  We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer
chain complex for restricted immersions of compact 1-manifolds into the
pillowcase, and apply it to the intersection diagram obtained by taking
traceless $SU(2)$ character varieties of 2-tangle decompositions of knots.
Calculations for torus knots are explained in terms of pictures in the
punctured plane. The relation to the reduced instanton homology of knots is
explored.
