Turaev-torsion Dehn-filling formula

Let $Y$ be obtained by Dehn filling a torus boundary of $X$, with core class $h\in H_1(Y)$. Choose a one-dimensional coefficient representation $\rho :H_1(Y)\to F^{\times }$ over a field, with $\rho (h)\ne 1$, and assume the associated twisted chain complexes are acyclic. Then, up to the usual torsion unit,With compatible Euler structures and homology orientations, the refined formula fixes that ambiguity. The denominator comes from the attached solid torus's circle complex. When $\rho (h)=1$, this division is unavailable; additional homology data are needed.^[ref]