function*]{} $${\mathcal Z}_{n,k}(q,v) {\stackrel{\mbox{\scriptsize def}}{=}}\sum_{G \in \Upsilon_{n,k}} Z_G(q,v)G \in
\Undir_{n,k}$$ and its “shaved” version $$\widehat{{\mathcal Z}}_{n,k}(q,v) {\stackrel{\mbox{\scriptsize def}}{=}}\sum_{G \in \Upsilon_{n,k}}
Z_{\widehat G}(q,v)G \in \Undir_{n,k}.$$

\[Pp:LaplTutte\] $$\Delta\widehat{{\mathcal Z}}_{n,k}(-1,1) = (-1)^k {\mathcal
Z}_{n,k}(-1,-1).$$

Note that by Proposition \[Pp:SpecVal\] the right-hand side of the equality contains only graphs without loops, as does the left-hand side.

Corollary \[Cr:NumSSC\] implies that $$\widehat{{\mathcal Z}}_{n,k}(-1,1) = (-1)^k k!\sum_{I \subset \{1 {, \dots ,} n\}}
\lmod \det_{n,k}^I\rmod.$$ Apply now the Laplace operator $\Delta$ to both sides of the equality. Apparently, $\Delta$ commutes with the forgetful map: $\lmod \Delta(x)\rmod
= \Delta(\lmod x\rmod)$ for any $x \in \Graph_{n,k}$. Therefore by Theorem \[Th:Diag\] and Proposition \[Pp:SpecVal\] $$\begin{aligned}
\Delta\widehat{{\mathcal Z}}_{n,k}(-1,1) &= (-1)^k k!\sum_{I \subset \{1
{, \dots ,} n\}} \lmod \Delta \det_{n,k}^I\rmod = (-1)^{n+k} \sum_{I \subset \{1
{, \dots ,} n\}} \lmod U(\AC_{n,k}^I)\rmod \\
&= (-1)^k {\mathcal Z}_{n,k}(-1,-1).
 \end{aligned}$$

Proposition \[Pp:LaplTutte\] admits several generalizations. The author is planning to write a separate paper considering action of the Laplace operator on the universal Potts functions and their oriented-graph versions.

An application: invariants of $3$-manifolds
-------------------------------------------

Universal determinants have an application in $3$-dimensional topology, due to M.Polyak. We describe it briefly here; see [@Toronto] and the MSc. thesis [@Epstein] for detailed definitions, formulations and 