Tangent circle bundles admit positive open book decompositions along arbitrary links. (Q1420506)

In [Osaka J. Math. 39, No.3, 681--703 (2002; Zbl 1030.57007)], \textit{W. Gibson} and \textit{M. Ishikawa} proved that, for any link \(L\) in \(\mathbb S^3\), there exists an additional knot \(K\) such that \(L \cup K\) is a fibered link in \(\mathbb S^3\) with positive monodromy. The aim of the present paper is to show the same assertion for tangent circle bundles to a compact, oriented, smooth surfaces, both in the closed and bounded case: Let \(ST(\Sigma_g)\) be the tangent circle bundle to compact, oriented, smooth surface \(\Sigma_g\) of genus \(g\geq 0.\) Then, for any link \(L\) in \(ST(\Sigma_g)\) there exists an additional knot \(K\) such that \(L\cup K\) is the binding of a positive open book decomposition of \(ST(\Sigma_g).\) For any link \(L\) in \(\#_m (\mathbb S^1 \times \mathbb S^2)\), where \(m\in \mathbb N,\) there exists an additional knot \(K\) such that \(L\cup K\) is the binding of a positive open book decomposition of \(\#_m (\mathbb S^1 \times \mathbb S^2)\). In order to obtain these results, the author studies the image of a generic, relative immersion of a finite number of copies of the unit interval or the unit circle into the genus \(g\) surface \(\Sigma_{g,n}\) with \(n\) boundary components (so generalizing the notion of \textit{divide}, originally introduced by \textit{N. A'Campo} in [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 8, No. 1, 5--23; erratum ibid. 8, No. 2, 343 (1999; Zbl 0962.32025)]) and proves a suitable fibration theorem for generalized divides.