## On Finite Type 3-Manifold Invariants I (1996)

Venue: | II, Math. Annalen |

Citations: | 32 - 9 self |

### BibTeX

@ARTICLE{Garoufalidis96onfinite,

author = {Stavros Garoufalidis},

title = {On Finite Type 3-Manifold Invariants I},

journal = {II, Math. Annalen},

year = {1996},

volume = {5},

pages = {691--718}

}

### OpenURL

### Abstract

. Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3-spheres (ZHS for short). In the present paper we propose another definition of finite type invariants of ZHS and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra and are of finite type in in the sense of Ohtsuki and thus conclude that the associated graded algebra is a priori finite dimensional in each degree. We discover a new set of restrictions that Ohtsuki's invariants satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3-manifold invariants with the (Vassiliev) knot invariants. Contents 1. Introduction 2 1.1. History 1.2. A review of Ohtsuki's definition 1.3. Variations for finite type 3-manifold invariants 1.4. Statement of the results 1.5. Plan of the proof 1.6. Acknowledgmen...

### Citations

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Citation Context ...ly and combinatorially) knot and 3-manifold invariants. A unifying approach to these invariants is the concept of a topological quantum field theory (TQFT for short) in 2 + 1 dimensions, [At]. Witten =-=[Wi2]-=-, using path integrals with a Chern-Simons action (a not yet defined infinite dimensional integration) gave examples of such theories depending on a semisimple compact Lie group and an integer. Shortl... |

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Citation Context ...f knots in S 3 one has: ffl an axiomatic definition, ffl a general existence theorem [B-N1], [Ko1], [LM], ffl a comparison theorem to the above mentioned nonperturbative kont invariants [B-N1], [Dr], =-=[Ka]-=-, and to the Chern-Simons theory perturbative knot invariants [BT], and finally ffl ways of calculating them, from combinatorics of chord diagrams [B-N2]. The situation with perturbative (or finite ty... |

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Citation Context ...rn-Simons action (a not yet defined infinite dimensional integration) gave examples of such theories depending on a semisimple compact Lie group and an integer. Shortly afterwards, Reshetikhin-Turaev =-=[RT1]-=-, [RT2], (and simultaneously many other authors [Koh], [Ku], [KR], [Po], [TW]), used equivalent initial data (namely semisimple Lie algebra and a primitive complex root of unity) as in Witten's ChernS... |

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Citation Context ...ants of knots in S 3 one has: ffl an axiomatic definition, ffl a general existence theorem [B-N1], [Ko1], [LM], ffl a comparison theorem to the above mentioned nonperturbative kont invariants [B-N1], =-=[Dr]-=-, [Ka], and to the Chern-Simons theory perturbative knot invariants [BT], and finally ffl ways of calculating them, from combinatorics of chord diagrams [B-N2]. The situation with perturbative (or fin... |

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Citation Context ... (geometrically and combinatorially) knot and 3-manifold invariants. A unifying approach to these invariants is the concept of a topological quantum field theory (TQFT for short) in 2 + 1 dimensions, =-=[At]-=-. Witten [Wi2], using path integrals with a Chern-Simons action (a not yet defined infinite dimensional integration) gave examples of such theories depending on a semisimple compact Lie group and an i... |

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Citation Context ..., [RT1]. Examples of nonperturbative 3-manifold invariants are the Reshetikhin-Turaev invariants [RT2]. Examples of perturbative (or finite type) knot invariants are the Vassiliev invariants, [B-N1], =-=[BL]-=-, [Va]. For the Vassiliev invariants of knots in S 3 one has: ffl an axiomatic definition, ffl a general existence theorem [B-N1], [Ko1], [LM], ffl a comparison theorem to the above mentioned nonpertu... |

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Citation Context ...pe) knot invariants are the Vassiliev invariants, [B-N1], [BL], [Va]. For the Vassiliev invariants of knots in S 3 one has: ffl an axiomatic definition, ffl a general existence theorem [B-N1], [Ko1], =-=[LM]-=-, ffl a comparison theorem to the above mentioned nonperturbative kont invariants [B-N1], [Dr], [Ka], and to the Chern-Simons theory perturbative knot invariants [BT], and finally ffl ways of calculat... |

