Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: (1, 2) and weak (1, 3) homotopies on knot projections   In this paper, we obtain the necessary and sufficient condition that two knot
projections are related by a finite sequence of the first and second flat
Reidemeister moves (Theorem 1). We also consider an equivalence relation that
is called weak (1, 3) homotopy. This equivalence relation occurs by the first
flat Reidemeister move and one of the third flat Reidemeister moves. We
introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes
(Sec. 3). Using the map, we determine which knot projections are trivialized
under weak (1, 3) homotopy (Corollary 3).

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The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom