An invariant of fiberwise Morse functions on surface bundle over $S^1$ by counting graphs

We apply Lescop's construction of $\mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of "AL-paths" given in arXiv:1403.8030. We obtain an invariant $\hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, which can be considered as a higher loop analogue of the Lefschetz zeta function and whose construction will be applied to that of finite type invariants of knots in such a 3-manifold. We also give a combinatorial formula for Lescop's equivariant invariant $\mathscr{Q}$ for 3-manifolds with $H_1=\mathbb{Z}$ fibered over $S^1$. Moreover, surgery formulas of $\hat{Z}_n$ and $\mathscr{Q}$ for alternating sums of surgeries are given. This gives another proof of Lescop's surgery formula of $\mathscr{Q}$ for special kind of 3-manifolds and surgeries, which is simple in the sense that the formula is obtained easily by counting certain graphs in a 3-manifold.

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