This paper compares the definitions of finite-type invariants due to Ohtsuki
and to Garoufalidis, showing that, residually, type 3m of the former equals
type m of the latter. It also shows that type 2m Ohtsuki invariants define knot
invariants of type 3m (first proved by Habegger).Comment: 13 pages, Postscrip

Garoufalidis, Stavros

Levine, Jerome

English

arXiv

. The present paper is a continuation of [Ga], [GL1] and [GO]. Using a key Lemma we compare the two currently existing definitions of finite type invariants of oriented integral homology spheres and show that type 3m invariants in the sense of Ohtsuki [Oh] are included in type m invariants in the sense of the first author [Ga]. This partially answers question 1 of [Ga]. We show that type 3m invariants of integral homology spheres in the sense of Ohtsuki map to type 2m invariants of knots in S  3  , thus answering question 2 from [Ga]. Contents  1. Introduction 2  1.1. Definitions 1.2. Statement of the results 1.3. Questions 1.4. Plan of the proof 1.5. Acknowledgement  2. A key Lemma 4 3. Proof of Theorems 1 and 2 6  3.1. Proof of Theorem 1 3.2. Proof of Theorem 2  4. Proof of Theorem 3 12  The authors were partially supported by NSF grants DMS-95-05106 and DMS-93-03489. This and related preprints can also be obtained by accessing the WEB in the address http://www.math. brown.edu/¸stavr..

On Finite Type 3-manifold invariants IV: Comparison of Definitions

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