2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134730We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical lifts of the higher Johnson homomorphisms $τ_m$, for $m\geq 1$, to the Torelli groupoid, and we provide a recursive combinatorial formula for tensor representatives of these lifts. In particular, we give an explicit 1-cocycle in the dual fatgraph complex which lifts $τ_2$ and thus answer affirmatively a question of Morita-Penner. To illustrate our techniques for calculating higher Johnson homomorphisms in general, we give explicit examples calculating $τ_m$, for $m\leq 3$.38 pages, 6 figuresGeometric Topology32G15; 57M99; 14H10; 57N05; 20F99text