Links and gordian numbers associated with generic immersions of intervals
A (relative) divide is a number of properly immersed arcs in the 2-disk with transverse intersections. The link of a divide, first defined by N. A'Campo, associates to a divide a link in \(S^3\) with one component for each arc in the divide. These arise from the study of isolated complex plane curve singularities. In a previous work, the present authors defined the notion of an oriented divide, which consists of several immersed oriented circles in the interior of the 2-disk, and an associated link. In the present paper, they define the notion of a free divide, which is the same as a divide, except that some of the end points are allowed to be in the interior of the disk. There is also an associated link after making some choices at the free end points. The main result of the paper is that the unknotting (Gordian) number of the associated link of a free divide is equal to the number of double points. This was originally proved by A'Campo for one-component relative divides. There is also a formula for the slice genus. The proof uses a doubling construction to go from a free divide to an oriented divide. The paper concludes with an extensive list of \textsl{simple} free divides with one component and small numbers of crossings and their associated knots.
scientific article; zbMATH DE number 1879076