We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings. We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte-Gabella-Goncharov [DGGo]. We exploit this structure to find conjectural open Gromov-Witten invariants for the non-exact filling, following Aganagic-Vafa [AV, AV2].

中文翻译：

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三次平面图和勒让德曲面理论

我们研究由三次平面图确定的勒让德曲面。具有不同色度多项式的图确定了不是勒让德同位素的表面，因此给出了许多具有相同经典不变量的非同位素勒让德表面的例子。Legendrians 没有精确的拉格朗日填料，但有许多有趣的非精确填料。我们通过研究表面有微支撑的三球上的滑轮来获得这些结果。这种层的模在图方面有具体的描述，并且作为辛周期域的全纯拉格朗日子流形的美丽嵌入，拉格朗日量出现在 Dimofte-Gabella-Goncharov [DGGo] 的工作中。我们利用这种结构来寻找非精确填充的推测性开放 Gromov-Witten 不变量，遵循 Aganagic-Vafa [AV, AV2]。