On parametrizations of angle structures of knot complements

In this paper, for a prime knot [Formula: see text] in [Formula: see text], we consider an ideal triangulation [Formula: see text] of the complement [Formula: see text] of [Formula: see text] associated to a diagram [Formula: see text] of [Formula: see text], and give a parametrization of the generalized angle structures for [Formula: see text] corresponding to crossings, edges and faces of [Formula: see text]. A direct calculation shows that the derivatives of the volume functional with respect to these parameters coincide with the nonlinear parts of the hyperbolicity equations.