Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Knotted Hamiltonian cycles in linear embedding of $K_7$ into
  $\mathbb{R}^3$   In 1983 Conway and Gordon proved that any embedding of the complete graph
$K_7$ into $\mathbb{R}^3$ contains at least one nontrivial knot as its
Hamiltonian cycle. After their work knots (also links) are considered as
intrinsic properties of abstract graphs, and numerous subsequent works have
been continued until recently. In this paper we are interested in knotted
Hamiltonian cycles in linear embedding of $K_7$. Concretely it is shown that
any linear embedding of $K_7$ contains at most three figure-8 knots as its
Hamiltonian cycles.

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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph