An upper bound on the first homology of spline complexes

Let $\Delta$ be a connected, pure $2$-dimensional simplicial complex embedded in $\mathbb{R}^2$ and let $C^{r}(\hat{\Delta})$ be the homogenized spline module of $\Delta$ with smoothness $r$. To study $C^{r}(\hat{\Delta})$, Schenck and Stillman developed the spline complex $S_\bullet/J_\bullet$. Schenck and Stiller conjectured that the regularity of $H_1(S_\bullet/J_\bullet)$ is less than $2r+1$. In this article, we first consider the case when $\Delta$ has only one totally interior edge, because it is the simplest non-trivial case. Then we may apply the formula we find here to get an upper bound on some more general cases.