#### Volume 10, issue 4 (2010)

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The stable $4$–genus of knots

### Charles Livingston

Algebraic & Geometric Topology 10 (2010) 2191–2202
##### 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, ${\mathsc{C}}_{Q}=\mathsc{C}\otimes Q$. Basic properties of ${g}_{st}$ are developed, as are examples focused on understanding the unit ball for ${g}_{st}$ on specified subspaces of ${\mathsc{C}}_{Q}$. 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.

##### Keywords
knot concordance, four-genus
Primary: 57M25