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

The refinement resolves sign and normalization choices present in Reidemeister torsion.

A homology orientation chooses an orientation of the real vector space ${⨁}_i{H}_i(M;\mathbb{R})$. It fixes the sign in refined Turaev torsion.

An Euler structure fixes the lift ambiguity in refined Turaev torsion. Changing it can multiply the torsion by an element of the first homology group.

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, $i_{\ast }\tau (X)\doteq (1-h)\tau (Y)$.