On the Stein framing number of a knot

Mark TE, Piccirillo L, Vafaee F. On the Stein framing number of a knot. 2020.


To appear in J. Symp. Geom.


For an integer \(n\), write \(X_n(K)\) for the 4-manifold obtained by attaching a 2-handle to the 4-ball along the knot \(K\subset S^3\) with framing \(n\). It is known that if \(n< \overline{\mathrm{tb}}(K)\), then \(X_n(K)\) admits the structure of a Stein domain, and moreover the adjunction inequality implies there is an upper bound on the value of \(n\) such that \(X_n(K)\) is Stein. We provide examples of knots \(K\) and integers \(n\geq \overline{\mathrm{tb}}(K)\) for which \(X_n(K)\) is Stein, answering an open question in the field. In fact, our family of examples shows that the largest framing such that the manifold \(X_n(K)\) admits a Stein structure can be arbitrarily larger than \(\overline{\mathrm{tb}}(K)\). We also provide an upper bound on the Stein framings for \(K\) that is typically stronger than that coming from the adjunction inequality.

Last updated on 03/06/2022