Label the vertices of the 3-simplex pyramid (1,2,3,4) by $1,2,3,4$. If one follows the brute force formula, we know the boundary equals $234-134+124-123$.
Traversing by the indices in the order $1,2,3,4$ (the default orientation), I am getting $123+234+134+124$. Rearranging gives me $234 + 134 + 124 + 123$.
I thought by default the simplex $1 \to 2 \to 3 \to 4$ goes in that order so why are $123$ and $134$ negative? Note that working backwards will get me the negative on $123$ because of how $341$ is defined.
