Kirby--Thompson Invariants of Distant Sums of Standard Surfaces

Blair–Campisi–Taylor–Tomova [BCTT22] 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 [APZ23] 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.