finite) then $$\label{Eq:3ManifDet}
    \lmod H_1(\Manif(G),\Integer)\rmod = \lmod \det L(G)\rmod = \lmod\langle
    L(G) \mid \det_{n,n}^{\varnothing}\rangle\rmod.$$

3.  If $L(G)$ is nondegenerate then $$\label{Eq:3ManifTheta}
    \langle W(G) \mid \Theta_n\rangle = 12 \det L(G)
    \bigl(\lambda_{CW}(\Manif(G)) - \frac{1}{4} \name{sign}(L(G))\bigr)$$ where $\lambda_{CW}$ is the Casson–Walker invariant [@Walker] of the raional homology sphere $\Manif(G)$, $\name{sign}$ is the signature of the symmetric matrix $L(G)$, and $\Theta_n$ is an element of $\Graph_{n,n+1}
    \oplus \Graph_{n,n-1}$ defined as $$\Theta_n \stackrel{\mathrm{def}}{=}\det_{n,n+1}^{\varnothing}- \sum_{1 \le i
    \ne j \le n} ([ij])*\det_{n,n-2}^{\{i,j\}} - \sum_{i=1}^n
    \det_{n,n-1}^{\{i\}}.$$

Conjecturally, and begin a series of formulas for invariants of $3$-manifolds. See [@Toronto] for details.

Applying $\Delta$ to the element $\Theta_n$ and using Theorem \[Th:Diag\] and Corollary \[Cr:Diag\] one obtains

$\Delta \Theta_n = -2 U(\AC_{n,n-1})$. Therefore if $G$ is balanced then $\langle L(G) \mid \Theta_n\rangle$ is equal to $-2$ times the codimension $1$ diagonal minor of $L(G)$.

The last assertion is [@Epstein Theorem 84].

Acknowledgements {#acknowledgements .unnumbered}
----------------

The research was inspired by numerous discussions with prof. Michael Polyak (Haifa Technion, Israel) whom the author wishes to express his most sincere gratitude.

The research was funded by the Russian Academic Excellence Project ‘5-100’ and by the grant No. 15-01-0031 “Hurwitz numbers and graph isomorphism” of the Scientific Fund of the Higher 