  In this paper, we consider a little-studied observable in classical pure AdS3
gravity: the twist along a geodesic. The motivation is that the twist only
supports on the geodesic so may be a candidate element in the center of the
algebra in either entanglement wedge associated to the geodesic. We study the
properties of the twist and get the following results. First, we get the
system's evolution generated by the twist, which exhibits a relative shift
along the geodesic. Second, we show that the twist commutes with the length of
the same geodesic, which supports the proposal that the twist is a candidate
element in the center.
