×

Milnor-Orr invariants from the Kontsevich invariant. (English) Zbl 1445.57004

In comparing the Milnor \(\mu\)-invariants, the Orr invariant and the Kontsevich invariant, the paper starts out by showing, for string links, the equivalence of the Milnor \(\mu\)-invariant of degree \(< \, 2k\) and the Orr invariant of degree \(k\), assuming a certain lifting property for the abelianization of the fundamental group of the link exterior. Next, for a based string link satisfying the lifting property above, the equivalence of the Orr invariant of degree \(k\) and the tree reduction of the Kontsevich invariant of degree \(< \, 2k\) is proved. A technical key role in all this is played by the kernel of a commuting operator in the context of abstract free groups whose generators correspond to meridians. Finally, an appendix displays a method to compute the Orr invariant from HOMFLYPT polynomials, a result based on work of J.-B. Meilhan and A. Yasuhara [Geom. Topol. 16, No. 2, 889–917 (2012; Zbl 1260.57026)].

MSC:

57K10 Knot theory
57K14 Knot polynomials
20F18 Nilpotent groups
55S30 Massey products
57M05 Fundamental group, presentations, free differential calculus
57K31 Invariants of 3-manifolds (including skein modules, character varieties)

Citations:

Zbl 1260.57026
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. C. Cha, Rational Whitney tower filtration of links, Math. Ann.379(2018) 963-992. Zbl 1404.57040MR 3770161 · Zbl 1404.57040
[2] S. Chmutov, S. Duzhin and J. Mostovoy,Introduction to Vassiliev knot invariants, Cambridge University Press, 2012.Zbl 1245.57003MR 2962302 · Zbl 1245.57003
[3] S. Garoufalidis and J. P. Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, inGraphs and patterns in mathematics and theoretical physics, Proceedings of Symposia in Pure Mathematics 73, American Mathematical Society, Providence, RI, 2005, 173-203.Zbl 1086.57013MR 2131016 · Zbl 1086.57013
[4] N. Habegger and G. Masbaum, The Kontsevich integral and Milnor’s invariants, Topology39(2000), 1253-1289.Zbl 0964.57011MR 1783857 · Zbl 0964.57011
[5] K. Igusa and K. Orr, Links, pictures and the homology of nilpotent groups, Topology 40(2001), 1125-1166.Zbl 1002.57012MR 1867241 · Zbl 1002.57012
[6] N. Kawazumi, Cohomological aspects of Magnus expansions,arXiv:math.GT/0505497 (2006).
[7] H. Kodani, Group-like expansions and invariants of string links,arXiv:1604.03213(2016).
[8] H. Kodani and T. Nosaka, Milnor invariants via unipotent Magnus embeddings, arXiv:1709.07335(2017). · Zbl 1432.57022
[9] J. P. Levine, The ¯µ-invariants of based links, inDifferential topology (Siegen, 1988), Lecture Notes in Mathematics 1350, Springer, Berlin, 1988, 87-103.Zbl 0677.57004 MR 0979334
[10] J. P. Levine, Addendum and correction to: Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol.2(2002) 1197-1204 (electronic). Zbl 1065.57501MR 1943338 · Zbl 1065.57501
[11] G. Massuyeau, Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant, Bull. Soc. Math. France140(2012), 101-161.Zbl 1248.57009MR 2903772 · Zbl 1248.57009
[12] G. Massuyeau, Formal descriptions of Turaev’s loop operations, Quantum Topol.9 (2018), 39-117.Zbl 1393.57016MR 3760878 · Zbl 1393.57016
[13] J. B. Meilhan and A. Yasuhara, Milnor invariants and the HOMFLYPT polynomial, Geom. Topol16(2012), 889-917.Zbl 1260.57026MR 2928984 · Zbl 1260.57026
[14] J. W. Milnor, Link groups, Ann. of Math. (2)59(1954) 177-195.Zbl 0055.16901 MR 0071020 · Zbl 0055.16901
[15] J. W. Milnor, Isotopy of links, inAlgebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton Mathematical Series 12, Princeton University Press, Princeton, NJ, 1957, 280-306.Zbl 0080.16901MR 0092150 · Zbl 0077.16602
[16] T. Nosaka, Cocycles of nilpotent quotients of free groups,arXiv:1706.01189(2018). · Zbl 1442.55014
[17] K. Orr, Homotopy invariants of links, Invent. Math.95(1989), 379-394.Zbl 0668.57014 MR 0974908 · Zbl 0668.57014
[18] A.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.