---
_id: '1522'
abstract:
- lang: eng
  text: 'We classify smooth Brunnian (i.e., unknotted on both components) embeddings
    (S2 × S1) ⊔ S3 → ℝ6. Any Brunnian embedding (S2 × S1) ⊔ S3 → ℝ6 is isotopic to
    an explicitly constructed embedding fk,m,n for some integers k, m, n such that
    m ≡ n (mod 2). Two embeddings fk,m,n and fk′ ,m′,n′ are isotopic if and only if
    k = k′, m ≡ m′ (mod 2k) and n ≡ n′ (mod 2k). We use Haefliger’s classification
    of embeddings S3 ⊔ S3 → ℝ6 in our proof. The relation between the embeddings (S2
    × S1) ⊔ S3 → ℝ6 and S3 ⊔ S3 → ℝ6 is not trivial, however. For example, we show
    that there exist embeddings f: (S2 ×S1) ⊔ S3 → ℝ6 and g, g′ : S3 ⊔ S3 → ℝ6 such
    that the componentwise embedded connected sum f # g is isotopic to f # g′ but
    g is not isotopic to g′.'
acknowledgement: "I thank A. Skopenkov for telling me about the problem and for his
  useful remarks.  I also thank A. Sossinsky,\r\nA. Zhubr, M. Skopenkov, P. Akhmetiev,
  and an anonymous referee for their feedback.  Author was partially\r\nsupported
  by Dobrushin fellowship, 2013, and by RFBR grant 15-01-06302."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Serhii
  full_name: Avvakumov, Serhii
  id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
  last_name: Avvakumov
citation:
  ama: Avvakumov S. The classification of certain linked 3-manifolds in 6-space. <i>Moscow
    Mathematical Journal</i>. 2016;16(1):1-25. doi:<a href="https://doi.org/10.17323/1609-4514-2016-16-1-1-25">10.17323/1609-4514-2016-16-1-1-25</a>
  apa: Avvakumov, S. (2016). The classification of certain linked 3-manifolds in 6-space.
    <i>Moscow Mathematical Journal</i>. Independent University of Moscow. <a href="https://doi.org/10.17323/1609-4514-2016-16-1-1-25">https://doi.org/10.17323/1609-4514-2016-16-1-1-25</a>
  chicago: Avvakumov, Sergey. “The Classification of Certain Linked 3-Manifolds in
    6-Space.” <i>Moscow Mathematical Journal</i>. Independent University of Moscow,
    2016. <a href="https://doi.org/10.17323/1609-4514-2016-16-1-1-25">https://doi.org/10.17323/1609-4514-2016-16-1-1-25</a>.
  ieee: S. Avvakumov, “The classification of certain linked 3-manifolds in 6-space,”
    <i>Moscow Mathematical Journal</i>, vol. 16, no. 1. Independent University of
    Moscow, pp. 1–25, 2016.
  ista: Avvakumov S. 2016. The classification of certain linked 3-manifolds in 6-space.
    Moscow Mathematical Journal. 16(1), 1–25.
  mla: Avvakumov, Sergey. “The Classification of Certain Linked 3-Manifolds in 6-Space.”
    <i>Moscow Mathematical Journal</i>, vol. 16, no. 1, Independent University of
    Moscow, 2016, pp. 1–25, doi:<a href="https://doi.org/10.17323/1609-4514-2016-16-1-1-25">10.17323/1609-4514-2016-16-1-1-25</a>.
  short: S. Avvakumov, Moscow Mathematical Journal 16 (2016) 1–25.
date_created: 2018-12-11T11:52:30Z
date_published: 2016-01-01T00:00:00Z
date_updated: 2022-02-25T10:15:57Z
day: '01'
department:
- _id: UlWa
doi: 10.17323/1609-4514-2016-16-1-1-25
external_id:
  arxiv:
  - '1408.3918'
intvolume: '        16'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://arxiv.org/abs/1408.3918
month: '01'
oa: 1
oa_version: Preprint
page: 1 - 25
publication: Moscow Mathematical Journal
publication_identifier:
  eissn:
  - 1609-4514
publication_status: published
publisher: Independent University of Moscow
publist_id: '5652'
quality_controlled: '1'
scopus_import: '1'
status: public
title: The classification of certain linked 3-manifolds in 6-space
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16
year: '2016'
...