Gnomonious Projections for Bend-Free Textures: Thoughts on the Splay-Twist Phase

TL;DR

This paper introduces a novel gnomonic projection of the Hopf fibration to create bend-free liquid crystalline textures characterized by splay and twist, inspired by geometric structures on the three sphere.

Contribution

It proposes a new projection method for Hopf fibrations to generate unique liquid crystal textures with specific deformation properties.

Findings

The structure is based on tangent vectors along hyperboloids.

The phase exhibits no bend deformations, similar to certain nematic phases.

Domains can be stabilized via anchoring and saddle-splay effects.

Abstract

The Hopf fibration has inspired any number of geometric structures in physical systems, in particular in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, $\mathbb{S}^3$, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic-preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new liquid crystalline texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay.