Links and gordian numbers associated with generic immersions of intervals

A divide is the image of a generic, relative immersion of intervals in the unit disk. In the present paper we remove the relative condition by introducing a generalized class of divide called free divides. We describe how to define the link of a free divide in a well-defined way and, further, show that its unknotting (gordian) number is still equal to the number of double points of the immersed intervals. This extends the result of AʹCampo concerning unknotting numbers of just relative divides. We conclude the paper with a table of free divides and their links which, by virtue of the main result, are tabulated according to their unknotting numbers.