A tight polyhedral immersion of the twisted surface of Euler characteristic −3

Early studies of tight surfaces showed that almost every surface can be immersed tightly in three-space. For these surfaces we can ask: How many significantly different tight immersions are there? If we consider two such immersions to be the same when they are image homotopic, then we can answer the question by determining the number of homotopy classes where tight immersions are possible. A complete description of all classes of immersions under image homotopy already exists, and tight examples are known in all but three of the classes where tight immersions are possible. In this paper we produce examples in two of those three missing classes, and conjecture that no tight immersion exists in the third.