The stable $4$–genus of knots

Abstract

We define the stable $4$–genus of a knot $K\subset S^3$, $g_{st}\left(K\right)$, to be the limiting value of $g_4\left(nK\right)∕n$, where $g_4$ denotes the $4$–genus and $n$ goes to infinity. This induces a seminorm on the rationalized knot concordance group, ${\mathcal{C}}_Q=\mathcal{C}\otimes Q$. Basic properties of $g_{st}$ are developed, as are examples focused on understanding the unit ball for $g_{st}$ on specified subspaces of ${\mathcal{C}}_Q$. 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 $g_{st}\left(K\right)$ can be a noninteger.