On the Polynomial of the Dunwoody (1, 1)-knots

There is a special connection between the Alexander polynomial of (1, 1)-knot and the certain polynomial associated to the Dunwoody 3-manifold ([3], [10] and [13]). We study the polynomial(called the Dunwoody polynomial) for the (1, 1)-knot obtained by the certain cyclically presented group of the D...

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Bibliographic Details
Published inKyungpook mathematical journal Vol. 52; no. 2; pp. 223 - 243
Main Authors Kim, Soo-Hwan, Kim, Yang-Kok
Format Journal Article
LanguageKorean
Published 2012
Subjects
Alexander polynomial
Torus knot
(1, 1)-decomposition
(1, 1)-knot
Heegaard diagram
Dunwoody 3-manifold
Heegaard splitting
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Summary:There is a special connection between the Alexander polynomial of (1, 1)-knot and the certain polynomial associated to the Dunwoody 3-manifold ([3], [10] and [13]). We study the polynomial(called the Dunwoody polynomial) for the (1, 1)-knot obtained by the certain cyclically presented group of the Dunwoody 3-manifold. We prove that the Dunwoody polynomial of (1, 1)-knot in $\mathbb{S}^3$ is to be the Alexander polynomial under the certain condition. Then we find an invariant for the certain class of torus knots and all 2-bridge knots by means of the Dunwoody polynomial.
Bibliography:KISTI1.1003/JNL.JAKO201225067513364
ISSN:1225-6951
0454-8124