  We introduce persistent Betti numbers to characterize topological structure
of jets. These topological invariants measure multiplicity and connectivity of
jet branches at a given scale threshold, while their persistence records
evolution of each topological feature as this threshold varies. With this
knowledge, in particular, we are able to reconstruct branch phylogenetic tree
of each jet. These points are demonstrated in the benchmark scenario of
light-quark versus gluon jets. This study provides a topological tool to
develop jet taggers, and opens a new angle to look into jet physics.
