Reidemeister torsion and Dehn surgery on twist knots. (English) Zbl 1367.57013
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 \(u\).
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 \(u\).
Reviewer: Jean Raimbault (Toulouse)
MSC:
57N10 | Topology of general \(3\)-manifolds (MSC2010) |
57M25 | Knots and links in the \(3\)-sphere (MSC2010) |