On the corner contributions to the heat coefficients of geodesic polygons

Let $\mathcal{O}$ be a compact Riemannian orbisurface. We compute formulas for the contribution of cone points of~$\mathcal{O}$ to the coefficient at $t^2$ of the asymptotic expansion of the heat trace of $\mathcal{O}$, the contributions at $t^0$ and $t^1$ being known from the literature. As an application, we compute the coefficient at $t^2$ of the contribution of interior angles of the form $\gamma =\pi \mathrm{/}k$ in geodesic polygons in surfaces to the asymptotic expansion of the Dirichlet heat kernel of the polygon, under a certain symmetry assumption locally near the corresponding corner. The main novelty here is the determination of the way in which the Laplacian of the Gauss curvature at the corner point enters into the coefficient at $t^2$. We finish with a conjecture concerning the analogous contribution of an arbitrary angle $\gamma$ in a geodesic polygon.