ETD

• cobordismi di link
• invariante di Jacobsson
• Jacobsson invariant
• Khovanov homology
• knot theory
• link cobordisms
• mosse di Reidemeister
• omologia di Khovanov
• Reidemeister moves
• teoria dei nodi

Khovanov homology is a bigraded algebraic invariant of links that refines the Jones polynomial, in the sense that the Jones polynomial can be recovered as its graded Euler characteristic. A central question is whether Khovanov homology is functorial with respect to link cobordisms, that is, whether a smooth surface connecting two links in four-dimensional space induces a well-defined map between the corresponding homology groups. This thesis presents an exposition of Jacobsson's 2004 paper on this question. The map is constructed by decomposing a cobordism through elementary local moves, each inducing a chain map on the Khovanov complex. Composing these maps gives a map on homology. Khovanov conjectured that this map is invariant under ambient isotopy of the cobordism up to sign, but Jacobsson showed that this fails in general, providing two explicit counterexamples. It becomes true once isotopies are required to fix the boundary setwise, and this thesis presents Jacobsson's proof. The thesis concludes with Lefschetz polynomials of link endocobordisms and an overview of the relative Khovanov-Jacobsson class due to Sundberg and Swann.