Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New deformations on spherical curves and \"Ostlund Conjecture   In 2018, Funakoshi, Hashizume, Ito, Kobayashi, and Murai used a deformation
of spherical curves called deformation type $\alpha$. Then, it was showed that
if two spherical curves $P$ and $P'$ are equivalent under the relation
consisting of deformations of type RI and type RIII up to ambient isotopy, and
satisfy certain conditions, then $P'$ is obtained from $P$ by a finite sequence
of deformations of type $\alpha$. In this paper, we introduce a new type of
deformations of spherical curves, called deformation of type $\beta$. The main
result of this paper is: Two spherical curves $P$ and $P'$ are equivalent under
(possibly empty) deformations of type RI and a single deformation of type RIII
up to ambient isotopy if and only if reduced(P) and reduced(P') are transformed
each other by exactly one deformation which is of type RIII, type $\alpha$, or
type $\beta$ up to ambient isotopy, where reduced(Q) is the spherical curve
which does not contain a $1$-gon obtained from a spherical curve $Q$ by
applying deformations of type RI up to ambient isotopy.

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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc