Limiting characters at ideal points detecting twice-punctured tori

The limiting character, introduced by Tillmann, has been studied recently in the context of Culler-Shalen theory. We extend the methods of the author’s previous work to show that certain families of essential twice-punctured tori are detected by an ideal point on the character variety and determine the limiting character at these ideal points. We then provide numerous explicit examples, including certain two-bridge knots, 3-strand pretzel knots, and knots with non-integral toroidal surgeries. We also prove that the union of a once- and a twice-punctured torus inside the $(-3,5,5)$ or $(3,-5,-5)$ pretzel knot, both essential, is detected by an ideal point of the character variety and explicitly determine its limiting character.