On spun-normal and twisted squares surfaces

TL;DR

This paper explores the relationship between surfaces constructed from ideal points in 3-manifolds with torus boundary, comparing Yoshida's twisted squares method and Tillmann's spun-normal surfaces, and extends a detection result to twisted squares.

Contribution

It extends Tillmann's detection result to surfaces generated from twisted squares, linking ideal points of the deformation and character varieties.

Findings

Extended detection result to twisted squares surfaces

Established relation between twisted squares and spun-normal surfaces

Connected ideal points of deformation and character varieties

Abstract

Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation between the generated surfaces and extend a result of Tillmann's (that if the ideal point of the deformation variety corresponds to an ideal point of the character variety then the generated spun-normal surface is detected by the character variety) to the generated twisted squares surfaces.