Torus Knots and 3-Manifolds

Abstract

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).

Two Dimensional Arrays for Alexander Polynomials of Torus Knots

“…Consequently we confirm that the complexities of (1,1)-diagrams of torus knots of type t(kq + 2, q) and t(kq + q − 2, q) in [5] agree with N kq+2,q and N kq+q−2,q respectively.…”

“…Through our expanding formula of ∆ p,q (t), we prove in Proposition 2.1 the Carlitz's four formulas stated without proof in [1]. Finally we show that the complexities of (1,1)-diagrams of torus knots t(p, q) in [5] agree with N p,q in Corolary 2.8.…”

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

“…We here determine a type of 4-tuples representing all 2-bridge knots and their dual and mirror images from a different point of view. We also recall that a type of 4-tuples representing the torus knot T (p, q) was determined in [1] and [15] when either q ≡ ±1 mod p or q ≡ ±2 mod p.…”

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

“…However, the subset of D representing all torus knots is not yet determined completely. In [1], [5], [13] and [6] we have Dunwoody (1, 1)-decompositions representing the certain class of torus knots.…”

“…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]. Furthermore, in [6], it has been obtained for all torus knots with bridge number at most three.…”