On the Stein framing number of a knot

TL;DR

This paper constructs examples of knots and framings where the associated 4-manifolds admit Stein structures beyond known bounds, challenging previous assumptions and providing new insights into Stein fillability.

Contribution

It demonstrates that the maximum Stein framing number for a knot can be arbitrarily larger than the maximal Thurston-Bennequin number, answering an open question.

Findings

Examples of knots with Stein structures beyond the known bounds.

The largest Stein framing can be arbitrarily larger than the maximal Thurston-Bennequin number.

An upper bound on Stein framings that is often stronger than the adjunction inequality.

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