  The comparison principle for scalar second order parabolic PDEs on functions
$u(t,x)$ admits a topological interpretation: pairs of solutions,
$u^1(t,\cdot)$ and $u^2(t,\cdot)$, evolve so as to not increase the
intersection number of their graphs. We generalize to the case of multiple
solutions $\{u^\alpha(t,\cdot)\}_{\alpha=1}^n$. By lifting the graphs to
Legendrian braids, we give a global version of the comparison principle: the
curves $u^\alpha(t,\cdot)$ evolve so as to (weakly) decrease the algebraic
length of the braid.
  We define a Morse-type theory on Legendrian braids which we demonstrate is
useful for detecting stationary and periodic solutions to scalar parabolic
PDEs. This is done via discretization to a finite dimensional system and a
suitable Conley index for discrete braids.
  The result is a toolbox of purely topological methods for finding invariant
sets of scalar parabolic PDEs. We give several examples of spatially
inhomogeneous systems possessing infinite collections of intricate stationary
and time-periodic solutions.
