# Equivalence of conditions for torsions of links to be defined

The torsion of a link complement $$S^3 \setminus L$$ is defined in terms of the twisted chain complex $$C_*(S^3 \setminus L; \rho)$$, where $$\rho : S^3 \setminus L \to \operatorname{GL}_k(\mathbb{k})$$ is a representation of the link complement into the group of matrices over a field $$\mathbb{k}$$. (For the usual Reidemeister torsion/Alexander polynomial, $$\rho$$ is the representation sending each meridian to a variable $$t \in \operatorname{GL}_1(\mathbb{Q}(t))$$, but more generally we can consider nonabelian representations.)

For the torsion to be defined, the complex $$C_*(S^3 \setminus L; \rho)$$ needs to be acyclic. There are two conditions usually given in the literature. One is that there is some meridian $$x$$ with $$\det(\rho(x) - I_k)$$ nonzero. Another is related to the reduced Burau representation. If $$L$$ is represented as the closure of a braid $$\beta$$ on $$n$$ strands, then we have $$n$$ meridians $$x_1, \dots, x_n$$ corresponding to the strands of $$\beta$$. The second condition says that $$C_*(S^3 \setminus L; \rho)$$ is acyclic if $$\det(\rho(x_1 \cdots x_n) - I_k)$$ is nonzero.

Either of these conditions is sufficient, but is either necessary? Are they equivalent?