# On the Stein framing number of a knot

Mark TE, Piccirillo L, Vafaee F.. 2020.

## NOTES

To appear in J. Symp. Geom.

## Abstract

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