The earring tangle consists of four strands 4 pt × I ⊂ S 2 $4\text{pt} \times \subset S^2 I$ and one meridian around the strands. Equipping this with a nontrivial O ( 3 ) $SO(3)$ bundle, we show that its traceless U $SU(2)$ flat moduli space is topologically smooth genus three surface. We also restriction map from surface to boundary particular Lagrangian immersion into product two pillowcases....