Topology detects holes. The fundamental group counts loops. Homology groups count voids of each dimension. Applied to a chaotic attractor in phase space, these tools can tell you that the trajectory wraps around something — a hole in the shape of the attractor that the dynamics never fills. But they can't tell you which way the trajectory goes around it. Two chaotic systems can have identical topology — same holes, same voids, same homological invariants — while their dynamics circulate in completely different patterns.
Sciamarella (arXiv 2602.18463, February 2026) fixes this by introducing directed topological invariants for chaotic attractors. The construction is called a templex: a cell complex (the shape of the branched manifold the attractor lives on) paired with a directed graph (the flow-compatible paths the dynamics actually follows). The cell complex captures what classical topology already captures — the geometry of voids. The directed graph adds something new: the causal structure of how the flow moves through the branched manifold.
The critical new objects are what she calls generatex semigroups. Where homology groups classify undirected loops — closed paths that can be continuously deformed into each other — generatex semigroups classify directed cycles. Two directed cycles are equivalent if they traverse the same joining loci (the points where multiple branches of the manifold meet) in the same sequence. The result is algebraic: a semigroup rather than a group, because directed paths compose associatively but don't generally have inverses. You can't run a chaotic trajectory backwards and arrive at the same joining locus sequence.
The framework detects what she calls “topological tipping points” — changes in the generatex structure of a system under parameter variation that are invisible to classical topological invariants. A wind-driven ocean circulation model shows six distinct generatex classes. As forcing changes, the number and connectivity of these classes change — the flow reorganizes its directed structure without necessarily changing the topology of the attractor. Homology sees the same shape. The templex sees a different circulation pattern.
The application to experimental data is notable. A recording of the vowel “a” — 1183 data points at 8 kHz — produces a three-dimensional phase portrait whose templex has three generatex classes connected by bonds of valence 2 and 3. The directed invariants extract causal branching structure from a speech signal. The topology of the attractor says “there are holes.” The templex says “the voice goes around them this way, not that way.” The direction is the information that standard topology throws away.