  We present a general method to analyze the topological nature of the domain
boundary connectivity that appeared in relaxed moir\'e superlattice patterns at
the interface of 2-dimensional (2D) van der Waals (vdW) materials. At large
enough moir\'e lengths, all moir\'e systems relax into commensurated 2D domains
separated by networks of dislocation lines. The nodes of the 2D dislocation
line network can be considered as vortex-like topological defects. We find that
a simple analogy to common topological systems with an $S^1$ order parameter,
such as a superconductor or planar ferromagnet, cannot correctly capture the
topological nature of these defects. For example, in twisted bilayer graphene,
the order parameter space for the relaxed moir\'e system is homotopy equivalent
to a punctured torus. Here, the nodes of the 2D dislocation network can be
characterized as elements of the fundamental group of the punctured torus, the
free group on two generators, endowing these network nodes with non-Abelian
properties. Extending this analysis to consider moir\'e patterns generated from
any relative strain, we find that antivortices occur in the presence of
anisotropic heterostrain, such as shear or anisotropic expansion, while arrays
of vortices appear under twist or isotropic expansion between vdW materials.
Experimentally, utilizing the dark field imaging capability of transmission
electron microscopy (TEM), we demonstrate the existence of vortex and
antivortex pair formation in a moir\'e system, caused by competition between
different types of heterostrains in the vdW interfaces. We also present a
methodology for mapping the underlying heterostrain of a moir\'e structure from
experimental TEM data, which provides a quantitative relation between the
various components of heterostrain and vortex-antivortex density in moir\'e
systems.
