On torsion in (bi)linearized Legendrian contact homology in dimension 3

TL;DR

This paper investigates the presence and characteristics of torsion in linearized Legendrian contact homology and bilinearized LCH for Legendrian knots in three-dimensional space, providing new insights into their algebraic structures over integers.

Contribution

It characterizes torsion phenomena in linearized LCH and maps out the possible algebraic structures of bilinearized LCH over integer coefficients.

Findings

Identifies properties of torsion in linearized LCH with integer coefficients.

Provides a complete classification of bilinearized LCH structures over integers.

Abstract

Linearized Legendrian contact homology (LCH) and bilinearized LCH are important homological invariants for Legendrian submanifolds in contact geometry. For legendrian knots in $\mathbb{R}^3$, very little was previously known about the possibility of having torsion in these invariants when they are defined over integer coefficients. In this paper, we give properties of torsion that can appear in linearized LCH with integer coefficients, and also give the full geography of bilinearized LCH with integer coefficients.