The lower bound of the \(w\)-indices of non-ribbon surface-links (Q1769148)

A (simple) surface braid of degree \(m\) is a locally flat properly embedded surface \(S \subset D^2_1 \times D^2_2\), which projects onto \(D^2_2\) by a (simple) branched covering and whose boundary \(\partial S\) coincides with \(Q_m \times \partial D^2_2\), for a fixed \(m\)-points set \(Q_m \subset \mathop{\text{Int}} D^2_1\). If we think of \(D^2_1\) as the product of two intervals \(I_1 \times I_2\) and assume \(S\) to be generic, then the projection \(S'\) of \(S\) into \(I_1 \times D^2_2\) is an immersed surface with double curves and a finite number of branch and triple points. Then, the multiple set of \(S'\) is a graph \(\Gamma\) with vertices of valence 1 (branch points) and 6 (triple points). The planar diagram of \(\Gamma\) in \(D^2_2\), together with a certain \(\{1, \dots, m- 1\}\)-labelling of its edges, completely determines \(S\) up to isotopy and is called a chart description of \(S\). The \(w\)-index \(w(S)\) of \(S\) is the minimum number of triple points needed for describing a surface braid equivalent to \(S\) (\(w\) stands for white, being customary to call white vertices of a chart those corresponding to triple points). A theorem by \textit{O. Viro} and \textit{S. Kamada} says that any orientable surface-link \(L \subset \mathbb R^4\) is isotopic to the closure \(\widehat S\) of some surface braid \(S\), obtained by trivially capping off \(\partial S\) outside \(D^2_1 \times D^2_2\) [see: \textit{S. Kamada}, Braid and knot theory in dimension four, Mathematical Surveys and Monographs 95, Providence, AMS (2002; Zbl 0993.57012)]. Hence, it makes sense to define the \(w\)-index \(w(L)\) of any orientable surface-link \(L \subset \mathbb R^4\) as the minimum \(w\)-index of a surface braid \(S\) such that \(\widehat S\) is isotopic to \(L\). An orientable surface-link is called ribbon if it is equivalent to a trivial \(S^ 2\)-link up to surgery along 1-handles. It is known that an orientable surface-link \(L\) is ribbon if and only if \(w(L) = 0\) [\textit{S. Kamada}, J. Knot Theory Ramifications 1, No. 2, 137--160 (1992; Zbl 0763.57013)]. Here the author derives a lower bound for the \(w\)-index in the non-ribbon case, from a combinatorial computation on charts and braid monodromies. Namely, he proves that \(w(L) \geq 4\) if \(L\) is a non-ribbon orientable surface-link and that \(w(L) \geq 6\) if \(L\) is a non-ribbon \(S^2\)-link. Both these inequalities are sharp. In particular, the 2-twist spun trefoil has \(w\)-index 6, although it admits a 3-dimensional diagram with only 4 triple points [\textit{S. Satoh} and \textit{A. Shima}, Trans. Am. Math. Soc. 356, No. 3, 1007-1024 (2004; Zbl 1037.57018)].