The stable 4–genus of knots

Abstract

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 D 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.