Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology   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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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an