A-polynomial and Bloch Invariants of Hyperbolic 3-manifolds

Let N be a complete, orientable, finite-volume, one-cusped hyperbolic 3-manifold with an ideal triangulation. Using combinatorics of the ideal triangulation of N we construct a plane curve in C×C which contains the squares of eigenvalues of PSL(2,C) representations of the meridian and longitude. We show that the defining polynomial of this curve is related to the PSL(2,C) A-polynomial and has properties similar to the classical A-polynomial. We further show that a factor of this polynomial, A0(l,m), associated to the discrete, faithful representation of π1(N) in PSL(2,C) is independent of the ideal triangulation. The Bloch invariant β(N) of N is related to the volume and Chern-Simons invariant of N . The variation of Bloch invariant is defined to be the change of β(N) under Dehn surgery on N . We relate A0(l,m) to the variation of the Bloch invariant of N . We show that A0(l,m) determines the variation of Bloch invariant in the case when A0(l, m) is a defining equation of a rational curve. We also show that in this case the Bloch invariant reads the symmetry of A0(l,m).