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

• Published in: Kyungpook mathematical journal Vol. 52; no. 2; pp. 223 - 243
• Main Authors: Kim, Soo-Hwan, Kim, Yang-Kok
• Format: Journal Article
• Language: Korean
• Published: 2012
• Subjects: Alexander polynomial; Torus knot; (1, 1)-decomposition; (1, 1)-knot; Heegaard diagram; Dunwoody 3-manifold; Heegaard splitting
• ISSN: 1225-6951

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.