The Hcircle times(gcircle plus2h) Jahn-Teller problem in the trigonal regime offers an example of a dynamic ground-state crossover from a degenerate to a singlet state. This crossover is connected to a change in tunneling paths which results in a loss of Berry phase. Previous studies tried to explain the entanglement which lies at the basis of this phenomenon using two-dimensional (2D) cuts through coordinate space but have reached opposite interpretations. In this paper we introduce a method which scans the topology of the full coordinate space using a net of triangular cross sections between the ten D-3d minima. It is shown that the Berry phase of every loop on the potential energy hypersurface can be expressed in terms of these 2D triangular cross sections. Using this method we show that the absence of Berry phase for the cycles containing five trigonal wells originates from two seams of conical intersections going through these cycles. The connection between this analysis and previous treatments is established and the apparent dilemma of interpretation is resolved.