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Citation Context ...nite type) knot invariants are the Vassiliev invariants, [B-N1], [BL], [Va]. For the Vassiliev invariants of knots in S 3 one has: ffl an axiomatic definition, ffl a general existence theorem [B-N1], =-=[Ko1]-=-, [LM], ffl a comparison theorem to the above mentioned nonperturbative kont invariants [B-N1], [Dr], [Ka], and to the Chern-Simons theory perturbative knot invariants [BT], and finally ffl ways of ca... |

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Citation Context ...mples of such theories depending on a semisimple compact Lie group and an integer. Shortly afterwards, Reshetikhin-Turaev [RT1], [RT2], (and simultaneously many other authors [Koh], [Ku], [KR], [Po], =-=[TW]-=-), used equivalent initial data (namely semisimple Lie algebra and a primitive complex root of unity) as in Witten's ChernSimons theory and combinatorially defined TQFT in 2 + 1 dimensions. TQFT s in ... |

40 | An extension of Casson’s invariant - Walker |

33 | Quantum SU(2)-invariants dominate Casson’s SU(2)-invariant.Math - Murakami - 1994 |

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Topological invariants for 3-manifolds using representations of mapping class groups I. Topology 31
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Citation Context ...nal integration) gave examples of such theories depending on a semisimple compact Lie group and an integer. Shortly afterwards, Reshetikhin-Turaev [RT1], [RT2], (and simultaneously many other authors =-=[Koh]-=-, [Ku], [KR], [Po], [TW]), used equivalent initial data (namely semisimple Lie algebra and a primitive complex root of unity) as in Witten's ChernSimons theory and combinatorially defined TQFT in 2 + ... |

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Citation Context ...egration) gave examples of such theories depending on a semisimple compact Lie group and an integer. Shortly afterwards, Reshetikhin-Turaev [RT1], [RT2], (and simultaneously many other authors [Koh], =-=[Ku]-=-, [KR], [Po], [TW]), used equivalent initial data (namely semisimple Lie algebra and a primitive complex root of unity) as in Witten's ChernSimons theory and combinatorially defined TQFT in 2 + 1 dime... |

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Citation Context ...on) gave examples of such theories depending on a semisimple compact Lie group and an integer. Shortly afterwards, Reshetikhin-Turaev [RT1], [RT2], (and simultaneously many other authors [Koh], [Ku], =-=[KR]-=-, [Po], [TW]), used equivalent initial data (namely semisimple Lie algebra and a primitive complex root of unity) as in Witten's ChernSimons theory and combinatorially defined TQFT in 2 + 1 dimensions... |

17 |
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(Show Context)
Citation Context ...]. Examples of nonperturbative 3-manifold invariants are the Reshetikhin-Turaev invariants [RT2]. Examples of perturbative (or finite type) knot invariants are the Vassiliev invariants, [B-N1], [BL], =-=[Va]-=-. For the Vassiliev invariants of knots in S 3 one has: ffl an axiomatic definition, ffl a general existence theorem [B-N1], [Ko1], [LM], ffl a comparison theorem to the above mentioned nonperturbativ... |

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of 3-manifolds via link polynomials and quantum
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Citation Context ...ns action (a not yet defined infinite dimensional integration) gave examples of such theories depending on a semisimple compact Lie group and an integer. Shortly afterwards, Reshetikhin-Turaev [RT1], =-=[RT2]-=-, (and simultaneously many other authors [Koh], [Ku], [KR], [Po], [TW]), used equivalent initial data (namely semisimple Lie algebra and a primitive complex root of unity) as in Witten's ChernSimons t... |

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Citation Context ...4 5 dimGmW 1 0 1 1 3 4 Let us denote by \Delta (m) (K) := d m dhm j h=0 \Delta(K )(e h ) the m th derivative of the AlexanderConway polynomial \Delta(K ) of a knot K [Ro] with the normalization as in =-=[B-NG]-=-, example 2:8. It is clear that \Delta (m) 2 FmV. Let us denote by F Special m V the vector space of special Vassiliev invariants, i.e., these whose degree m part (i.e., whose image in GmV) consists o... |

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homotopical algebra and low-dimensional topology
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Citation Context ...nal homology 3-sphere and ae 2 Hom( 1 (M); G) ON FINITE TYPE 3-MANIFOLD INVARIANTS I 3 (G is a fixed compact semisimple Lie group here). In cases of acyclic ae one has such invariants [AxS1], [AxS2], =-=[Ko2]-=-. However, these invariants do not solve any of the above mentioned problems,essentially due to the absence of surgery formulas. 1.2. A review of Ohtsuki's definition. On the other hand, Ohtsuki [Oh1]... |

