Consider the 3-sphere $S_3$ with an unlink loop $L$ whose tubular neighborhood is identified with the solid torus $B_2\times S_1$ with one twist, i.e., such that the image of $x\times S_1$ (where $x$ is a fixed point on the boundary of $B_2$) has linking number $1$ with $L$. Now, cut the solid torus and glue it back in with a modular $S$-transformation in between. In other words, apply a regular 1-surgery (I hope "1" is the correct numbering convention). What is the resulting 3-manifold?
Now add a fourth dimension, considering $S_3=\partial B_4$. Attach a 2-handle along the solid torus from the previous paragraph. What is the resulting 4-manifold with boundary? The boundary should be the result from 1).
My motivation behind the question is that this surgery should compute the chiral central charge of a modular tensor category. I would therefore expect the resulting 4-manifold to be $\mathbb{C}P(2)$ with something simple removed, like a 4-ball, or the tubular neighborhood of an unlink.