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