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

• Journal title : Kyungpook mathematical journal
• Volume 52, Issue 2,  2012, pp.223-243
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2012.52.2.223
Title & Authors
On the Polynomial of the Dunwoody (1, 1)-knots
Kim, Soo-Hwan; Kim, Yang-Kok;

Abstract
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 $\small{\mathbb{S}^3}$ 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.
Keywords
Torus knot;(1, 1)-knot;(1, 1)-decomposition;Dunwoody 3-manifold;Alexander polynomial;Heegaard splitting;Heegaard diagram;
Language
English
Cited by
1.
The Dual and Mirror Images of the Dunwoody 3-Manifolds, International Journal of Mathematics and Mathematical Sciences, 2013, 2013, 1
References
1.
H. Aydin, I. Gultekin and M. Mulazzani, Torus knots and Dunwoody manifolds, Siberian Math. J., 45(2004), 1-6.

2.
J. W. Alexander, Topological invariants of knots and links, Amer. Math. Soc., 20(1923), 275-306.

3.
A. Cattabriga, The Alexander polynomial of (1, 1)-knots, J. Knot Theory its Ramifications, 15(9)(2006), 1119-1129.

4.
A. Cattabriga and M. Mulazzani, Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups, Math. Proc. Camb. Phil. Soc., 135(2003), 137-146.

5.
A. Cattabriga and M. Mulazzani, All strongly-cyclic branched coverings of (1,1)-knots are Dunwoody manifolds, J. London Math. Soc., 70(2004), 512-528.

6.
A. Cattabriga and M. Mulazzani, Representations of (1, 1)-knots, Fund. Math., 118(2005), 45-57.

7.
A. Cattabriga, M. Mulazzani and A. Vesnin, Complexity, Heegaard diagrams and generalized Dunwoody manifolds, J. Korean Math.Soc., 47(3)(2010), 585-599.

8.
A. Cavicchioli, B. Ruini and F. Spaggiari, On a conjecture of M.J.Dunwoody, Algebra Coll., 8(2)(2001), 169-218.

9.
R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Springer-Verlag, Berlin, (1963).

10.
M. J. Dunwoody, Cyclic presentations and 3-manifolds, Proc. Inter. Conf., Group-Korea '94, Walter de Gruyter, Berlin-New York (1995), 47-55.

11.
L. Grasselli and M. Mulazzani, Genus one 1-bridge knots and Dunwoody manifolds, Forum Math., 13(2001), 379-397.

12.
S. H. Kim, On spatial theta-curves with the same (\$Z_2\$ \${\oplus}\$ \$Z_2\$)-fold and 2-fold branched covering, Note di Mathematica, 23(1)(2004/2005), 111-122.

13.
S. H. Kim and Y. Kim, Torus knots and 3-manifolds, J. Knot Theory and its Ram., 13(8)(2004), 1103-1119.

14.
S. H. Kim and Y. Kim, On the 2-bridge knots of Dunwoody (1, 1)-knots, Bull. Korean Math. Soc., 48(1)(2011), 197-211.

15.
S. H. Kim and Y. Kim, On the generalized Dunwoody 3-manifolds, Osaka Journal of Mathematics, to appear.

16.
M. Mulazzani, Cyclic presentations of groups and cyclic branched coverings of (1,1)-knots, Bull. Korean Math. Soc., 40(1)(2003), 101-108.

17.
L. Neuwirth, An algorithm for the construction of 3-manifolds from 2-complexes, Proc. Camb. Phil. Soc., 64(1)(1968), 603-613.

18.
D. Rolfsen, Knots and Links, Math. Lect. Series, Vol.7, Berkeley, Publish or Perish inc., (1976).

19.
H. J. Song and S. H. Kim, Dunwoody 3-manifolds and (1, 1)-decomposiable knots, Proc. Workshop in pure math(edited by Jongsu Kim and Sungbok Hong), 19(2000), 193-211.

20.
V. G. Turaev, The Alexander polynomial of a three-dimensional manifold, USSR Sbornik, 26(1975), 313-329.

21.
V. G. Turaev, Introduction to Combinatorial Torsions, Birkhauser, Basel, Boston, Berlin, 2001.