$n$, discarding low-Q modes with $\kappa\ge0.05$ (for $n=1.6$, only modes with $\kappa\ge0.2$ are shown). The deformation is $\epsilon=0.444$. The appearance of distinct Q-factor peaks as a function of quasi-bound state wavenumber is most pronounced at the largest refractive index, $n=2.4$, and becomes nearly unobservable at $n=1.6$ (therefore, data for $n=1.6$ were not collected beyond $nk\approx106.5$). The peaks that do remain observable are approximately at the same values of $nk$ for all $n$.\[fig:Wavenumber-sweeps\]](Figure5){width="0.75\columnwidth"}

By definition, the boundary-integral method assumes that regions of uniform dielectric constant are bounded by closed curves, so we model the waveguides as finite-length attachments, see Fig. \[fig:Q-factor-versusEps\]. By varying the lengths of these waveguides (either together or independently of each other) we verified that the finiteness of the stubs has no significant effect on the mode structure: neither real parts nor imaginary parts of $k_{\text{QB}}$ for the modes with figure-eight topology were affected unless their Q factor was below approximately $200$ to begin with. Because the modes are built on classical ray orbits, they will not be found (with appreciable lifetimes) in small cavities. This is illustrated in Fig. \[fig:Small-cavity-limit\] where the self-intersecting topology is barely discernible in the wave intensities. This raises the question whether the 