The correspondence induced on the pillowcase by the earring tangle

Abstract

The earring tangle consists of four strands $4\text{pt}\times I\subset S^2\times I$ and one meridian around one of the strands. Equipping this tangle with a nontrivial $SO\left(3\right)$ bundle, we show that its traceless $SU\left(2\right)$ 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.