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I am working with a simplicial complex $K$ and wish to determine whether a given $k$-cycle is non-trivial in homology—that is, whether it represents a homology class that is not a boundary.

I am already familiar with how to compute homology generators using SageMath (as I had previously inquired on this platform). For instance, given the simplicial complex

K = SimplicialComplex([[0, 1], [1, 2], [2, 0]])
K.set_immutable()
K.homology(generators=True)

the output is:

{0: [], 1: [(Z, (0, 1) - (0, 2) + (1, 2))]}

My goal is to input a specific cycle and check whether it is non-trivial in homology. The reason for this approach is that my actual simplicial complex is quite large, and Sage struggles to compute the full set of generators. However, I do have a candidate cycle that I believe may be a generator.

I am working with a simplicial complex $K$ and wish to determine whether a given $k$-cycle is non-trivial in homology—that is, whether it represents a homology class that is not a boundary.

I am already familiar with how to compute homology generators using SageMath (as I had previously inquired on this platform). For instance, given the simplicial complex

K = SimplicialComplex([[0, 1], [1, 2], [2, 0]])
K.set_immutable()
K.homology(generators=True)

the output is:

{0: [], 1: [(Z, (0, 1) - (0, 2) + (1, 2))]}

My goal is to input a specific cycle and check whether it is non-trivial in homology. The reason for this approach is that my actual simplicial complex is quite large, and Sage struggles to compute the full set of generators. However, I do have a candidate cycle that I believe may be a generator.