The comparison principle for scalar second order parabolic PDEs on functions admits a topological interpretation: pairs of solutions, and , evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions . By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves 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.