On unknotting numbers and four-dimensional clasp numbers of links (Q2750911)

The author introduces the four-dimensional clasp number of a link \(L\), denoted by \(c_s(L)\), that is the minimum number of double points for transversely immersed disks in \(D^4\) with boundary \(L\) and with only finite double points as singularities. This link invariant, together with the greatest Euler characteristic for an oriented 2-manifold in \(D^4\) bounded by the link \(L\), and denoted by \(\chi_s (L)\), is introduced to give lower bounds for the unknotting number \(u(L)\) of the link \(L\). In fact, he proves that, if \(L\) is an oriented link with \(r\) components, then the following inequalities hold \(u(L)\geq c_s(L)\geq 1/2(r- \chi_s(L))\). Combining this result with a result announced by \textit{L. Rudolph} [Geom. Topol. Monogr. 2, 555-562 (1999; Zbl 0962.57004)] the author proves that an inequality stronger than the Bennequin unknotting inequality [\textit{D. Bennequin}, Astérisque 107/108, 87-161 (1983; Zbl 0573.58022)] holds for any link diagram and he also shows the equality conjectured by \textit{M. Boileau} and \textit{C. Weber} [Enseign. Math., II. Sér. 30, 173-222 (1984; Zbl 0556.57002)] for a closed positive braid diagram.