Turaev torsion

Turaev torsion is a sign-refined form of Reidemeister torsion for manifolds and chain complexes. For a knot exterior with meridian $\mu$ and first Betti number one, its group-ring normalization satisfies

If filling a boundary torus along a slope produces $Y$ and the filling core represents $h\in H_1(Y)$, then, up to the standard torsion unit,Equivalently, $i_{\ast }\tau (X)\doteq (1-h)\tau (Y)$.