Examples of non-trivial roots of unity at ideal points of hyperbolic 3-manifolds

This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points whose associated roots of unity are not 1 or -1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than 1 and -1 occur.Comment: 12 pages, 1 figure, LaTeX2e. Minor changes, additional remarks, new description of 2nd example. To appear in_Topology

Dialectical Teaching Strategies for First and Second-Year Students

An oppositional or bi-polar understanding of meaning (Rychlak, 1976). This ancient understanding of meaning has many applications for instructors.https://digitalscholarship.unlv.edu/btp_expo/1020/thumbnail.jp