Abstract: Metrics in computational topology are often either (i) themselves in the form of the interleaving distance d(F,G) between certain order-preserving maps F and G between posets or (ii) admit d(F,G) as a tractable lower bound. In this talk, assuming that the target poset of F and G admits a join-dense subset, we propose certain join representations of F and G which facilitate the computation of d(F,G). We leverage this result in order to (i) elucidate the structure and computational complexity of the interleaving distance for poset-indexed clusterings (i.e. poset-indexed subpartition-valued functors), (ii) to clarify the relationship between the erosion distance by Patel and the graded rank function by Betthauser, Bubenik, and Edwards, and (iii) to reformulate and generalize the tripod distance by the second author. This is joint work with Facundo Memoli and Anastasios Stefanou. https://arxiv.org/abs/1912.04366