Isotopy and homeomorphism of closed surface braids

Mark Grant; Agata Sienicka

doi:10.1017/S0017089520000208

Isotopy and homeomorphism of closed surface braids

Mark Grant, Agata Sienicka

Mathematical Science

University of Bonn

Research output: Contribution to journal › Article › peer-review

8 Downloads (Pure)

Abstract

The closure of a braid in a closed orientable surface Σ is a link in Σ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.

Original language	English
Pages (from-to)	297-306
Number of pages	10
Journal	Glasgow Mathematical Journal
Volume	63
Issue number	2
Early online date	15 May 2020
DOIs	https://doi.org/10.1017/S0017089520000208
Publication status	Published - May 2021

Keywords

Surface braids
link isotopy
link homeomorphism
Birman exact sequence
Dehn–Nielsen–Baer Theorem
57N05
57M27
2010 Mathematics Subject Classification
20F36

Access to Document

10.1017/S0017089520000208Licence: Unspecified

Grant_Sienicka_GMJ_IsotopyAndHomeomorphism_AAM
This article has been published in a revised form in Glasgow Mathematical Journal https://doi.org/10.1017/S0017089520000208. This version is free to view and download for private research and study only. Not for re-distribution or re-use. © copyright holder.
Accepted author manuscript, 335 KBLicence: Unspecified

Cite this

@article{8023ebf377fd47aca3b8cfcb97dcbd87,

title = "Isotopy and homeomorphism of closed surface braids",

abstract = "The closure of a braid in a closed orientable surface Σ is a link in Σ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.",

keywords = "Surface braids, link isotopy, link homeomorphism, Birman exact sequence, Dehn–Nielsen–Baer Theorem, 57N05, 57M27, 2010 Mathematics Subject Classification, 20F36",

author = "Mark Grant and Agata Sienicka",

year = "2021",

month = may,

doi = "10.1017/S0017089520000208",

language = "English",

volume = "63",

pages = "297--306",

journal = "Glasgow Mathematical Journal",

issn = "0017-0895",

publisher = "Cambridge University Press",

number = "2",

}

TY - JOUR

T1 - Isotopy and homeomorphism of closed surface braids

AU - Grant, Mark

AU - Sienicka, Agata

PY - 2021/5

Y1 - 2021/5

N2 - The closure of a braid in a closed orientable surface Σ is a link in Σ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.

AB - The closure of a braid in a closed orientable surface Σ is a link in Σ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.

KW - Surface braids

KW - link isotopy

KW - link homeomorphism

KW - Birman exact sequence

KW - Dehn–Nielsen–Baer Theorem

KW - 57N05

KW - 57M27

KW - 2010 Mathematics Subject Classification

KW - 20F36

UR - http://www.scopus.com/inward/record.url?scp=85085380748&partnerID=8YFLogxK

U2 - 10.1017/S0017089520000208

DO - 10.1017/S0017089520000208

M3 - Article

SN - 0017-0895

VL - 63

SP - 297

EP - 306

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

IS - 2

ER -

Isotopy and homeomorphism of closed surface braids

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this