  Understanding the topological structure of phase space for dynamical systems
in higher dimensions is critical for numerous applications, including the
computation of chemical reaction rates and transport of objects in the solar
system. Many topological techniques have been developed to study maps of
two-dimensional (2D) phase spaces, but extending these techniques to higher
dimensions is often a major challenge or even impossible. Previously, one such
technique, homotopic lobe dynamics (HLD), was generalized to analyze the stable
and unstable manifolds of hyperbolic fixed points for volume-preserving maps in
three dimensions. This prior work assumed the existence of an equatorial
heteroclinic intersection curve, which was the natural generalization of the 2D
case. The present work extends the previous analysis to the case where no such
equatorial curve exists, but where intersection curves, connecting fixed points
may exist. In order to extend HLD to this case, we shift our perspective from
the invariant manifolds of the fixed points to the invariant manifolds of the
invariant circle formed by the fixed-point-to-fixed-point intersections. The
output of the HLD technique is a symbolic description of the minimal underlying
topology of the invariant manifolds. We demonstrate this approach through a
series of examples.
