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Augmentations, Fillings, and Clusters for 2-Bridge Links

  • Orsola Capovilla-Searle [1] ; James Hughes [2] ; Daping Weng [3]
    1. [1] Oregon State University

      Oregon State University

      Estados Unidos

    2. [2] Duke University

      Duke University

      Township of Durham, Estados Unidos

    3. [3] Department of Mathematics, University of North Carolina, USA
    Mostrar afiliaciones +
  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
  • Idioma: inglés
  • DOI: 10.1007/s00029-025-01092-z
  • Enlaces
  • Resumen

    • We produce the first examples relating non-orientable exact Lagrangian fillings of Legendrian links to cluster theory, showing that the ungraded augmentation variety of certain max-tb representatives of Legendrian 2-bridge links is isomorphic to a product of An-type cluster varieties. As part of this construction, we describe a surjective map from the set of (possibly non-orientable) exact Lagrangian fillings to cluster seeds, producing a product of Catalan numbers of distinct fillings. We also relate the ruling stratification of the ungraded augmentation variety to Lam and Speyer’s anticlique stratification of acyclic cluster varieties, showing that the two coincide in this context.

      As a corollary, we apply a result of Rutherford to show that the cluster-theoretic stratification encodes the information of the lowest a-degree term of the Kauffman polynomial of the smooth isotopy class of the 2-bridge links we study

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