Commensurability of knots and \(L^2\)-invariants (Q2867795)

Two knots in \(S^3\) are called commensurable if there exists a diffeomorphism between some finite covers of the knot complements. By a theorem of \textit{M. Boileau, S. Boyer, R. Cebanu} and \textit{G. S. Walsh} [Geom. Topol. 16, No. 2, 625--664 (2012; Zbl 1258.57001)], two hyperbolic knots (without hidden symmetries) are commensurable if and only if there are finite cyclic covers of their complements which are diffeomorphic. If \(K\) is a knot and \(\Delta_K\) is its Alexander polynomial, then \(\tau(K):=\int_{S^1} \log|\Delta_K(z)|\) denotes the Mahler measure of the Alexander polynomial.NEWLINENEWLINEDefine moreover \(\rho(K):=\int_{S^1} \text{sign}(A(1-z)+A^t(1-\overline{z}))\).NEWLINENEWLINEThe main result of the paper is:NEWLINENEWLINETheorem. If \(K_1\) and \(K_2\) have diffeomorphic finite cyclic covers of their complements (with degrees \(n_1\) and \(n_2\), respectively) and if no root of unity is a zero of their Alexander polynomials, then \(n_1\tau(K_1) = n_2\tau(K_2)\) and \(n_1\rho(K_1) +m_1 = \pm (n_2 \rho(K_2) +m_2)\) for suitable integers \(m_1,m_2\). The sign is \(+1\) if the diffeomorphism in question is orientation preserving, and \(-1\) if it is orientation reversing.NEWLINENEWLINEThis is proved by interpreting \(\tau(K)\) as the \(L^2\)-torsion of the infinite cyclic covering of the knot complement and \(\rho(K)\) as the \(L^2\)-rho invariant of the infinite cyclic covering of the knot complement, then using rather standard properties of \(L^2\)-invariants. The paper also indicates an alternative route to prove the theorem which avoids the use of \(L^2\)-invariants.NEWLINENEWLINEFriedl also shows how to compute \(\rho(K)\) and \(\tau(K)\) explicitly in terms of the zeros of the Alexander polynomial and finitely many integer calculations involving a Seifert matrix.NEWLINENEWLINEFinally, using these computation techniques, he studies a couple of examples for commensurability and non-commensurability, in the former case also checking which coverings are diffeomorphic (distinguishing also cases with orientation preserving and orientation reversing diffeomorphisms).NEWLINENEWLINEFor the entire collection see [Zbl 1272.57002].