• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.869-877
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• 2022

In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let M be a homology lens space with H₁(M; ℤ) ≅ ℤ_p and K a not null-homologous knot in M. We show that, K is determined by its complement if M is non-hyperbolic, K is hyperbolic, and p is a prime greater than 7, or, if M is actually a lens space L(p, q) and K represents a generator of H₁(L(p, q)).

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.239-250
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• 2018

A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called w-variables. In this paper, we consider the case when pinched octahedra appear as a boundary parabolic solution in this decomposition. The w-solution with pinched octahedra induces a solution for a new knot obtained by changing the crossing or inserting a tangle at the pinched place. We discuss this phenomenon with corresponding holonomy representations and give some examples including ones obtained from connected sum.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.113-122
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• 2010

In [4], we gave a condition for a pair of unknotting tunnels of a non-trivial tunnel number one link to give a genus three Heegaard splitting of the link complement. In this paper we prove the corresponding result for tunnel number one knots.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.115-128
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• 2015

Plumbing surfaces of links were introduced to study the geometry of the complement of the links. A basket surface is one of these plumbing surfaces and it can be presented by two sequential presentations, the first sequence is the flat plumbing basket code found by Furihata, Hirasawa and Kobayashi and the second sequence presents the number of the full twists for each of annuli. The minimum number of plumbings to obtain a basket surface of a knot is defined to be the basket number of the given knot. In present article, we first find a classification theorem about the basket number of knots. We use these sequential presentations and the classification theorem to find the basket number of all prime knots whose crossing number is 7 or less except two knots $7_1$ and $7_5$.