Hypothesis 3 The path from a visited state to its parent (that is, the first state on the traceback at either the start or end of a substitution edge) does not contains insertion edges or does not contain deletion edges. Hypothesis 4 Every time a traceback has an insertion, then matches, and then a deletion, the start of the deletion is also the start of a critical substitution edge (i.e. coloured red in our figures). Algorithm 3 (affine traceback) Given a state \(f_{s,k}\) and the last gap-open state on its path \(f_{s’, k’}\), the path in between can be found as follows: Walk diagonally with matches and mismatch until reaching the row or column of the parent.