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.

中文翻译：

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关于结的斯坦因帧数

对于整数 $n$，将 2-handle 连接到 4-ball 获得的 4-manifold 写为 $X_n(K)$，沿着结 $K\subset S^3$ 和框架 $n$。已知如果 $n< \overline{\text{tb}}(K)$，则 $X_n(K)$ 承认 Stein 域的结构，而且附加不等式意味着存在上界$n$ 的值，使得 $X_n(K)$ 是 Stein。我们提供了结 $K$ 和整数 $n\geq \overline{\text{tb}}(K)$ 的示例，其中 $X_n(K)$ 是 Stein，回答了该领域的一个悬而未决的问题。事实上，我们的一系列例子表明，流形 $X_n(K)$ 承认 Stein 结构的最大框架可以任意大于 $\overline{\text{tb}}(K)$。我们还提供了 $K$ 的 Stein 框架的上限，该上限通常强于来自附加不等式的上限。