Whitehead double and milnor invariants

Jean Baptiste Meilhan, Akira Yasuhara

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length ≤ k are all zero into a link with vanishing Milnor invariants of length ≤ 2k +1, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Δ-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if the link L with a single component Whitehead 1 doubled is self Δ-equivalent to the unlink.

Original languageEnglish
Pages (from-to)371-381
Number of pages11
JournalOsaka Journal of Mathematics
Volume48
Issue number2
Publication statusPublished - 2011 Jun 1
Externally publishedYes

Fingerprint

Invariant
Link Homotopy
Equivalence
If and only if
Zero

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Whitehead double and milnor invariants. / Meilhan, Jean Baptiste; Yasuhara, Akira.

In: Osaka Journal of Mathematics, Vol. 48, No. 2, 01.06.2011, p. 371-381.

Research output: Contribution to journalArticle

Meilhan, Jean Baptiste ; Yasuhara, Akira. / Whitehead double and milnor invariants. In: Osaka Journal of Mathematics. 2011 ; Vol. 48, No. 2. pp. 371-381.
@article{49a22fee81874430a970ddf49ce8fe46,
title = "Whitehead double and milnor invariants",
abstract = "We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length ≤ k are all zero into a link with vanishing Milnor invariants of length ≤ 2k +1, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Δ-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if the link L with a single component Whitehead 1 doubled is self Δ-equivalent to the unlink.",
author = "Meilhan, {Jean Baptiste} and Akira Yasuhara",
year = "2011",
month = "6",
day = "1",
language = "English",
volume = "48",
pages = "371--381",
journal = "Osaka Journal of Mathematics",
issn = "0030-6126",
publisher = "Osaka University",
number = "2",

}

TY - JOUR

T1 - Whitehead double and milnor invariants

AU - Meilhan, Jean Baptiste

AU - Yasuhara, Akira

PY - 2011/6/1

Y1 - 2011/6/1

N2 - We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length ≤ k are all zero into a link with vanishing Milnor invariants of length ≤ 2k +1, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Δ-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if the link L with a single component Whitehead 1 doubled is self Δ-equivalent to the unlink.

AB - We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length ≤ k are all zero into a link with vanishing Milnor invariants of length ≤ 2k +1, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Δ-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if the link L with a single component Whitehead 1 doubled is self Δ-equivalent to the unlink.

UR - http://www.scopus.com/inward/record.url?scp=80052766832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052766832&partnerID=8YFLogxK

M3 - Article

VL - 48

SP - 371

EP - 381

JO - Osaka Journal of Mathematics

JF - Osaka Journal of Mathematics

SN - 0030-6126

IS - 2

ER -