On the Stein framing number of a knot

Original · EN

For an integer n, write Xₙ(K) for the 4-manifold obtained by attaching a 2-handle to the 4-ball along the knot K⊂ S³ with framing n. It is known that if n< tb(K), then Xₙ(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 Xₙ(K) is Stein. We provide examples of knots K and integers n≥ tb(K) for which Xₙ(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 Xₙ(K) admits a Stein structure can be arbitrarily larger than 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.