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Adinkras are graphs that encode the supersymmetric pairings between particles in physics. However they are also cubical complexes, which are structures built not only from points and lines, but also squares, cubes, and hypercubes, and therefore have higher dimensional topological structure. Additionally, Adinkras are equivalent to N-cubes quotiented by doubly-even codes of length N, which are a specific type of subgroup of Z2^N. Within this paper, we consider generalized Adinkras, which are N-cubes quotiented by any codes of length N.
Homology is an algebraic invariant of topological spaces. In order to learn more about their topology, we compute the homology of a variety of generalized Adinkras. Within this paper we examine the relationship between the codes by which we are quotienting and the homology of the cubes quotiented by these codes. Our main result is that the first homology group of any N-cube quotiented by a code of length N is isomorphic to the code itself.
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Stone, Jacqueline A., "Generalized Adinkra Homology" (2011). Senior Projects Spring 2011. 3.
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