Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology

Details Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology

2010-10-18

arxiv.org/abs/1010.3742v1

Mathematics

Geometric Topology

39 pages, 32 figures

Scientific paper

In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ 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 $5_2$ knot and a trefoil knot.

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