The six-vertex model is an exactly solvable model in statistical mechanics that has been widely studied by both physicists and combinatorialists for its many lovely properties. We introduce a symmetrized version of the six-vertex model that adapts nicely from the square lattice to arbitrary 4-regular graphs embedded in a disk, along with and an alter ego we call hourglass plabic graphs. We use these new objects to give a bijection between move equivalence classes of these graphs and 4-row tableaux in such a way that promotion corresponds to rotation, yielding a rotation-invariant web basis for SL_4. Along the way, we find move equivalence classes corresponding to n x n alternating sign matrices, plane partitions, and their symmetry classes (joint with Ashleigh Adams). This talk is primarily based on joint work with Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, and Joshua P. Swanson.
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