Milnor invariants and the HOMFLYPT Polynomial

Meilhan, Jean Baptiste; Yasuhara, Akira

doi:10.2140/gt.2012.16.889

Cited by 18 publications

(39 citation statements)

References 22 publications

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“…We can generalize Theorems 1.1 and 1.2 about all repeated sequences by the same arguments as those in [5,Introduction]. That is, we have formulae for not only Milnor link-homotopy invariants but also Milnor isotopy invariants.…”

Section: Introductionmentioning

confidence: 86%

“…With the same assumption as in Theorem 1.1, the same formula but δ L (I) = 0 holds for a sequence I with 3 ≤ |I| ≤ 2k + 1 [5], in which a sign (−1) 2k+1 of Theorem 1.1 seems to be missing. (See Remark 2.3.…”

Section: Introductionmentioning

confidence: 97%

“…In [8], M. Polyak gave a formula expressing Milnor invariant of length 3, and in [5], J.-B. Meilhan and the second author generalized it.…”

Section: Introductionmentioning

confidence: 99%

“…Meilhan and the second author generalized it. More precisely, in [5] they showed that any Milnor invariant of length between ✩ The first author is partially supported by Platform for Dynamic Approaches to Living System from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The second author is partially supported by a Grant-in-Aid for Scientific Research (C) (#26400081) of the Japan Society for the Promotion of Science.…”

Section: Introductionmentioning

confidence: 99%

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Milnor invariants of length 2k+2 for links with vanishing Milnor invariants of length ≤k

Kotorii

Yasuhara

2015

Topology and its Applications

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Available online xxxx Keywords:Milnor μ-invariant HOMFLYPT polynomial Clasper J.-B. Meilhan and the second author showed that any Milnor μ-invariant of length between 3 and 2k + 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all μ-invariants of length ≤ k vanish. They also showed that their formula does not hold for length 2k + 2. In this paper, we improve their formula to give the μ-invariants of length 2k + 2 by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined by μ-invariants of length k + 1. In particular, for any 4-component link the μ-invariants of length 4 are given by our formula, since all μ-invariants of length 1 vanish.

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 97%

“…In [8], M. Polyak gave a formula expressing Milnor invariant of length 3, and in [5], J.-B. Meilhan and the second author generalized it.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Milnor invariants of length 2k+2 for links with vanishing Milnor invariants of length ≤k

Kotorii

Yasuhara

2015

Topology and its Applications

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“…An important step was taken by Habegger and G. Masbaum, who showed explicitly in [10] how Milnor invariants are related to the Kontsevich integral. More recently, A. Yasuhara and the first author gave explicit formulas relating Milnor invariants to the HOMFLYPT polynomial [22].Key words and phrases. quantum and finite type invariants, weight system, link concordance, link-homotopy.…”

mentioning

confidence: 99%

The universal sl2 invariant and Milnor invariants

Meilhan

Suzuki

2016

Int. J. Math.

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Abstract. The universal sl 2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the -adic completed tensor powers of the quantized enveloping algebra of sl 2 . In this paper, we exhibit explicit relationships between the universal sl 2 invariant and Milnor invariants, which are classical invariants generalizing the linking number, providing some new topological insight into quantum invariants. More precisely, we define a reduction of the universal sl 2 invariant, and show how it is captured by Milnor concordance invariants. We also show how a stronger reduction corresponds to Milnor link-homotopy invariants. As a byproduct, we give explicit criterions for invariance under concordance and link-homotopy of the universal sl 2 invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal sl 2 invariant. IntroductionThe theory of quantum invariants of knots and links emerged in the middle of the eighties, after the fundamental work of V. F. R. Jones. Instead of the classical tools of topology, such as algebraic topology, used until then, this new class of invariants was derived from interactions of knot theory with other fields of mathematics, such as operator algebras and representation of quantum groups, and revealed close relationships with theoretical physics. Although this gave rise to a whole new class of powerful tools in knot theory, we still lack a proper understanding of the topological information carried by quantum invariants. One way to attack this fundamental question is to exhibit explicit relationships with classical link invariants. The purpose of this paper is to give such a relation, by showing how a certain reduction of the universal sl 2 invariant is captured by Milnor invariants.Milnor invariants were originally defined by J. Milnor for links in S 3 [23,24]. Their definition contains an intricate indeterminacy, which was shown by N. Habegger and X. S. Lin to be equivalent to the indeterminacy in representing a link as the closure of a string link, i.e. of a pure tangle without closed components [8]. Milnor invariants are actually well-defined integer-valued invariants of framed string links, and the first non-vanishing Milnor string link invariants can be assembled into a single Milnor map µ k . See Section 2 for a review of Milnor string link invariants.Milnor invariants constitute an important family of classical (string) link invariants, and as such, their connection with quantum invariants has already been the subject of several works. The first attempt seems to be due to L. Rozansky, who conjectured a formula relating Milnor invariants to the Jones polynomial [29]. An important step was taken by Habegger and G. Masbaum, who showed explicitly in [10] how Milnor invariants are related to the Kontsevich integral. More recently, A. Yasuhara and the first author gave explicit formulas relating Milnor invariants to the HOMFLYPT polynomial [22].Key wor...

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HL-homotopy of handlebody-links and Milnor's invariants

Kotorii¹,

Mizusawa²

2017

Topology and its Applications

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Abstract. A handlebody-link is a disjoint union of embeddings of handlebodies in S 3 and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of HLhomotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this paper, we construct a family of invariants for HL-homotopy classes of general handlebody-links, by using Milnor's µ-invariants. Moreover, we give a bijection between the set of HL-homotopy classes of almost trivial handlebody-links and tensor product space modulo some general linear actions, especially for 3-or more component handlebody-links. Through this bijection we construct comparable invariants of HL-homotopy classes.

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Milnor invariants and the HOMFLYPT Polynomial

Cited by 18 publications

References 22 publications

Milnor invariants of length 2k+2 for links with vanishing Milnor invariants of length ≤k

Milnor invariants of length 2k+2 for links with vanishing Milnor invariants of length ≤k

The universal sl2 invariant and Milnor invariants

HL-homotopy of handlebody-links and Milnor's invariants

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