On the Stein framing number of a knot

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.

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Received 2 June 2018

Accepted 1 October 2018

Published 25 March 2020