Uniqueness of holomorphic quilts lifted from holomorphic bigons on surfaces

In the author’s previous paper, the author constructed holomorphic quilts from the bigons of the Lagrangian Floer chain group after performing Lagrangian composition. This paper proves the uniqueness of such holomorphic quilts. As a consequence, it provides a combinatorial method for computing the boundary map of immersed Lagrangian Floer chain groups when the symplectic manifolds are closed surfaces. One outcome is the construction of many examples exhibiting figure eight bubbling, which also confirms a conjecture of Cazassus Herald Kirk Kotelskiy.