Properly immersed curves in arbitrary surfaces via apparent contours on spines of traversing flows

Let Σ be a compact surface with boundary and F be the set of the orbits of a traversing flow on Σ. If the flow is generic, its orbit space is a spine G of Σ, namely G is a graph embedded in Σ and Σ is a regular neighbourhood of G. Moreover an extra structure on G turns it into a flow-spine, from which one can reconstruct Σ and F. In this paper we study properly immersed curves C in Σ. We do this by considering generic C’s and their apparent contour relative to F, namely the set of points of G corresponding to orbits that either are tangent to C, or go through a self-intersection of C, or meet the boundary of C. We translate this apparent contour into a decoration of G that allows one to reconstruct C, and then we allow C to vary up to homotopy within a fixed generic F, and next also F to vary up to homotopy, and we identify a finite set of local moves on decorated graphs that translate these homotopies.