On the 2-Bridge Knots of Dunwoody (1, 1)-Knots

Abstract

Abstract. Every (1, 1)-knot is represented by a 4-tuple of integers (a, b, c, r), where a > 0, b ≥ 0, c ≥ 0, d = 2a+ b + c, r ∈ Z d , and it is well known that all 2-bridge knots and torus knots are (1, 1)-knots. In this paper, we describe some conditions for 4-tuples which determine 2 -bridge knots and determine all 4-tuples representing any given 2-bridge knot.

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

“…For example, both K(1, 3, 4, 7) and K(2, 1, 4, 4) represent the pretzel knot P (−2, 3, 7) which is a (1, 1)-knot as was mentioned in [19]. The subset of D representing all 2-bridge knots was determined by [11], [14] and [19]. However, the subset of D representing all torus knots is not yet determined completely.…”

“…Conversely, for a given (1, 1)-knot K, it is an interesting problem to determine a type of the Dunwoody 3-manifold representing K even if it is not unique. Until now these problems for all 2-bridge knots and some torus knots were solved in [11], [14] and [19]. For example, the explicit type for the torus knot T (p, q) satisfying q ≡ ±1 mod p has been obtained in [1] and [5], and for the torus knot T (p, q) satisfying q ≡ ±2 mod p, the type has been obtained in [13].…”

The Dual and Mirror Images of the Dunwoody 3-Manifolds

“…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 ( , , , ).…”

“…For example, (1, 3, 4, 7) and (2, 1, 4, 4) represent the pretzel knot (−2, 3, 7) as a (1, 1)-knot. 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].…”

“…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].…”