Invariants of knotted surfaces from link homology and bridge trisections

2018/09/17 by Adam Saltz, Saltz, Adam
Mathematics · #57Q45 (Primary) 57M27 (Secondary) #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1809.06327

openalex publication_date 2018/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use link homology to construct an invariant of knotted surfaces (up to isotopy) which distinguishes the unknotted sphere from certain knotted spheres. We also construct an invariant of a bridge-trisected surface in the the form of an A_∞-algebra. Both invariants are defined by a novel connection between A_∞-algebras and Manolescu and Ozsváth's hyperboxes of chain complexes.

Citations

• Holomorphic disks and topological invariants for closed three-manifolds
• Strong Khovanov-Floer Theories and Functoriality

Related

• Kirby-Thompson invariants of distant sums of standard surfaces
• Lecture notes on link homologies and knotted surfaces
• The algebra of knotted trivalent graphs and Turaev's shadow world
• On closed oriented surfaces in the 3-sphere
• Knotted surfaces in 4-manifolds