The Stable 4–genus of Knots

We define the stable 4–genus of a knot K ⊂ S, gst(K), to be the limiting value of g4(nK)/n, where g4 denotes the 4–genus and n goes to infinity. This induces a seminorm on the rationalized knot concordance group, CQ = C ⊗ Q. Basic properties of gst are developed, as are examples focused on understanding the unit ball for gst on specified subspaces of CQ. Subspaces spanned by torus knots are used to illustrate the distinction between the smooth and topological categories. A final example is given in which Casson-Gordon invariants are used to demonstrate that gst(K) can be a noninteger. 1. Summary. In order to better understand the smooth 4–genus of knots K ⊂ S, denoted g4(K), we introduce and study here the stable 4–genus, gst(K) = lim n→∞ g4(nK)/n. As will be seen in Section 2, the existence of the limit and its basic properties follow from the subadditivity of g4 as a function on the classical knot concordance group C; that is, g4(K# J) ≤ g4(K) + g4(J) for all K and J . Neither classical knot invariants nor the invariants that arise from HeegaardFloer theory [12] or Khovanov homology [13] can be used to demonstrate that gst(K) / ∈ Z for some K. One result of this paper is the construction of a knots K for which gst(K) is close to 1 2 . Perhaps of greater interest is the exploration of the new perspective on the 4–genus and knot concordance offered from the stable viewpoint. In particular, a number of interesting and challenging new questions arise naturally. For example, we note that finding a knot K with 0 < gst(K) < 1 2 is closely related to the existence of torsion in C of order greater than 2. We will also consider the distinction between the smooth and topological categories from the perspective of the stable genus. Acknowledgements Thanks are due to Pat Gilmer for conversations related to his results on 4–genus, which play a key role in Section 7. Thanks are also due to Ian Agol and Danny Calegari for discussing with me the analogy between the stable genus and the stable commutator length, described in Section 8. 2. Algebraic preliminaries. The existence of the limiting value and its basic properties are summarized in the following general theorem. This work was supported by a grant from the NSF.