A diagrammatic approach to the Rasmussen invariant via tangles and cobordisms

Abstract: We introduce a diagrammatic approach to Rasmussen's $s$-invariant via tangles and cobordisms, combining Bar-Natan's formulation of Khovanov homology for tangles and cobordisms with the characterization of $s$ via the divisibility of the Lee class, as developed in the author's previous works. This framework enables a "divide-and-conquer" method for computing $s$ from a tangle decomposition of a given knot diagram, making it suitable for both pen-and-paper calculations and algorithmic implementations. As an application, we determine the $s$-invariants of pretzel knots of the form $P(p_1, -p_2, \ldots, -p_l)$, where $l \geq 3$ is odd, all $p_i$ are positive and odd, and $p_1 < \min{p_2, \ldots, p_l}$.