4 |
A polynomial invariant of integral homology spheres
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Citation Context ...[Ko2]. However, these invariants do not solve any of the above mentioned problems,essentially due to the absence of surgery formulas. 1.2. A review of Ohtsuki's definition. On the other hand, Ohtsuki =-=[Oh1]-=- recently defined finite type invariants for integral homology 3-spheres (ZHS for short). His definition was inspired by the notion of finite type knot invariants. Let us review his definition and int... |

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On the Vassiliev Knot Invariants Preprint
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Citation Context ...of knots, [RT1]. Examples of nonperturbative 3-manifold invariants are the Reshetikhin-Turaev invariants [RT2]. Examples of perturbative (or finite type) knot invariants are the Vassiliev invariants, =-=[B-N1]-=-, [BL], [Va]. For the Vassiliev invariants of knots in S 3 one has: ffl an axiomatic definition, ffl a general existence theorem [B-N1], [Ko1], [LM], ffl a comparison theorem to the above mentioned no... |

3 |
Surgery on knots
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Citation Context ... type 3 knot invariant satisfying the assumptions of proposition 1.4. Figure 15 shows two knots K 3 and K 4 with the property that \Gamma1 surgery on them gives diffeomorphic ZHS. The knots appear in =-=[Li]-=- as an example of distinct knots in S 3 whose \Gamma1 surgery gives diffeomorphicsZhomology spheres. We are indebted to R. Kirby for pointing out this reference to us. Since (K \Gamma1 3 ; S 3 ) and (... |

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Perturbative aspects of Chern-Simons topological quantum field theory
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Citation Context ... a rational homology 3-sphere and ae 2 Hom( 1 (M); G) ON FINITE TYPE 3-MANIFOLD INVARIANTS I 3 (G is a fixed compact semisimple Lie group here). In cases of acyclic ae one has such invariants [AxS1], =-=[AxS2]-=-, [Ko2]. However, these invariants do not solve any of the above mentioned problems,essentially due to the absence of surgery formulas. 1.2. A review of Ohtsuki's definition. On the other hand, Ohtsuk... |

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Chern-Simons pertrubation theory
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Citation Context ...ere M is a rational homology 3-sphere and ae 2 Hom( 1 (M); G) ON FINITE TYPE 3-MANIFOLD INVARIANTS I 3 (G is a fixed compact semisimple Lie group here). In cases of acyclic ae one has such invariants =-=[AxS1]-=-, [AxS2], [Ko2]. However, these invariants do not solve any of the above mentioned problems,essentially due to the absence of surgery formulas. 1.2. A review of Ohtsuki's definition. On the other hand... |

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Citation Context ... existence theorem [B-N1], [Ko1], [LM], ffl a comparison theorem to the above mentioned nonperturbative kont invariants [B-N1], [Dr], [Ka], and to the Chern-Simons theory perturbative knot invariants =-=[BT]-=-, and finally ffl ways of calculating them, from combinatorics of chord diagrams [B-N2]. The situation with perturbative (or finite type) 3-manifold invariants is puzzling. On the one hand perturbativ... |

1 |
poly.c, available at millett@math.ucsb.edu
- Ewing, Millett
(Show Context)
Citation Context ...at J (2) (K 3 ) = J (2) (K 4 ) = \Gamma6 (this is not a surprise, since the Casson invariant exists!) but J (3) (K 3 ) = 0 6= J (3) (K 4 ) = \Gamma180. Therefore, 3 There are various programs [B-N2], =-=[EM]-=-, [Och] that calculate the Jones polynomial of knots. As a check, we used all of the above mentioned and got the same results. We thank D. Dar-Natan, L. Kauffman, K. Millet and M. Ochiai for their hel... |

1 |
Knot theory by computer, available at ochai@ics.nara-wu.ac.jp
- Ochiai
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Citation Context ...2) (K 3 ) = J (2) (K 4 ) = \Gamma6 (this is not a surprise, since the Casson invariant exists!) but J (3) (K 3 ) = 0 6= J (3) (K 4 ) = \Gamma180. Therefore, 3 There are various programs [B-N2], [EM], =-=[Och]-=- that calculate the Jones polynomial of knots. As a check, we used all of the above mentioned and got the same results. We thank D. Dar-Natan, L. Kauffman, K. Millet and M. Ochiai for their help in di... |