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Here's a partial answer that works when $p=2$. If $\Sigma (a_0,a_1, \dots ,a_n)$ is defined as the link of the the singularity $\sum z_i ^{a_i}$, the map $\pi_0:\Bbb C^{n+1}\to \Bbb C^n$ with $\pi_0(z_0,\dots,z_n)=(z_1,\dots, z_n) $ restricts to an $a_0$-fold branched contact covering $\Sigma(a_0,a_1,\dots,a_n)\to S^{2n-1}$ with branch set $\Sigma(a_1,\dots, ...


The Witten-Reshetikhin-Turaev approach to constructing quantum topological invariants of $3$-manifolds is to define them on framed links and to prove invariance under Kirby moves. There is a paper of Broda which presents a $4$--manifold version of this strategy to construct "Witten-Reshetikhin-Turaev invariants" for $4$-manifolds.

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