In Table 1 we list the invariants for each class. Hence our invariants are not complete. Of course, one can simply take orientability as an additional invariant. The frustrating point is that the motivation behind all the new invariants was precisely to capture orientation data. However, we have not been able to tell if they form a single twist class or not.
Several features stand out.
However, certain groups do not seem to show up as Bowen-Franks groups of double cover flows. Finally, we present two twist matrices where the Bowen-Franks groups of the double covers is the only distinguishing invariant. The author would like to thank John Franks for several helpful conversations. Ann. of Math. There is just this one file.