Reidemeister torsion \(\tau_\rho(X)\) is an invariant associated to a finite CW-complex \(X\) and a representation \(\rho: \pi_1(X) \to \mathrm{SL}(V)\) on a finite-dimensional vector space \(V\) which is acyclic (that is \(H^*(X; \rho) = 0\)). By convention it is 0 if \(\rho\) is not acyclic.
Twist knots are an infinite family of knots given by very simple diagrams. For such a knot \(K\) with complement \(E_K = \mathbb S^3 \setminus K\) the fundamental group \(\pi_1(E_K)\) has many representations to \(\mathrm{SL}_2(\mathbb C)\), parametrised by the character variety. In fact in this case the representations can be explicitely parametrised by the affine curve in \(\mathbb C^2\) given by the “Riley polynomial” associated to the diagram.
In this paper the author computes the Reidemeister torsion \(\tau_\rho(E_K)\) for these representations, as an explicit function in the trace functions associated to explicit words in the generators. That is, he gives elements \(g, h \in \pi_1(E_K)\) and a rational function \(F\) such that \(\tau_\rho(E_K) = F\left(\mathrm{tr}(\rho(g)), \mathrm{tr}(\rho(h))\right)\).
For \(q \geq 1\) performing \(1/q\)-Dehn filling on \(E_K\) produces a closed 3-manifold \(M\). The \(\mathrm{SL}_2(\mathbb C)\)-character variety for \(M\) is a subvariety of that of \(E_K\), and the paper also computes the Reidemeister torsion for \(M\) in terms of the same traces and an additional parameter \(u\).