Flat connections at infinity on knot surgery manifolds

$\rm SL(2,\mathbb{C})$ Chern-Simons theory on a closed 3-manifold is one of the most interesting, yet tractable examples of a QFT. On one hand, its non-perturbative structure is not yet fully understood; on the other, the mathematical structure turns out to be very rich. In this work we explore the new phenomenon of flat connections at infinity on various knot surgery manifolds. Such flat connections can be understood as asymptotic ends in the non-compact moduli space of flat $\rm SL(2,\mathbb{C})$ connections. We focus on the examples of $\pm 1/r$-surgeries on torus, twist and some double twist knot complements in $S^3$. Surprisingly, our findings suggest that flat connections at infinity are abundant even for simple low-crossing knot surgeries. We therefore believe that their presence would shed light on the resurgent nature of the path integral.