Kirby–Thompson invariants of distant sums of standard surfaces

Blair–Campisi–Taylor–Tomova [Blair, Campisi, Taylor and Tomova, Kirby-Thompson distance for trisections of knotted surfaces, J. Lond. Math. Soc. (2) 105(2) (2022) 765–793.] defined the [Formula: see text]-invariant [Formula: see text] of a knotted surface [Formula: see text], using pants complexes of trisection surfaces of bridge trisections of [Formula: see text]. After that, Aranda–Pongtanapaisan–Zhang [Aranda, Pongtanapaisan and Zhang, Bounds for Kirby-Thompson invariants of knotted surfaces, Geom. Dedicata 217(6) (2023) 30] introduced the [Formula: see text]-invariant [Formula: see text] using dual curve complexes instead of pants complexes. In this paper, we determine both the [Formula: see text]-invariant and the [Formula: see text]-invariant of any finite distant sum of standard surfaces, and this is the first example of knotted surfaces whose bridge numbers and these invariants can be arbitrarily large.