The correspondence induced on the pillowcase by the earring tangle

Abstract

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