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A bridge trisection of a smooth surface in S4 is a decomposition analogous to a bridge splitting of a link in S3. The Kirby–Thompson invariant of a bridge trisection measures its complexity in terms of distances between disk sets in the pants complex of the trisection surface. We give the first significant bounds for the Kirby–Thompson invariant of spun knots. In particular, we show that the Kirby–Thompson invariant of the spun trefoil is 15. 

WISE J224607.6–052634.9 (W2246–0526) is a hot dust-obscured galaxy atz = 4.601, and the most luminous obscured quasar known to date. W2246–0526 harbors a heavily obscured supermassive black hole that is most likely accreting above the Eddington limit. We present observations with the Atacama Large Millimeter/submillimeter Array (ALMA) in seven bands, including band 10, of the brightest far-infrared (FIR) fine-structure emission lines of this galaxy: [OI]_63 μm, [OIII]_88 μm, [NII]_122 μm, [OI]_145 μm, [CII]_158 μm, [NII]_205 μm, [CI]_370 μm, and [CI]_609 μm. A comparison of the data to a large grid of CLOUDYradiative transfer models reveals that a high hydrogen density (n_H ∼ 3 × 10³cm⁻³) and extinction (A_V ∼ 300 mag), together with extreme ionization (log(U) = − 0.5) and a high X-ray to UV ratio (α_ox ≥ −0.8) are required to reproduce the observed nuclear line ratios. The values ofα_oxandUare among the largest found in the literature and imply the existence of an X-ray-dominated region (XDR). In fact, this component explains the a priori very surprising non-detection of the [OIII]_88 μmemission line, which is actually suppressed, instead of boosted, in XDR environments. Interestingly, the best-fitted model implies higher X-ray emission and lower CO content than what is detected observationally, suggesting the presence of a molecular gas component that should be further obscuring the X-ray emission over larger spatial scales than the central region that is being modeled. These results highlight the need for multiline infrared observations to characterize the multiphase gas in high redshift quasars and, in particular, W2246–0526 serves as an extreme benchmark for comparisons of interstellar medium conditions with other quasar populations at cosmic noon and beyond.