Generalized virtualization on welded links

Haruko A. Miyazawa; Kodai Wada; Akira Yasuhara

doi:10.2969/JMSJ/82248224

Generalized virtualization on welded links

Haruko A. Miyazawa, Kodai Wada, Akira Yasuhara

School of Commerce

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

Abstract

The aim of this paper is to study two local moves V (n) and V ⁿ on welded links for a positive integer n, which are generalizations of the crossing virtualization. We show that the V (n)-move is an unknotting operation on welded knots for any n, and give a classification of welded links up to V (n)-moves. On the other hand, we give a necessary condition for two welded links to be equivalent up to V ⁿ-moves. This leads us to show that the V ⁿ-move is not an unknotting operation on welded knots except for n = 1. We also discuss relations among V ⁿ-moves, associated core groups and the multiplexing of crossings.

Original language	English
Pages (from-to)	923-944
Number of pages	22
Journal	Journal of the Mathematical Society of Japan
Volume	72
Issue number	3
DOIs	https://doi.org/10.2969/JMSJ/82248224
Publication status	Published - 2020

Keywords

Alexander polynomial
Arrow calculus
Associated core group
Multiplexing of crossings
Unknotting operation
Virtualization
Welded knot
Welded link

ASJC Scopus subject areas

Mathematics(all)

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10.2969/JMSJ/82248224

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@article{bd1df6a646ff4dae999b9d3854db8021,

title = "Generalized virtualization on welded links",

abstract = "The aim of this paper is to study two local moves V (n) and V n on welded links for a positive integer n, which are generalizations of the crossing virtualization. We show that the V (n)-move is an unknotting operation on welded knots for any n, and give a classification of welded links up to V (n)-moves. On the other hand, we give a necessary condition for two welded links to be equivalent up to V n-moves. This leads us to show that the V n-move is not an unknotting operation on welded knots except for n = 1. We also discuss relations among V n-moves, associated core groups and the multiplexing of crossings.",

keywords = "Alexander polynomial, Arrow calculus, Associated core group, Multiplexing of crossings, Unknotting operation, Virtualization, Welded knot, Welded link",

author = "Miyazawa, {Haruko A.} and Kodai Wada and Akira Yasuhara",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary 57M25; Secondary 57M27. Key Words and Phrases. welded knot, welded link, virtualization, unknotting operation, arrow calculus, Alexander polynomial, associated core group, multiplexing of crossings. The second author was supported by Grants-in-Aid for JSPS Research Fellow (#17J08186, #19J00006) of the Japan Society for the Promotion of Science. The third author was partially supported by Grant-in-Aid for Scientific Research (C) (#17K05264) of the Japan Society for the Promotion of Science and Waseda University Grant for Special Research Projects (#2018S-077). 1For simplicity, we do not use here the usual drawing convention for virtual crossings, which is a small circle around the corresponding double point.",

year = "2020",

doi = "10.2969/JMSJ/82248224",

language = "English",

volume = "72",

pages = "923--944",

journal = "Journal of the Mathematical Society of Japan",

issn = "0025-5645",

publisher = "Mathematical Society of Japan - Kobe University",

number = "3",

}

TY - JOUR

T1 - Generalized virtualization on welded links

AU - Miyazawa, Haruko A.

AU - Wada, Kodai

AU - Yasuhara, Akira

N1 - Funding Information: 2010 Mathematics Subject Classification. Primary 57M25; Secondary 57M27. Key Words and Phrases. welded knot, welded link, virtualization, unknotting operation, arrow calculus, Alexander polynomial, associated core group, multiplexing of crossings. The second author was supported by Grants-in-Aid for JSPS Research Fellow (#17J08186, #19J00006) of the Japan Society for the Promotion of Science. The third author was partially supported by Grant-in-Aid for Scientific Research (C) (#17K05264) of the Japan Society for the Promotion of Science and Waseda University Grant for Special Research Projects (#2018S-077). 1For simplicity, we do not use here the usual drawing convention for virtual crossings, which is a small circle around the corresponding double point.

PY - 2020

Y1 - 2020

N2 - The aim of this paper is to study two local moves V (n) and V n on welded links for a positive integer n, which are generalizations of the crossing virtualization. We show that the V (n)-move is an unknotting operation on welded knots for any n, and give a classification of welded links up to V (n)-moves. On the other hand, we give a necessary condition for two welded links to be equivalent up to V n-moves. This leads us to show that the V n-move is not an unknotting operation on welded knots except for n = 1. We also discuss relations among V n-moves, associated core groups and the multiplexing of crossings.

AB - The aim of this paper is to study two local moves V (n) and V n on welded links for a positive integer n, which are generalizations of the crossing virtualization. We show that the V (n)-move is an unknotting operation on welded knots for any n, and give a classification of welded links up to V (n)-moves. On the other hand, we give a necessary condition for two welded links to be equivalent up to V n-moves. This leads us to show that the V n-move is not an unknotting operation on welded knots except for n = 1. We also discuss relations among V n-moves, associated core groups and the multiplexing of crossings.

KW - Alexander polynomial

KW - Arrow calculus

KW - Associated core group

KW - Multiplexing of crossings

KW - Unknotting operation

KW - Virtualization

KW - Welded knot

KW - Welded link

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