2 Ju l 1 99 9 BRAID COMMUTATORS AND DELTA FINITE - TYPE INVARIANTS

Delta finite-type invariants are defined analogously to finite-type invariants, using delta moves instead of crossing changes. We show that they are closely related to the lower central series of the commutator subgroup of the pure braid group. 0. INTRODUCTION We consider in this paper delta finite-type invariants of knots and links (∆FT invari-ants). In the case of links, these are the same invariants as defined by Mellor [4]. We shall prove some properties of ∆FT invariants which closely resemble properties of finite-type invariants in the usual sense (FT invariants). In the same way that FT invariants are based on sets of crossing changes in knot or link diagrams, ∆FT invariants are based on sets of delta moves in knot or link diagrams. In the same way that FT invariants are closely related to γ n (P), the the lower central series of the pure braid group P , ∆FT invariants are closely related to γ n (P ′), the lower central series of the commutator subgroup of P. Here P = P k , the pure braid group on k strands. We will show that two knots have matching ∆FT invariants of order < n if and only if they are equivalent modulo γ n (P ′), just as two knots have matching FT invariants of order < n if and only if they are equivalent modulo γ n (P) (see [8]). Since γ n (P ′) ⊂ γ 2n (P), it follows that an FT invariant of order < 2n is a ∆FT invariant of order < n. It may turn out that all ∆FT invariants of order < n occur this way. However, there is a difference between γ n (P) and γ n (P ′) which may make the ∆FT invariants more than just a relabeling of the FT invariants: For any k and any n, the quotient group γ n (P k)/γ n+1 (P k) is finitely-generated, whereas even γ 1 (P ′ 3)/γ 2 (P ′ 3) is not finitely-generated. For links of more than one component, there are known advantages to working with ∆FT invariants instead of FT invariants. Murakami and Nakanishi [5] showed that two links are equivalent by a sequence of delta moves if and only if they have the same pairwise linking numbers. Thus, ∆FT invariants detect linking number right at order 0, and so essentially each link homology class gets …