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In a previous article, a construction of the smooth Deligne-Beilinsoncohomology groups $H^p_D(M)$ on a closed $3$-manifold $M$ represented by aHeegaard splitting $X_L \cup_f X_R$ was presented. Then, a determination of thepartition functions of the $U(1)$ Chern-Simons and BF Quantum Field theorieswas deduced from this construction. In this second and concluding article westay in the context of a Heegaard spitting of $M$ to define Deligne-Beilinson$1$-currents whose equivalent classes form the elements of $H^1_D(M)^\star$,the Pontryagin dual of $H^1_D(M)$. Finally, we use singular fields to firstrecover the partition functions of the $U(1)$ Chern-Simons and BF quantum fieldtheories, and next to determine the link invariants defined by these theories.The difference between the use of smooth and singular fields is also discussed.

3D Topological Models and Heegaard Splitting II: Pontryagin duality and Observables