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

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 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.

The Dual and Mirror Images of the Dunwoody 3-Manifolds

“…In 1995, Dunwoody introduced the 6-tuples yielding a family of genus Heegaard diagrams of closed orientable 3manifolds called the Dunwoody 3-manifolds [4]. Moreover, the Dunwoody 3-manifolds are determined by the -fold strongly cyclic coverings of lens spaces branched over (1, 1)-knots , defined by the monodromy : 1 ( − ) → Z , where Z is the cyclic group of order , ≥ 2 [3,[5][6][7][8]. In fact, such branched sets in the quotient spaces of the Dunwoody 3-manifolds by a cyclic action of order are representing the (1, 1)-knots in lens spaces (possibly the 3sphere) [7,9], and some classes of such knots represented by the Dunwoody 3-manifolds contain all (1, 1)-knots in S 3 [6].…”

“…was introduced in [3,8,10] (see Figure 1), where each cycle of corresponds to the end points of line segments in the Heegaard diagram as in Figure 1, and each cycle of corresponds to a pair of end points which is identified in forming the handlebody 1 . For each ( , , , ) ∈ D, we denote the Dunwoody (1, 1)-decomposition of ( , ) by ( , , , ) and the Dunwoody (1, 1)-knot , represented from ( , , , ), by ( , , , ).…”

“…which forms exactly an orbit of [8]. It is well known that all (1, 1)-knots in S 3 are to be the Dunwoody (1, 1)-knots, but the representation of (1, 1)-knots by 4-tuples ( , , , ) is not unique.…”

“…However all types of the Dunwoody (1, 1)-decompositions representing all 2bridge knots were determined completely in [7,10], and moreover, the types of the Dunwoody (1, 1)-decompositions representing the certain class of torus knots are given in [6,11,12]. For each ≥ 2, the 6-tuples ( , , , , , ) satisfying conditions + ≡ 0 mod and ( , , , ) ∈ D induce the Dunwoody 3-manifolds, denoted by ( , , , , ), as closed orientable 3-manifolds, where , , and are some integers defined in [4,8]. Thus, the Dunwoody 3-manifold ( , , , , ) is the -fold strongly cyclic covering space of a lens space branched over the Dunwoody (1, 1) knot ( , , , ).…”

“…Therefore, the concept of the Dunwoody 3-manifolds is important in knots, branched coverings, and graph theories. Moreover, recent manuscripts of the Dunwoody 3-manifolds can be found in [3,8,10,12,[16][17][18][19].…”