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For a 3-manifold $M$ with $b_1(M)=1$ fibered over $S^1$ and the fiberwisegradient $\xi$ of a fiberwise Morse function on $M$, we introduce the notion ofamidakuji-like path (AL-path) on $M$. An AL-path is a piecewise smooth path on$M$ consisting of edges each of which is either a part of a critical locus of$\xi$ or a flow line of $-\xi$. Counting closed AL-paths with signs gives theLefschetz zeta function of $M$. The "moduli space" of AL-paths on $M$ givesexplicitly Lescop's equivariant propagator, which can be used to define$\mathbb{Z}$-equivariant version of Chern--Simons perturbation theory for $M$.

Morse theory and Lescop's equivariant propagator for 3-manifolds with $b_1=1$ fibered over $S^1$