle is attached along a Legendrian knot \(\mathcal{L}\) with framing one less than the Thurston-Bennequin framing \(\operatorname{tb}(\mathcal{L})\). Given a \(4\)-manifold with a random handle decomposition, it is not always clear whether the given handle decomposition can be modified so that it satisfies Eliashberg's criterion. In the paper under review, the authors consider the case of a single \(2\)-handle attached to a \(4\)-ball along a knot \(K\subset S^3\). It is elementary to see that the resulting \(4\)-manifold, \(X_n(K)\), admits a Stein structure if \(n<\overline{\operatorname{tb}}(K)\), where \(\overline{\operatorname{tb}}(K)\) denotes the maximum Thurston-Bennequin number of all Legendrian representatives of \(K\). The authors show that the converse of the above result is false. In particular, they show that the largest \(n\) such that \(X_n(K)\) is Stein, a number they call the \textit{Stein framing number} and denote \(\operatorname{Sf}(K)\), can be arbitrarily larger than \(\overline{\operatorname{tb}}(K)\). This answers negatively a question stated explicitly by \textit{K. Yasui} [J. Symplectic Geom. 15, No. 1, 91--105 (2017; Zbl 1369.57030)]. In order to prove their result, the authors use the work of \textit{T. Abe} et al. [Int. Math. Res. Not. 2015, No. 22, 11667--11693 (2015; Zbl 1331.57004)] to construct a family of pairs of distinct knots \(P_m\) and \(Q_m\) such that \(X_{-m}(P_m)\) is diffeomorphic to \(X_{-m}(Q_m)\) for \(m\in\mathbb{N}\). A large part of the paper is devoted to using techniques from Khovanov homology to obtain estimates for \(\overline{\operatorname{tb}}\) that are used in the proof. Using knot Floer homology, the authors also give new upper bounds on \(\operatorname{Sf}\).