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?


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