Asymmetric TP and TN completion problems

We first give a complete list of polynomial conditions on the data for TP (TN) completability of partial TP (TN) matrices with just one or two unspecified entries in either or both of the upper right or lower left entries. These results are used to identify which such patterns, and related patterns, are TP (TN) completable. Then, the TN completable echelon and TP completable jagged patterns are characterized. This generalizes earlier work on combinatorially symmetric TN completable patterns.