UC Berkeley

The traditional way to study smooth 4-manifolds up to diffeomorphism, after the groundbreaking work of S. Donaldson, is to employ gauge-theoretic or Floer-theoretic invariants, including Donaldson invariants [Don90], Seiberg–Witten invariants [SW94], and Heegaard Floer invariants [OS04]. These tools, powerful as they are, typically come with constraints on the topology of the 4-manifolds, and there is a sense in which their applicability to new phenomena is approaching their natural limits. This thesis presents several studies on smooth 4-manifolds via Khovanov homology [Kho00] and the associated Rasmussen sinvariants [Ras10], tools that originate in representation theory and are combinatorial in nature.In the first chapter, we establish (relative) adjunction-type inequalities for surfaces in #kCP2\ int(B4 ), with a correction term coming from the s-invariant of the boundary link [Ren24a; Ren25]. Compared to similar inequalities in Heegaard Floer theory, the adjunction inequality for the s-invariant has the advantage of potentially detecting an exotic copy of #kCP2 , whose existence is not currently known. The inequality is established as a consequence of a full computation of the filtration structure of the Lee homology [Lee05] of all torus links. Of independent interest, we also propose a conjectural formula for the rational Khovanov homology of torus links T(n, n).In the second chapter, we review the construction of skein lasagna modules, a package of smooth 4-manifold invariants introduced by Morrison–Walker–Wedrich [MWW22]. We review some formal properties, make some general observations, and introduce some natural variants of skein lasagna modules, including skein lasagna modules with 1-dimensional inputs. Then we specialize to the theory of Khovanov and Lee skein lasagna modules.In the third chapter, we present joint work with Michael Willis [RW24] on detecting pairs of exotic 4-manifolds (with boundary) using Khovanov and Lee skein lasagna modules. We introduce a lasagna analogue of Rasmussen’s s-invariants. We present some vanishing and nonvanishing results for Khovanov skein lasagna modules and lasagna s-invariants. The techniques developed in the first chapter are crucial for the main nonvanishing results.In the last chapter, we present joint work with Ian Sullivan, Paul Wedrich, Michael Willis, and Melissa Zhang [Ren+25] on using Khovanov skein lasagna modules to prove excellent functoriality properties for the theory of Khovanov homology in connected sums of copies of S 1 × S 2 as defined by Rozansky [Roz10] and Willis [Wil21]. This provides an interesting example of the theory of skein lasagna modules with 1-dimensional inputs; this theory admits a forgetful map from the usual theory of Khovanov skein lasagna modules.Compared with the original sources [Ren24a; Ren25; RW24; Ren+25], this thesis contains significant additions in Sections 1.6.2, 2.4, 2.5, 2.6.2.2, 2.6.4, 3.4.4, 3.5.1, 3.5.2, 4.1.1, 4.1.3, 4.2.3, 4.3.1; simplifications or significant expositional changes have been made in Section 1.4, Chapter 2, Sections 3.3.1, 3.3.2, 3.4.1, 3.4.3, 3.4.4, 3.5.5, 3.5.9, 4.2.4, and Appendix D.