Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simplifying branched covering surface-knots by chart moves involving
  black vertices   A branched covering surface-knot is a surface-knot in the form of a branched
covering over an oriented surface-knot $F$, where we include the case when the
covering has no branch points. A branched covering surface-knot is presented by
a graph called a chart on a surface diagram of $F$. We can simplify a branched
covering surface-knot by an addition of 1-handles with chart loops to a form
such that its chart is the union of free edges and 1-handles with chart loops.
We investigate properties of such simplifications for the case when branched
covering surface-knots have a non-zero number of branch points, using chart
moves involving black vertices.

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The categories of this paper are: bayes-an
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The categories of this paper are: econ
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
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The categories of this paper are: nucl-ex