  We examine the holonomy-perturbed traceless SU(2) character variety of the
trivial four-stranded tangle {p_1,p_2,p_3,p_4} X [0,1] in S^2 X [0,1] equipped
with a strong marking, either an earring or a bypass. Viewing these marked
tangles as endomorphisms in the cobordism category from the four-punctured
sphere to itself, we identify the images of these endomorphisms in the
Weinstein symplectic partial category under the partially defined
holonomy-perturbed traceless character variety functor. We express these
endomorphisms on immersed curves in the pillowcase in terms of doubling and
figure eight operations and prove they have the same image.
