Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Self-overlapping Curves Revisited   A surface embedded in space, in such a way that each point has a neighborhood
within which the surface is a terrain, projects to an immersed surface in the
plane, the boundary of which is a self-intersecting curve. Under what
circumstances can we reverse these mappings algorithmically? Shor and van Wyk
considered one such problem, determining whether a curve is the boundary of an
immersed disk; they showed that the self-overlapping curves defined in this way
can be recognized in polynomial time. We show that several related problems are
more difficult: it is NP-complete to determine whether an immersed disk is the
projection of a surface embedded in space, or whether a curve is the boundary
of an immersed surface in the plane that is not constrained to be a disk.
However, when a casing is supplied with a self-intersecting curve, describing
which component of the curve lies above and which below at each crossing, we
may determine in time linear in the number of crossings whether the cased curve
forms the projected boundary of a surface in space. As a related result, we
show that an immersed surface with a single boundary curve that crosses itself
n times has at most 2^{n/2} combinatorially distinct spatial embeddings, and we
discuss the existence of fixed-parameter tractable algorithms for related
problems.

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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
2
The categories of this paper are: math
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org