Stabilization distance between surfaces (Q2024039)

In this paper, the authors consider surfaces smoothly and properly embedded in a compact, oriented, smooth 4-manifold. The 1-handle stabilization distance is the minimal number of 1-handle additions necessary to make two surfaces with the same genera ambiently isotopic relative to the boundary. The generalized stabilization distance is the minimal number \(k\) of 1-handle additions such that two surfaces with the same genera become ambiently isotopic relative to the boundary, after each surface had additions of \(k\) 1-handles and additions of arbitrarily many knotted 2-spheres; where an addition of a knotted 2-sphere \(K\) is to take the connected sum with \(K\). A slice disc for a 1-knot \(J\) in \(S^3\) is a smoothly embedded disc in \(D^4\) that bounds \(J\). \newline The main results are as follows. For every nonnegative integer \(m\), there exists a knotted 2-sphere \(K\) in \(S^4\) such that the pair of \(K\) and an unknotted 2-sphere has the 1-handle stabilization distance \(m\). As the authors remark in the paper, the original credit for this result belongs to \textit{K. Miyazaki} [Kobe J. Math. 3, 77--85 (1986; Zbl 0622.57016)]. For every nonnegative integer \(m\), there exist a knot \(J\) in \(S^3\) and a pair of slice discs for \(J\) with generalized stabilization distance \(m\). For every nonnegative integer \(m\), there exist a knot \(J\) in \(S^3\) and a pair of slice disks \(D_1\) and \(D_2\) for \(J\) with generalized stabilization distance at least \(m\) such that for the inclusion-induced maps carrying the Alexander module with rational coefficients of \(J\) to that of \(D_i\) \((i=1,2)\), their kernels coincide. The authors' examples of the second result demonstrate that for every integer \(m\), there is a knot \(J\) with \(n_s(J) \geq m\), where \(n_s(J)\) for a knot \(J\) denotes the number of equivalence classes of slice discs for \(J\), up to additions of knotted 2-spheres and ambient isotopy relative to the boundary; and the authors propose a problem to determine the value of \(n_s(J)\) for some nontrivial knot \(J\). The authors show the results using Alexander modules and twisted homology coming from metabelian representations that factor through the dihedral group \(D_{2n} \cong \mathbb{Z}/2\mathbb{Z} \ltimes \mathbb{Z}/n\mathbb{Z}\).