For simplicial complexes of dimension 2 or lower, the boundary matrix is totally uni- modular when the complex does not have a M¨ obius subcomplex. Theorem 2.4.4. [14, Theorem 5.13] For d ≤ 1 , the boundary matrix [ ∂$_{d}$$_{+1}$ ] is totally unimodular if and only if the finite simplicial complex has no ( d +1) -dimensional M¨ obius subcomplex. It is appropriate to mention here that the connection between total unimodularity of boundary matrices and torsion in the complex has been observed as early as in 1895 by Poincar´ e [36]. However, the result in [14] connecting the total unimodularity with relative torsion is different and has led to a polynomial time algorithm for the OHCP problem. Notice that a complex can be torsion-free, but have non-trivial relative torsion. The M¨ obius strip is such an example. We illustrate the implications of the results above for the efficient computation of the multiscale simplicial flat norm by considering certain sets. When the input set T is of dimension 1, and is described on an orientable 2-manifold to which the choices of 2-dimensional set S are also restricted, we can always compute its multiscale simplicial flat norm by solving the multiscale simplicial flat norm LP (Equation 2.9) in polynomial time. A similar result holds when T is a set of dimension 2 described as a subcomplex of a 3-complex sitting in R $^{3}$. For a 1-dimensional set T with choices of S restricted to a 2-complex K , we can always compute the multiscale simplicial flat norm of T efficiently as long as K does not have a 2-dimensional M¨ obius subcomplex. Notice that K itself need not be embedded in R 3 for this result to work – it could be sitting in some higher dimensional space. 28