Homfly polynomials as vassiliev link invariants

Abstract

We prove that the number of linearly independent Vassiliev invariants for an r-component link of order n, which derived from the HOMFLY polynomial, is greater than or equal to min{n, [(n+ r − 1)/2]}. Introduction. Let Vn denote the vector space consisting of all Vassiliev knot invariants of order less than or equal to n. There is a filtration V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn ⊂ · · · in the entire space of Vassiliev knot invariants. Each Vn is finite-dimensional. Vassiliev [V] studied for the special cases when n is small: V0 = V1, which consists of a constant map (Propositions 3 and 5), and V2/V1 is a one-dimensional vector space, whose basis is the second coefficient of the Conway polynomial. The dimensions for small n are found by using the computer by Bar-Natan and Stanford (cf. [BN; B1, p. 282 ]): For n = 1, 2, 3, 4, 5, 6, 7, dimVn/Vn−1 = 0, 1, 1, 3, 4, 9, 14, respectively. On the other hand, Bar-Natan (cf. [BN]) showed that the nth coefficient of the Conway polynomial is of order less than or equal to n. Birman and Lin [BL] and Gusarov [G] proved that the Jones, HOMFLY, and Kauffman polynomials of a knot can be interpreted as an infinite sequence of Vassiliev knot invariants, and as a corollary they proved that dimVn/Vn−1 ≥ 1 for every n ≥ 2 using the HOMFLY polynomial [BL, Corollary 4.2 (i)]. 1991 Mathematics Subject Classification: Primary 57M25.