  We use our semi-analytic solution of the nonlinear force-free field equation
to construct three-dimensional magnetic fields that are applicable to the solar
corona and study their statistical properties for estimating the degree of
braiding exhibited by these fields. We present a new formula for calculating
the winding number and compare it with the formula for the crossing number. The
comparison is shown for a toy model of two helices and for realistic cases of
nonlinear force-free fields; conceptually the formulae are nearly the same but
the resulting distributions calculated for a given topology can be different.
We also calculate linkages, which are useful topological quantities that are
independent measures of the contribution of magnetic braiding to the total free
energy and relative helicity of the field. Finally, we derive new analytical
bounds for the free energy and relative helicity for the field configurations
in terms of the linking number. These bounds will be of utility in estimating
the braided energy available for nano-flares or for eruptions.
