For a knot $K$, let $J\_N(K)$ denote the $N$th Kashaev invariant of $K$. This [is the same as](https://arxiv.org/abs/math/9905075) the $N$th colored Jones polynomial evaluated at an $N$th root of unity (or $2N$th depending on your conventions for what the variable is).


In the literature on the volume conjecture it is common to use closed-form expressions for $J\_N(K)$. For example, [Kashev gives](https://arxiv.org/abs/q-alg/9601025) the formulas
$$
J\_N(4\_1) = \sum\_{k = 0}^{N-1} |(\omega)\_k|^2
$$
and
$$
J\_N(5\_2) = \sum\_{0 \le k \le l \le N-1} \frac{(\omega)\_k^2}{(\omega)\_l^\*} \omega^{-k(l+1)}
$$
where $\omega = e^{2\pi i/N}$ and $(q)\_k$ is the Pochammer symbol.


How can you compute these formulas? A number of authors mention that they follow from presenting the $4\_1$ and $5\_2$ knots in terms of surgery on the Borromean rings, but I can't find a precise derivation. More accurately, I once found a paper on the arXiv by Habiro and Le (?) that I think gives the computation, but I can't find it again. (I just checked again and their only joint paper is not about this, so I must be mistaken.)


Another possibility is to use the definition of the Kashaev invariant in terms of triangulations, not braid group representations. There are well-known ideal triangulations of the $4\_1$ complement with 2 tetrahedra and of the $5\_2$ complement with 3, so this seems like it might give the above expressions. However, I've never seen a detailed derivation of this approach either.



Closed-form expressions for the Kashaev invariant via surgery
I think the reference I was looking for was [H], which indeed uses presentations of various knots as surgery on the Borromean rings to derive closed forms for their Kashaev invariants.
Skein modules also show up, which I think is where the surgery formula comes from.


In [H] the author cites another paper "to appear" expanding on it, so now I'm curious if that paper was ever published.


[H] *Habiro, Kazuo*, On the colored Jones polynomials of some simple links, RIMS Kokyuroku 1172, 34-43 (2000). [ZBL0969.57503](https://zbmath.org/?q=an:0969.57503) [direct link](https://www.kurims.kyoto-u.ac.jp/%7Ekyodo/kokyuroku/contents/pdf/1172-3.pdf).


