Links of oriented divides and fibrations in link exteriors
An oriented divide is defined as the image of a self transverse immersion of a finite union of oriented circles into the unit disk \(D\). Considering the tangent bundle \(T D\) as a subset of \(T{\mathbb R}^2={\mathbb R}^4\) one gets \(TD=D\times{\mathbb R}^2\) and a divide introduces a map from the positive tangent half lines of the oriented circles into \(TD=D\times{\mathbb R}^2\). Intersecting the image of that map with the unit sphere \(S^3\subset{\mathbb R}^4\) one associates a link \(L(C)\subset S^3\) to each oriented divide. The authors prove that if \(D_L\subset D\) is a regular link projection of a link \(L\subset S^3\) then there exists an oriented divide \(C\), obtained from \(D_L\) by attaching small loops to it, such that \(L(C)\) is isotopic to \(L\), and a bound on the number of loops is given. (This theorem was proved in the setting of Legendrian knots by \textit{S. Chmutov, V. Goryunov}, and \textit{H. Murakami} [Math. Ann. 317, 389-413 (2000; Zbl 0957.57005)]). The authors also apply oriented divides to construct fibered knots with certain properties in the complement of a given link in \(S^3\).
scientific article; zbMATH DE number 1836042