Journal of Symplectic Geometry

Volume 18 (2020)

Number 1

On the Stein framing number of a knot

Pages: 191 – 215



Thomas E. Mark (Department of Mathematics, University of Virginia, Charlottesville, Va., U.S.A.)

Lisa Piccirillo (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Faramarz Vafaee (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)


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 $n \lt \overline{tb}(K)$, 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 $n \geq \overline{tb}(K)$ 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 $\overline{tb}(K)$. We also provide an upper bound on the Stein framings for $K$ that is typically stronger than that coming from the adjunction inequality.

Received 2 June 2018

Accepted 1 October 2018

Published 25 March 2020