Katherine Betts, Troy Larsen, Jeffrey Utley, Avalon Vanis, The Tri-Pants graph of the twice-punctured torus, preprint arXiv:2111.07136.
Abstract: We investigate the structure of the tri-pants graph, a simplicial graph introduced by Maloni and Palesi, whose vertices correspond to particular collections of homotopy classes of simple closed curves of the twice-punctured torus, called tri-pants, and whose edges connect two vertices whenever the corresponding pants differ by suitable elementary moves. In particular, by examining the relationship between the tri-pants graph and the dual of the Farey complex, we prove that the tri-pants graph is connected and it has infinite diameter.