  Coronal loops in active regions are often characterized by quasi-circular and
helically twisted (sigmoidal) geometries, which are consistent with dipolar
potential field models in the former case, and with nonlinear force-free field
models with vertical currents in the latter case. Alternatively, Parker-type
nanoflare models of the solar corona hypothesize that a braiding mechanism
operates between unresolved loop strands, which is a more complex topological
model. In this study we use the vertical-current approximation of a
nonpotential magnetic field solution (that fulfills the divergence-free and
force-free conditions) to characterize the number of helical turns $N_{twist}$
in twisted coronal loops. We measure the helical twist in 15 active regions
observed with AIA and HMI/SDO and find a mean nonpotentiality angle (between
the potential and nonpotential field directions) of $\mu_{NP} = 15^\circ \pm
3^\circ$. The resulting mean rotational twist angle is $\varphi = 49^\circ \pm
11^\circ$, which corresponds to $N_{twist}=\varphi/360^\circ = 0.14\pm0.03$
turns with respect to the untwisted potential field, with an absolute upper
limit of $N_{twist} \lapprox 0.5$, which is far below the kink instability
limit of $|N_{twist}| \gapprox 1$. The number of twist turns $N_{twist}$
corresponds to the Gauss linkage number $N_{link}$ in braiding topologies. We
conclude that any braided topology (with $|N_{link}| \ge 1$) cannot explain the
observed stability of loops in a force-free corona, nor the observed low twist
number. Parker-type nanoflaring can thus occur in non-forcefree environments
only, such as in the chromosphere and transition region.
