Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Alexander and Markov theorems for virtual doodles   Study of certain isotopy classes of a finite collection of immersed circles
without triple or higher intersections on closed oriented surfaces can be
thought of as a planar analogue of virtual knot theory where the genus zero
case corresponds to classical knot theory. Alexander and Markov theorems for
the genus zero case are known where the role of groups is played by twin
groups, a class of right angled Coxeter groups with only far commutativity
relations. The purpose of this paper is to prove Alexander and Markov theorems
for higher genus case where the role of groups is played by a new class of
groups called virtual twin groups which extends twin groups in a natural way.

23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys