Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries   Suppose $\alpha$ and $R$ are disjoint simple closed curves in the boundary of
a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of
a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $\alpha$ is
Seifert in $H$(i.e., a 2-handle addition $H[\alpha]$ is a Seifert-fibered
space) and not the meridian of $H[R]$. Then for a slope $\gamma$ of $k$
represented by $\alpha$, $\gamma$-Dehn surgery $k(\gamma)$ is a Seifert-fibered
space. Such a construction of Seifert-fibered Dehn surgeries generalizes that
of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a
knot, which was introduced in [D03].
  In this paper, we show that there exists a meridional curve $M$ of $k$ (or
$H[R]$) in $\partial H$ such that $\alpha$ intersects $M$ transversely in
exactly one point. It follows that such a construction of a Seifert-fibered
Dehn surgery $k(\gamma)$ can arise from a primtive/Seifert position of $k$ with
$\gamma$ its surface-slope. This result supports partially the two conjectures:
(1) any Seifert-fibered surgery on a hyperbolic knot in $S^3$ is integral, and
(2) any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises
from a primitive/Seifert position whose surface slope corresponds to the
surgery slope.

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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex