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This article introduces a new method to construct volume-filling symplecticembeddings of 4-dimensional ellipsoids by employing polytope mutations in toricand almost-toric varieties. The construction uniformly recovers the fullsequences for the Fibonacci Staircase of McDuff-Schlenk, the Pell Staircase ofFrenkel-Muller and the Cristofaro-Gardiner-Kleinman's Staircase, and adds newinfinite sequences of ellipsoid embeddings. In addition, we initiate the studyof symplectic tropical curves for almost-toric fibrations and emphasize theconnection to quiver combinatorics.

Full Ellipsoid Embeddings and Toric Mutations