Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Prime Decomposition and Non-Commutativity in the Monoid of Long Virtual
  Knots   It is well-known that the monoid of long virtual knots is not commutative.
This contrasts with the case of classical long knots, where $A \# B
\leftrightharpoons B \# A$ for all $A,B$. In the present paper, we present a
new proof that two inequivalent non-classical prime long virtual knots never
commute. The original result is due to Manturov. The techniques used here are
mostly geometric. First, a slightly strengthened version of Kuperberg's theorem
is established. We then show that a well-defined concatenation of two long
knots in a thickened surface is preserved by stabilization when both long knots
are non-classical. Finally, it is proved that if $A,B,C,D$ are prime
non-classical long virtual knots such that $A \# B$ is non-classical and $A \#
B \leftrightharpoons C \# D$, then $A \leftrightharpoons C$ and $B
\leftrightharpoons D$.

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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc