Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the usual Jones Polynomial which when evaluated at $i$, the 4th root of unity, is equal to the Arf invariant as per the bottom of this page. So the Jones polynomial will have value $0$ or $1$. This agrees with the Theorem 19 of this paper.
On the other hand if we look at the answer to this question and it's comment, where we take number of components to be $1$ for the link to be a knot, then the Jones polynomial takes the values $1$ or $-1$, depending on on the Arf invariant being $0$ or $1$, when evaluated at $i$. This is actually in agreement with the fact that if I consider the trefoil knot with Jones polynomial $t+t^3-t^4$ then at $t = i$ we get $-1$.So where is the discrepancy coming from?
More generally, is there any reference or understanding of the coloured Jones polynomial when evaluated at $i$ for other representations? For $N=1$ its just the trivial representation so $J_1^K(t)=1$ for any $t$. For $N=2$ at $i$, its the standard Jones polynomial so is $J_2^K(t)|_{t=i} =$$0$ or $1$, or, $= 1$ or $-1$ (up to discrepancy). Very specifically, what about for $N=3$ at $i$, i.e., $J_3^K(t)|_{t=i}$?