A note on surfaces in \(\mathbb{CP}^2\) and \(\mathbb{CP}^2\#\mathbb{CP}^2\) (Q6562823)

The \(\mathbb{CP}^2\)-genus of a knot \(K\) in \(S^3\), denoted by \(g_{\mathbb{CP}^2}(K)\), is the least genus of a smooth, compact, connected, orientable, properly embedded surface \(F\) in \(\mathbb{CP}^2 \backslash \mathrm{Int} B^4\) bounded by \(K\) in \(\partial (\mathbb{CP}^2 \backslash \mathrm{Int} B^4)\cong S^3\). The topological \(\mathbb{CP}^2\)-genus of a knot \(K\) in \(S^3\), denoted by \(g^{\mathrm{top}}_{\mathbb{CP}^2}(K)\), is the least genus of a locally flat, embedded surface \(F\) in \(\mathbb{CP}^2 \backslash \mathrm{Int} B^4\) bounded by \(K\) in \(\partial (\mathbb{CP}^2 \backslash \mathrm{Int} B^4)\). For a smooth 4-manifold \(X\), the genus function \(G_X\) is a function \(H_2(X; \mathbb{Z}) \to \mathbb{Z}_{\geq 0}\), given by \(G_X(\alpha)=\min\{g(F) \mid i: F \to X, i_*([F])=\alpha\}\), where \(g(F)\) is the genus and the minimum is taken over all smooth, closed, oriented surfaces \(F\) and all smooth embeddings \(i: F \to X\). \newline In this paper, the main results are as follows. For a non-negative integer \(n\), there exists a topologically slice knot \(K\) with \(g_{\mathbb{CP}^2}(K) \geq n\). For a knot \(K\) in \(S^3\), if \(\mathrm{Arf}(K)=0\), then \(g^{\mathrm{top}}_{\mathbb{CP}^2}(K)=0\). The authors give an upper bound of the \(\mathbb{CP}^2\)-genus of an \((n, n-1)\)-torus knot \(T_{n,n-1}\) for every positive integer \(n\), in terms of the 4-genus of \(T_{n,n-1}\). For integers \(n>d \geq 0\), the authors construct a certain surface \(\Sigma\) in \(\mathbb{CP}^2 \backslash \mathrm{Int} B^4\) bounded by the torus knot \(T_{n,n-1}\) such that \(\Sigma\) is of degree \(d\) and the genus \(g(\Sigma)\) is given by a certain formula of \(n\) and \(d\); as a corollary, the authors consider the minimal genus problem in \(\mathbb{CP}^2\# \mathbb{CP}^2\) and show that for a suitably chosen second homology class \((n,d)\) in \(H_2(\mathbb{CP}^2\# \mathbb{CP}^2; \mathbb{Z}) \cong \mathbb{Z} \oplus \mathbb{Z}\), the value \((G_{\mathbb{CP}^2}(n)+G_{\mathbb{CP}^2}(d))-G_{\mathbb{CP}^2\# \mathbb{CP}^2} (n,d)\) can be arbitrarily large. The first result is shown by constructing knots with arbitrarily large \(\mathbb{CP}^2\)-genus.