The stable 4-genus of knots
The 4-genus \(g_4(K)\) of a knot \(K\) in the 3-sphere is the minimal genus of a compact orientable surface in the 4-disk bounded by \(K\). The author introduces the stable 4-genus \(g_{\text{st}}(K)\) of \(K\) as the limit of \(g_4(nK)/n\), for \(n\) tending to \(\infty\), where \(nK\) is the \(n\)-fold knot sum of \(K\). As the 4-genus, the stable 4-genus is bounded below by \(\frac{1}{2}\) of the absolute value of the knot signature, is subadditive, and leads to a seminorm on the rational knot concordance group \(\mathcal C _{\mathbb Q}\). The unit ball with respect to this seminorm is investigated on 2-dimensional subspaces of \(\mathcal C _{\mathbb Q}\), spanned by torus knots, using the Ozsváth-Szabó invariant, the Khovanov-Rasmussen invariant, and the Tristram-Levine signature. The stable 4-genus is not necessarily an integer: By a construction using Casson-Gordon invariants, it is proved that the value of the stable 4-genus can be arbitrarily close to \(\frac{1}{2}\).
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