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Diassociative Algebras and Milnor’s Invariants for Tangles

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Abstract

We extend Milnor’s μ-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for μ-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday’s diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.

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Authors and Affiliations

  1. Université de Lyon, Université Lyon 1, ICJ, UMR 5208 CNRS, 43 blvd 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Olga Kravchenko

  2. Department of Mathematics, Technion, Haifa, 32000, Israel

    Michael Polyak

Authors
  1. Olga Kravchenko
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  2. Michael Polyak
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Correspondence to Michael Polyak.

Additional information

The second author was partially supported by the ISF grant 1343/10.

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Kravchenko, O., Polyak, M. Diassociative Algebras and Milnor’s Invariants for Tangles. Lett Math Phys 95, 297–316 (2011). https://doi.org/10.1007/s11005-010-0459-4

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  • DOI: https://doi.org/10.1007/s11005-010-0459-4

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