Colored Alexander invariants and cone-manifolds

Abstract

In this paper, we reconstruct the link invariant of framed links introduced in [1] by the universal $R$-matrix of $\mathcal{U}_{q}(\sl_{2})$ and name it the colored Alexander invariant. We check that the optimistic limit $\mathop{\mathrm{o-lim}}$ of this invariant is determined by the volume of the knot and link cone-manifold for figure eight knot, Whitehead link and Borromean rings. We also propose the $A$-polynomials of these examples obtained from the colored Alexander invariant.