The correspondence induced on the pillowcase by the earring tangle

Abstract

The earring tangle consists of four strands 4pt × I ⊂ S 2 × I and one meridian around one of the strands. Equipping this tangle with a nontrivial SO(3) bundle, we show that its traceless 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.

Singularities of Lagrangian Immersions and its Applications in Lagrangian Floer Theory

“…Embedded Lagrangian Floer theory has been studied by many people such as Fukaya, Ohta, Oh, Ono [17], and Seidel [26]. However, examples show that the Lagrangians R(H 0 ) and R(H 1 ) can be Lagrangian immersions into R(Σ g ) [7] [21]. Akaho and Joyce give the definition of immersed Lagrangian Floer theory in [2].…”

“…To generalize this theorem to the case where the map F → F 1 ×F 2 contains bisingular points, it is necessary to compare the boundary maps of CF(L 1 , L 12 • L 2 ) and CF(L 1 • L 12 , L 2 ) near the bisingular points. The nontrivial example comes from Cazassus, Herald, Kirk, Kotelskiy [7], which indicates the importance of studying maps with bisingular points.…”