Von Neumann rho invariants and torsion in the topological knot concordance group (Q422120)

Let \(K\) be a knot with exterior \(X\) and group \(\pi\). If \(p\in\mathbb{Q}\Lambda=\mathbb{Q}[t,t^{-1}]\) is an irreducible Laurent polynomial such that \(p(t^{-1})=t^kp(t)\) for some \(k\in\mathbb{Z}\) let \(A_o^p(K)\) be the localization of the knot module \(H_1(X;\mathbb{Q}\Lambda)\) at the prime \((p)\). The knot \(K\) is \(p\)-\textit{anisotropic} if \(A_o^p(K)\) has no nontrivial submodule which is self-annihilating with respect to the localization of the Blanchfield pairing. Let \(\rho_p^1(K)\) be the von Neumann \(\rho\)-invariant associated to the epimorphism \(\pi\to\pi/G\), where \(G\) is the preimage in \(\pi\) of the kernel of localization \(\pi'/\pi''\to{A_o^p(K)}\). It is shown first that these invariants are additive, and that there are slice knots \(K\) and polynomials \(p\) with \(\rho_p^1(K)\not=0\). Nevertheless, these invariants define homomorphisms on the subgroup of the TOP knot concordance group \(\mathcal{C}=\mathcal{C}_1^{TOP}\) generated by \(p\)-anisotropic knots. This is used to show that if \(K_i\) are \(p\)-anisotropic knots and \(\Sigma{m_i}[K_i]=0\) in \(\mathcal{C}\) then \(\Sigma{m_i}\rho_p^1(K_i)=0\) in \(\mathbb{R}\). In particular, if \(K\) has finite order in \(\mathcal{C}\) then \(\rho_p^1(K)=0\). Although these invariants are difficult to compute in general, if the image of \(K\) in the algebraic concordance group (the Witt group of Blanchfield pairings) has finite order there is a useful estimate. This estimate is used to show that a certain infinite family of twist knots (all of order 2 in the algebraic concordance group) is linearly independent in \(\mathcal{C}\).