The Dual and Mirror Images of the Dunwoody 3-Manifolds

الملخص EN

Recently, in 2013, we proved that certain presentations present the Dunwoody 3-manifold groups.

Since the Dunwoody 3-manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody 3-manifolds.

It is well known that every (1,1)-knots in a lens space can be represented by the set ? of the 4-tuples (a,b,c,r) (Cattabriga and Mulazzani (2004); S.

H.

Kim and Y.

Kim (2012, 2013)).

In particular, to determine a unique Heegaard diagram of the Dunwoody 3-manifolds, we proved the fact that the certain subset of ? representing all 2-bridge knots of (1,1)-knots is determined completely by using the dual and mirror (1,1)-decompositions (S.

H.

Kim and Y.

Kim (2011)).

In this paper, we show how to obtain the dual and mirror images of all elements in ? as the generalization of some results by Grasselli and Mulazzani (2001); S.

H.

Kim and Y.

Kim (2011).