Homotopy classification of ribbon tubes and welded string links

Audoux, Benjamin; Bellingeri, Paolo; Meilhan, Jean–Baptiste; Wagner, Emmanuel

doi:10.2422/2036-2145.201507_003

Cited by 27 publications

(100 citation statements)

References 45 publications

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“…As we shall explain in Section 7.1, the more general case of 1-dimensional cut-diagrams relates to the theory of welded knotted objects. Then, the case where Σ is a union of circles recovers the welded extension of Milnor's link invariant of [8,22], while the case of intervals recovers Milnor invariants of welded string links defined in [2], see also [15].…”

Section: 3mentioning

confidence: 99%

“…In terms of link diagrams, the 1-dimensional move amounts to replacing a classical crossing of two strands of a same component, by a virtual one. This is the self-virtualization move, which is known to imply link-homotopy for classical links, see [2]. In dimension 2, Roseman moves have been extended to the case of self-singular surfaces in [4], where three self-singular Roseman moves were introduced.…”

Section: Local Moves For Cut-diagramsmentioning

confidence: 99%

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Milnor concordance invariant for knotted surfaces and beyond

Audoux,

Meilhan,

Yasuhara

2021

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We generalize Milnor link invariants to all types of knotted surfaces in 4-space, and more generally to all codimension 2 embeddings. This is achieved by using the notion of cut-diagram, which is a higher dimensional generalization of Gauss diagrams, associated to codimension 2 embeddings in Euclidian spaces. We define a notion of group for cut-diagrams, which generalizes the fundamental group of the complement, and we extract Milnor-type invariants from the successive nilpotent quotients of this group. We show that the latter are invariant under an equivalence relation called cut-concordance, which encompasses the topological notion of concordance. We give several concrete applications of the resulting Milnor concordance invariants for knotted surfaces, comparing their relative strength with previously known concordance invariants, and providing realization results. We also obtain classification results for Spun links up to link-homotopy, as well as a criterion for a knotted surface to be ribbon. Finally, the theory of cut-diagrams is further investigated, heading towards a combinatorial approach to the study of surfaces in 4-space.

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Section: 3mentioning

confidence: 99%

Section: Local Moves For Cut-diagramsmentioning

confidence: 99%

Milnor concordance invariant for knotted surfaces and beyond

Audoux,

Meilhan,

Yasuhara

2021

Preprint

Self Cite

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“…Indeed the characterization will be made through the use of welded objects, which can be seen as a diagrammatical intermediary between classical knots and (the ribbon subclass of) knotted surfaces in R 4 . In [1], B. Audoux, P. Bellingeri, J-B. Meilhan and E. Wagner gave a diagrammatical characterization of the non-repeating Milnor invariants in terms of link-homotopy on welded string links, which consists in allowing strands to cross themselves.…”

Section: Introductionmentioning

confidence: 99%

A diagrammatical characterization of Milnor invariants

Colombari¹

2022

Preprint

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The goal of this paper is to give a diagrammatical characterization of the information given by the Milnor invariants of links and string links. More precisely, we describe when two string links have equal Milnor invariants of length ≤ q and when a link has trivial Milnor invariants of lenght ≤ q. This will be done through the use of welded knot theory, involving the notions of arrow calculus and wq-concordance introduced by J-B. Meilhan and A. Yasuhara. These results is to be compared to the classification of links up to Cq-concordance obtained by J. Conant, R. Schneiderman and P. Teichner.

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“…We can verify that µ σ (112) = 1 by using a computer program written by Y. Takabatake, T. Kuboyama and H. Sakamoto [29]. (1) While σ is 4-move equivalent to 1 2 , µ σ (112) is not congruent to 0 modulo 2.…”

Section: Milnor Isotopy Invariants and 2p-movesmentioning

confidence: 99%

“…≡ 1 + (terms of degree p). (1) Using the technique of "grammar compression", Takabatake, Kuboyama and Sakamoto [29] made a computer program in the program language C++, based on Milnor's algorithm, which is able to give us µ-invariants of length 16.…”

Section: Milnor Isotopy Invariants and 2p-movesmentioning

confidence: 99%