Immersive Contact Geometry: A VR-Based Approach to Embodied Understanding of Legendrian and Transverse Knots

Abstract

One of the main challenges to teaching advanced mathematics is understanding high dimensional geometric structures. Students' spatial understanding is limited by contact geometry's dependency on static 2D projections, which includes abstract concepts like Legendrian and transverse knots. The goal of this study is to present an immersive Virtual Reality (VR) environment that allows students to see and engage with three-dimensional contact manifolds and tight vs. overtwisted contact structures in them. To model contact planes and their relationship to Legendrian and transverse knots, a virtual reality learning environment was created by using Gravity Sketch. Participants in a pilot implementation explored knots in both standard and overtwisted contact structures in instructors supervision during a graduate summer school. Observations, informal interviews, and fieldnotes were collected to capture student engagement and conceptual understanding. According to participants, the VR experience allowed them to "see" tangency, cusps, and invariants for the first time, turning abstract concepts into real-world experiences. By using this method, teachers were able to spot misconceptions that were previously undetectable in 2D environments. Even if the sample is small, immersive virtual reality holds great promise for use as a precision pedagogical tool that facilitates embodied and conceptual learning in advanced mathematics, rather than just as a visualization tool. Formal pre/post assessments will be used in future research to measure conceptual advancements and spatial reasoning.