Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?

Question

For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined modulo $1/2$, i.e. as an element of $\mathbb R / \frac{1}{2}\mathbb Z$. (This is mentioned here.)

In the cusped case, is there some extra data that lets us refine $CS(M)$ to an element of $\mathbb R/\mathbb Z$? Maybe something like a choice of class in $H^1(M, \mathbb Z / 2\mathbb Z)$? I'm particularly interested in the case of $M = S^3 \setminus K$ a knot complement.

Answer 1

Remark 7.2 in Christian's paper suggests that for the choice of a lift of the discrete-faithful $PSL_2(\mathbb{C})$ representation to $SL_2(\mathbb{C})$, there ought to be a lift of the Chern-Simons invariant to $\mathbb{R}/\mathbb{Z}$. As he points out, this is equivalent to the choice of a spin structure (which is only a torsor over $H^1(M;\mathbb{Z}/2)$). This should be sandwiched in the diagram on p. 30:

$\require{AMScd}$

$$\begin{CD} H_3(SL_2(\mathbb{C}),P) @>>> H_3(SL_2(\mathbb{C}),\pm P) @>>> H_3(PSL_2(\mathbb{C}),P) \\ @V V V @V VV @V VV \\ \mathbb{C /4\pi^2 Z} @>>> \mathbb{C /2\pi^2 Z} @>>> \mathbb{C /\pi^2 Z}. \end{CD}$$

See also the discussion on p. 2, paragraph 2 to see why I think this normalization corresponds to the normalization given in your question.