  Unlike the standard entanglement entropy considered in the holographic
context, entwinement measures entanglement between degrees of freedom that are
not associated to a spatial subregion. Entwinement is defined for
two-dimensional CFTs with a discrete $\mathbb{Z}_N$ gauge symmetry. Since the
Hilbert space of these CFTs does not factorize into tensor products, even the
entanglement entropy associated to a spatial subregion cannot be defined as the
von Neumann entropy of a reduced density matrix. While earlier works considered
embedding the density matrix into a larger, factorizing Hilbert space, we apply
a gauge invariant approach by using a density matrix uniquely defined through
its relation to the local algebra of observables. We furthermore obtain a fully
gauge invariant definition of entwinement valid for general CFTs with
$\mathbb{Z}_N$ gauge symmetry in terms of all observables acting on the degrees
of freedom considered. Holographically, entwinement is dual to the length of
non-minimal geodesics present for conical defects or black holes. In this
context, we propose a definition of entwinement for thermal states dual to the
BTZ black hole. Our results show that "entwinement is enough" to describe the
full bulk geometry for the conical defect and provide strong hints that the
same holds true for the BTZ black hole. Thus, it provides an alternative to
holographic complexity for the theories considered.
