Embedded and Lagrangian Knotted Tori in \BR⁴ and Hypercube Homology

2010/10/18 by Scott Baldridge, Baldridge, Scott
Mathematics · #57Q45 #57R40 #57R58 #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.1010.3742

openalex publication_date 2010/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in \BR⁴ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology. We provide examples of hypercube diagrams and hypercube homology, including using the new invariant to distinguish (up to cube moves) two "Hopf linked" tori. We also give examples of a "Trefoil" torus and an immersed knotted torus that is an amalgamation of the 5₂ knot and a trefoil knot.

Citations

• Floer homology and knot complements
• Holomorphic disks and topological invariants for closed three-manifolds

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