Quandle and hyperbolic volume (Q968921)

One of the possible future directions for the study of 3-dimensional topology is to study connections between geometric structures on 3-manifolds and other topological or algebraic structures. In the paper under review, it is shown that, for an oriented hyperbolic knot in the 3-sphere, there exists a quandle 2-cocyle of the knot quandle and a shadow coloring of a diagram of the knot such that they give the hyperbolic volume of its complement. As an application, a criterion for such a knot to be invertible or positive/negative amphicheiral is given.