$\hat Z$- invariant for $SO(3)$ and $OSp(1|2)$ Groups

Three-manifold invariants $\hat{Z}$ (‘‘ $Z$ -hat’’), also known as homological blocks, are $q$ -series with integer coefficients. Explicit $q$ -series form for $\hat{Z}$ is known for $S U (2)$ group, supergroup $S U (2 ∣ 1)$ and ortho-symplectic supergroup $O S p (2 ∣ 2)$ . We focus on $\hat{Z}$ for $S O (3)$ group and orthosymplectic supergroup $O S p (1 ∣ 2)$ in this paper. Particularly, the change of variable relating $S U (2)$ link invariants to the $S O (3)$ & $O S p (1 ∣ 2)$ link invariants plays a crucial role in explicitly writing the $q$ -series.

$\hat{Z}$ - invariant for $S O (3)$ and $O S p (1 ∣ 2)$ Groups

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$\hat{Z}$ - invariant for $S O (3)$ and $O S p (1 ∣ 2)$ Groups

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Z^Z^- invariant for SO(3)SO(3) and OSp(1∣2)OSp(1∣2) Groups

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Z^Z^- invariant for SO(3)SO(3) and OSp(1∣2)OSp(1∣2) Groups

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$\hat{Z}$ - invariant for $S O (3)$ and $O S p (1 ∣ 2)$ Groups

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$\hat{Z}$ - invariant for $S O (3)$ and $O S p (1 ∣ 2)$ Groups

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