Classification of string links up to self delta-moves and concordance

Yasuhara, Akira

doi:10.2140/agt.2009.9.265

Cited by 12 publications

(20 citation statements)

References 17 publications

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“…(We refer the reader to [7] or Yasuhara [26] for a precise definition of Milnor invariants .I / of string links.) Furthermore, Milnor invariants of length k are known to be finite-type invariants of degree k 1 for string links by Bar-Natan [1] and Lin [15].…”

Section: String Linksmentioning

confidence: 99%

Milnor invariants and the HOMFLYPT Polynomial

Meilhan

Yasuhara

2012

Geom. Topol.

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We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant x L .J / vanishes for any sequence J with length at most k , then any Milnor x -invariant x L .I / with length between 3 and 2k C 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the "first nonvanishing" Milnor invariants can be always represented as such a linear combination. 57M25, 57M27

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Section: String Linksmentioning

confidence: 99%

Milnor invariants and the HOMFLYPT Polynomial

Meilhan

Yasuhara

2012

Geom. Topol.

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“…We refer the reader to [12] or [35] for a precise definition of Milnor invariants µ(I) of string links. The smallest length Milnor invariants µ σ (ij) of a string link σ coincide with the linking numbers lk(σ i ,σ j ).…”

Section: Milnor Invariants Given Anmentioning

confidence: 99%

“…Finally, we apply some of the techniques developed in this paper to previous works by the authors [26,36]. We first give a classification of 2-string links up to self C 3 -moves and concordance, where a self C k -move is a C k -move with all k + 1 strands in a single component.…”

Section: Conjecture (Goussarov-habiromentioning

confidence: 99%

“…Habegger and Lin showed that Milnor invariants are actually well defined integer-valued invariants of string links [12], and that the indeterminacy in Milnor invariants of a link is equivalent to the indeterminacy in regarding it as the closure of a string link. We refer the reader to [12] or [36] for a precise definition of Milnor invariants μ(I ) of string links. The smallest length Milnor invariants μ σ (i j) of a string link σ coincide with the linking numbers lk(σ i ,σ j ).…”

Section: •1•4 Milnor Invariantsmentioning

confidence: 99%

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Characterization of finite type string link invariants of degree <5

Meilhan¹,

Yasuhara²

2010

Math. Proc. Camb. Phil. Soc.

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Abstract. In this paper, we give a complete set of finite type string link invariants of degree < 5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to C k -moves for k ≤ 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar characterization of finite type concordance invariants of degree < 6.

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“…Links up to self ∆-equivalence and concordance have been studied in [28], and [32]. Classification of string links up to self-∆ concordance is given by the last author [32]. In [27] and [28], the second author defined an equivalence relation, ∆-cobordism.…”

Section: ¢-Movementioning

confidence: 99%

Homotopy, Δ-equivalence and concordance for knots in the complement of a trivial link

Fleming

Shibuya

Tsukamoto

et al. 2010

Topology and its Applications

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Link-homotopy and self ∆-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self ∆-equivalent) to a trivial link. We study link-homotopy and self ∆-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and ∆-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be ∆-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to ∆-equivalence and concordance.

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Classification of string links up to self delta-moves and concordance

Cited by 12 publications

References 17 publications

Milnor invariants and the HOMFLYPT Polynomial

Milnor invariants and the HOMFLYPT Polynomial

Characterization of finite type string link invariants of degree <5

Homotopy, Δ-equivalence and concordance for knots in the complement of a trivial link

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