Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topology of spaces of knots in dimension 3   This paper is a computation of the homotopy type of K, the space of long
knots in R^3, the same space of knots studied by Vassiliev via singularity
theory. Each component of K corresponds to an isotopy class of long knot, and
we `enumerate' the components via the companionship trees associated to the
knot. The knots with the simplest companionship trees are: the unknot, torus
knots, and hyperbolic knots. The homotopy-type of these components of K were
computed by Hatcher. In the case the companionship tree has height, we give a
fibre-bundle description of those components of K, recursively, in terms of the
homotopy types of `simpler' components of K, in the sense that they correspond
to knots with shorter companionship trees. The primary case studied in this
paper is the case of a knot which has a hyperbolic manifold contained in the
JSJ-decomposition of its complement.

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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg