Two Dimensional Arrays for Alexander Polynomials of Torus Knots

Abstract

Abstract. Given a pair p, q of relative prime positive integers, we have uniquely determined positive integers x, y, u and v such that vx − uy = 1, p = x + y and q = u + v. Using this property, we show that 1≤i≤x,1≤j≤vis the Alexander polynomial ∆p,q(t) of a torus knot t(p, q). Hence the number Np,q of non-zero terms of ∆p,q(t) is equal to vx + uy = 2vx − 1.Owing to well known results in knot Floer homology theory, our expanding formula of the Alexander polynomial of a torus knot provides a method of algorithmically determining the total rank of its knot Floer homology or equivalently the complexity of its (1,1)-diagram. In particular we prove (see Corollary 2.8);Let q be a positive integer> 1 and let k be a positive integer. Then we have(1)(k + 1)(q 2 − 1) − q where we further assume q is odd in formula (3) and (4).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.

Cable knots are not thin

“…Corollary 2.6 in [29] states that the number of non-zero terms in the Alexander polynomial of T p,q is 2vx − 1, where x, y, u, v are unique positive integers such that p = x + y, q = u + v such that vx − uy = 1. Now since T p,q are L-space knots (since positive surgery along torus knots with certain coefficient produces lens space, by [19]), Ozsváth and Szabó showed in [24] that the L-space knots forms a 'staircase' complex and hence each such j for which HFK (K, j) = 0 is of dimension 1 i.e dim F ( HFK (T p,q ) is equal the number of non-zero terms in ∆ p,q , which is 2vx − 1.…”

“…Now comparing gr(a ⊗ ξ 2 ) and gr(b ⊗ µ 1 ), we get that Equality (3) happens iff vx = −1. Now dim( HF K(S 3 , T p,q )) = 2vx − 1 ≥ 3 (see [29,Corollary 2.6]). That implies that vx should be greater than or equal to 2.…”