For a given boundary-parabolic representation of a link group to
PSL(2,
),
Inoue and Kabaya suggested a combinatorial method to obtain the developing map
of the representation using the octahedral triangulation and the shadow-coloring of
certain quandles. A quandle is an algebraic system closely related to the
Reidemeister moves, so their method changes quite naturally under the Reidemeister
moves.
We apply their method to the potential function, which was used to define the
optimistic limit, and construct a saddle point of the function. This construction
works for any boundary-parabolic representation, and it shows that the
octahedral triangulation is good enough to study all possible boundary-parabolic
representations of the link group. Furthermore, the evaluation of the potential
function at the saddle point becomes the complex volume of the representation, and
this saddle point changes naturally under the Reidemeister moves because it is
constructed using the quandle.
Keywords
optimistic limit, quandle, hyperbolic volume,
boundary-parabolic representation, link group
