The earring tangle consists of four strands $4\text{pt}\text{◂⊂▸}\times I\subset S^2\times I$ and one meridian around one of the strands. Equipping this tangle with a nontrivial $\text{◂⋅▸}SO(3)$ bundle, we show that its traceless $\text{◂⋅▸}SU(2)$ flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties.

中文翻译：

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耳环缠结在枕套上引起的对应

耳环缠结由四股线组成$\mathrm{4个}\text{点}\text{◂⊂▸}\times 我\subset {\mathrm{小号}}^{\mathrm{2个}}\times 我$一根子午线围绕其中一根股线。为这个缠结配备一个不平凡的$\text{◂⋅▸}\mathrm{小号}欧(\mathrm{3个})$捆绑，我们证明它无痕$\text{◂⋅▸}\mathrm{小号}ü(\mathrm{2个})$平坦模空间在拓扑上是光滑的亏格三曲面。我们还表明，从该表面到耳环缠结边界的无痕平坦模空间的限制映射是两个枕套产品的特定拉格朗日浸入。后一种计算表明，数字 8 冒泡——一种由 Bottman 和 Wehrheim 预测的微妙退化现象——出现在无踪字符变体的背景下。