Quasipositivity of links of divides and free divides
A divide \(P\) is a generic relative immersion in the unit disk \((D, \partial D)\) of a finite number of \(1\)-manifolds, i.e. copies of the unit interval \((I, \partial I)\) and the unit circle. In [Ann. Fac. Sci. Toulouse 8, 5-23 (1999; Zbl 0962.32025)], \textit{N. A'Campo} constructed links of divides as an extension of the class of algebraic links, that is, links of singularities of algebraic curves. Identifying the 3-sphere \(\mathbb S^3\) with the set \(ST\mathbb R^2=\{(x,u)\in T\mathbb R^2\mid x\in \mathbb R^2\), \(u\in T_x\mathbb R^2\), \(|x|^2+|u|^2=1\}\), where \(T_x\mathbb R\) and \(T\mathbb R\) are the tangent space at a point \(x\) of \(\mathbb R\) and the tangent bundle over \(\mathbb R\), respectively, \textsl{the link of a divide P} is the set given by \(L(P)=\{(x,u)\in ST\mathbb R^2 \mid x\in P\), \(u\in T_x P\}\). In this paper the author considers an algorithm given by \textit{M. Hirasawa} [Topology Appl. 121, 287-304 (2002; Zbl 1016.57006)] to draw a diagram of the link of a given divide and makes use of this algorithm to show that the link of any divide is quasipositive, that is, it admits a representation as the closure of the product of conjugates of positive braids. Then the same result is extended to links of free divides, i.e., links associated with non-relatively immersed arcs and loops, as defined in \textit{W. Gibson} and \textit{M. Ishikawa} [Topology Appl. 123, No. 3, 609-636 (2002; Zbl 1028.57006)].
scientific article; zbMATH DE number 1879114
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