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In this paper, we introduce a broad family of group homomorphisms that wename the Gauss-Epple homomorphisms. In the setting of braid groups, theGauss-Epple invariant was originally defined by Epple based on a note of Gaussas an action of the braid group $B_n$ on the set $\{1, \dots,n\}\times\mathbb{Z}$; we prove that it is well-defined. We consider theassociated group homomorphism from $B_n$ to the symmetric group$\text{Sym}(\{1, \dots, n\}\times\mathbb{Z})$. We prove that this homomorphismfactors through $\mathbb{Z}^n\rtimes S_n$ (in fact, its image is an order 2subgroup of the previous group). We also describe the kernel of thehomomorphism and calculate the asymptotic probability that it contains a randombraid of a given length. Furthermore, we discuss the super-Gauss-Epplehomomorphism, a homomorphism which extends the generalization of theGauss-Epple homomorphism and describe a related 1-cocycle of the symmetricgroup $S_n$ on the set of antisymmetric $n\times n$ matrices over the integers.We then generalize the super-Gauss-Epple homomorphism and the associated1-cocycle to Artin groups of finite type. For future work, we suggest studyingpossible generalizations to complex reflection groups and computing the vectorspaces of Gauss-Epple analogues.

On the Gauss-Epple homomorphism of the braid group $B_n$, and generalizations to Artin groups of crystallographic type