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.

Keywords: Torus knot, (1, 1)-knot, (1, 1)-decomposition, Dunwoody $3$-manifold, Alexander polynomial, Heegaard splitting, Heegaard diagram