When orientable surfaces having at least one edge are immersed into the plane, there often exists regions where two parts of the surface must occupy the same space in the plane. If these regions are considered overlaps rather than intersections, the surface remains embedded in three-space but appears to be flattened. This form of the surface, called a "flattening," can be subjected to certain deformations without leaving its flattened state. Called "flat deformations," these deformations can be used to show that two apparently different flattenings are sometimes just two forms of the same flattening; in that case, the two original forms are called "indistinct." This paper attempts to determine a formula relating the properties of a given surface with the number of distinct flattenings which can possibly be formed from it. Specifically, it focuses on the flattenings of tori with n disks removed. Although an explicit formula is not derived, the paper outlines a method for determining the correct number of flattenings, and charts this number against n for values up to ten.
"The 'Flattened' Projections of Orientable Surfaces,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 2
, Article 2.
Available at: http://scholar.rose-hulman.edu/rhumj/vol2/iss1/2