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A CLASSIFICATION OF LINKS OF THE FLAT PLUMBING BASKET NUMBERS 4 OR LESS
Kim, Dongseok The Kangwon-Kyungki Mathematical Society 2014 한국수학논문집 Vol.22 No.2
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. In present article, we study links of the flat plumbing basket numbers 4 or less using a special presentation of the flat plumbing basket surfaces. We find a complete classification theorem of links of the flat plumbing basket numbers 4 or less.
EVERY LINK IS A BOUNDARY OF A COMPLETE BIPARTITE GRAPH K<sub>2,n</sub>
Jang, Yongjun,Jeon, Sang-Min,Kim, Dongseok The Kangwon-Kyungki Mathematical Society 2012 한국수학논문집 Vol.20 No.4
A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the ($m,n$) torus link. In the present article, we prove that every link is the boundary of a complete bipartite multi-graph $K_{m,n}$ for which voltage assignments are either -1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n}$ for which voltage assignments are either -1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.
THE BOUNDARIES OF DIPOLE GRAPHS AND THE COMPLETE BIPARTITE GRAPHS K2,n
김동석 호남수학회 2014 호남수학학술지 Vol.36 No.2
We study the Seifert surfaces of a link by relating the embeddings of graphs with induced graphs. As applications, we prove that every link L is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph K2,n, where all voltage assignments on the edges of K2,n are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links 412 and 52.
THE BOUNDARIES OF DIPOLE GRAPHS AND THE COMPLETE BIPARTITE GRAPHS K<sub>2,n</sub>
Kim, Dongseok The Honam Mathematical Society 2014 호남수학학술지 Vol.36 No.2
We study the Seifert surfaces of a link by relating the embeddings of graphs with induced graphs. As applications, we prove that every link L is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2,n}$ are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links $4^2_1$ and $5_2$.
Bang, Je-Jun,Do, Jun-Ho,Kim, Dongseok,Kim, Tae-Hyung,Park, Se-Han The Kangwon-Kyungki Mathematical Society 2015 한국수학논문집 Vol.23 No.1
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 $7_5$.
THE BASKET NUMBERS OF LINKS OF 6 CROSSINGS OR LESS
KIM, DONGSEOK,KIM, GUNWOO,LEE, MINWOO,PARK, SUNG-HAN,SON, DONGBUM,BAE, CHEOLMIN The Youngnam Mathematical Society 2016 East Asian mathematical journal Vol.32 No.1
In present article, we find a complete classification theorem of links of basket numbers 2 or less. As an application, we study the basket numbers of links of 6 crossings or less.
THE BASKET NUMBERS OF LINKS OF 6 CROSSINGS OR LESS
김동석,김건우,이민우,박성한,손동범,배철민 영남수학회 2016 East Asian mathematical journal Vol.32 No.1
In present article, we find a complete classification theorem of links ofbasket numbers 2 or less. As an application, we study the basket numbers of links of6 crossings or less.
OH, SEUNGSANG 호남수학회 2001 한국수학학술지 Vol.23 No.1
Brittenham has shown how an incompressible Seifert surface F for a knot in S^3 can be used to find an infinite class of persistently laminar knots. We generalize this to create larger class of persistently laminar knots which therefore have property P.
OH, SEUNGSANG The Honam Mathematical Society 2001 호남수학학술지 Vol.23 No.1
Brittenham has shown how an incompressible Seifert surface F for a knot in $S^3$ can be used to find an infinite class of persistently laminar knots. We generalize this to create larger class of persistently laminar knots which therefore have property P.
Moon, Myoung-Ho Korean Mathematical Society 1996 대한수학회논문집 Vol.11 No.3
We prove that if G is the fundamental group of a closed surface or a Seifert fibered space and K is a finitely generated subgroup of G, and if for any element g in G there exists an integer $n_g$ such that $g^{n_g}$ belongs to K, then K is of finite index in G.