Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parity and Exotic Combinatorial Formulae for Finite-Type Invariants of
  Virtual Knots   The present paper produces examples of Gauss diagram formulae for virtual
knot invariants which have no analogue in the classical knot case. These
combinatorial formulae contain additional information about how a subdiagram is
embedded in a virtual knot diagram. The additional information comes from the
second author's recently discovered notion of parity. For a parity of flat
virtual knots, the new combinatorial formulae are Kauffman finite-type
invariants. However, many of the combinatorial formulae possess exotic
properties. It is shown that there exists an integer valued virtualization
invariant combinatorial formula of order n for every n (i.e. it is stable under
the map which changes the direction of one arrow but preserves the sign).
Hence, it is not of Goussarov-Polyak-Viro finite-type. Moreover, every
homogeneous Polyak-Viro combinatorial formula admits a decomposition into an
"even" part and an "odd" part. For the Gaussian parity, neither part of the
formula is of GPV finite-type when it is nonconstant on the set of classical
knots. In addition, eleven new non-trivial combinatorial formulae of order 2
are presented which are not of GPV finite-type.

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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc