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We consider a family of words in a free group of rank n which determine 3-manifolds Mn(p, q). We prove that the fundamental groups of Mn(p, q) are cyclically presented, and that Mn(p, q) is the n-fold cyclic covering of the 3-sphere branched over the torus knots T (p, q) if p is odd and q ≡ ±2(mod p). We also obtain an explicit Dunwoody parameters for the torus knots T (p, q) for odd p and q ≡ ±2(mod p).

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 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.

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