Splitting the spectral flow and the SU(3) Casson invariant for spliced sums

Bodén, Hans; Himpel, Benjamin

doi:10.2140/agt.2009.9.865

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…. We note that the same non-additivity formula is satisfied for the "SU(3) Casson invariant" in the case when K 1 and K 2 are torus knots in S 3 [BH09], but the relation between this gauge-theoretical invariant and finite-type invariants does not seem to be known yet. To deduce from Theorem 3.1 the above splicing formulas for λ W and λ 2 , we need first to determine the low-degree terms of the wheeled Kontsevich-LMO invariant Z .…”

Section: Low Degree Formulasmentioning

confidence: 79%

A splicing formula for the LMO invariant

Massuyeau¹,

Moussard²

2020

Can. J. Math.-J. Can. Math.

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We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology 3-spheres. Specifically, if a rational homology 3sphere M is obtained by gluing the exteriors of two framed knots K1 ⊂ M1 and K2 ⊂ M2 in rational homology 3-spheres, our formula expresses the LMO invariant of M in terms of the Kontsevich-LMO invariants of (M1, K1) and (M2, K2). The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's formula for the Casson-Walker invariant and we observe that the second term of the Ohtsuki series is not additive under "standard" splicing. The splicing formula also works when each Mi comes with a link Li in addition to the knot Ki, hence we get a "satellite formula" for the Kontsevich-LMO invariant. Contents 1. Introduction 1 2. Splicing and surgery 3 3. Splicing formula 7 4. Low degree formulas 13 5. Splicing and satellite operations 16 6. The case of knots that are not trivial in homology 19 Appendix A. Signatures of tridiagonal matrices 22 References 25

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Section: Low Degree Formulasmentioning

confidence: 79%

A splicing formula for the LMO invariant

Massuyeau¹,

Moussard²

2020

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

show abstract

“…(1). We note that the same non-additivity formula is satisfied for the "SU(3) Casson invariant" in the case when K 1 and K 2 are torus knots in S 3 [BH09], but the relation between this gauge-theoretical invariant and finite-type invariants does not seem to be known yet.…”

Section: Low Degree Formulasmentioning

confidence: 79%

A splicing formula for the LMO invariant

Massuyeau¹,

Moussard²

2020

Preprint

View full text Add to dashboard Cite

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology 3-spheres. Specifically, if a rational homology 3sphere M is obtained by gluing the exteriors of two framed knots K1 ⊂ M1 and K2 ⊂ M2 in rational homology 3-spheres, our formula expresses the LMO invariant of M in terms of the Kontsevich-LMO invariants of (M1, K1) and (M2, K2). The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's formula for the Casson-Walker invariant and we observe that the second term of the Ohtsuki series is not additive under "standard" splicing. The splicing formula also works when each Mi comes with a link Li in addition to the knot Ki, hence we get a "satellite formula" for the Kontsevich-LMO invariant. Contents 1. Introduction 1 2. Splicing and surgery 3 3. Splicing formula 7 4. Low degree formulas 12 5. Splicing and satellite operations 15 6. The case of knots that are not trivial in homology 18 Appendix A. Signatures of tridiagonal matrices 21 References 24

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Splitting the spectral flow and the SU(3) Casson invariant for spliced sums

Abstract: We show that the SU.3/ Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU.2/ Casson knot invariants. The key step is a splitting formula for su.n/ spectral flow for closed 3-manifolds split along a torus.

Cited by 2 publications

References 17 publications

A splicing formula for the LMO invariant

A splicing formula for the LMO invariant

A splicing formula for the LMO invariant

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