# On Legendrian products and twist spuns

## TL;DR

This paper investigates the relationship between Legendrian products and twist spuns, showing conditions under which they coincide and providing examples where they do not, using Legendrian contact homology techniques.

## Contribution

It establishes that Legendrian products are twist spuns when one component is large and analyzes cases where they are not, using bilinearised Legendrian contact homology.

## Key findings

- Legendrian product equals twist spun for large components.
- Examples of Legendrian products not isotopic to twist spuns.
- Bohr-Sommerfeld covers of certain tori are not twist spuns.

## Abstract

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In order to do this, we prove a few structural results on the bilinearised Legendrian contact homology and augmentation variety of a twist spun. In addition, we show that the threefold Bohr-Sommerfeld covers of the Clifford torus and Chekanov torus are not twist spuns.