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Abstract
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As one of the background papers of the classification project of hyperbolic primitive/Seifert (or
P/SF) knots in
whose complete list is given in [Berge and Kang 2020], this paper
classifies pairs of two disjoint nonseparating simple closed curves
and
lying in the boundary of
a genus two handlebody
such that
is a proper power curve and a 2-handle addition
along
embeds in
so that
is the exterior of a tunnel-number-one knot. As a consequence, if
is a
nonseparating simple closed curve on the boundary of a genus two handlebody such that
embeds in
,
then there exists a proper power curve disjoint from
if and
only if
is the exterior of the unknot, a torus knot, or a tunnel-number-one cable of a torus
knot.
The results of this paper will be mainly used in proving the hyperbolicity
of P/SF knots and in classifying P/SF knots in once-punctured tori in
, which
is one of the types of P/SF knots in [Berge and Kang 2020]. Together with
these results, the preliminary of this paper which consists of three parts: the
three diagrams which are Heegaard diagrams, R-R diagrams, and hybrid
diagrams, the “culling lemma”, and locating waves into an R-R diagrams,
will also be used in the classification of hyperbolic primitive/Seifert knots
in .
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Keywords
tunnel-number-one knots, primitive/Seifert knots, proper
power curves, 2-handle additions, Heegaard and R-R diagrams
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Mathematical Subject Classification
Primary: 57K10
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Milestones
Received: 24 May 2020
Revised: 23 September 2021
Accepted: 17 October 2021
Published: 19 January 2022
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