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Burnside groups and $ n$-moves for links


Authors: Haruko A. Miyazawa, Kodai Wada and Akira Yasuhara
Journal: Proc. Amer. Math. Soc. 147 (2019), 3595-3602
MSC (2010): Primary 57M25, 57M27; Secondary 20F50
DOI: https://doi.org/10.1090/proc/14470
Published electronically: March 26, 2019
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Abstract: M. K. Dabkowski and J. H. Przytycki introduced the $ n$th Burnside group of a link, which is an invariant preserved by $ n$-moves. Using this invariant, for an odd prime $ p$, they proved that there are links which cannot be reduced to trivial links via $ p$-moves. It is generally difficult to determine if $ p$th Burnside groups associated to a link and the corresponding trivial link are isomorphic. In this paper, we give a necessary condition for the existence of such an isomorphism. Using this condition we give a simple proof for their results that concern $ p$-move reducibility of links.


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Additional Information

Haruko A. Miyazawa
Affiliation: Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda-Machi, Kodaira, Tokyo, 187-8577, Japan
Email: aida@tsuda.ac.jp

Kodai Wada
Affiliation: Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan
Email: k.wada8@kurenai.waseda.jp

Akira Yasuhara
Affiliation: Faculty of Commerce, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan
Email: yasuhara@waseda.jp

DOI: https://doi.org/10.1090/proc/14470
Keywords: Link, Burnside group, Magnus expansion, Montesinos--Nakanishi $3$-move conjecture, Fox coloring, virtual link, welded link
Received by editor(s): February 18, 2018
Received by editor(s) in revised form: August 27, 2018, and October 31, 2018
Published electronically: March 26, 2019
Additional Notes: The second author was supported by a Grant-in-Aid for JSPS Research Fellow (#17J08186) of the Japan Society for the Promotion of Science.
The third author was partially supported by a Grant-in-Aid for Scientific Research (C) (#17K05264) of the Japan Society for the Promotion of Science.
Dedicated: Dedicated to Professor Shin’ichi Suzuki on his 77th birthday
Communicated by: David Futer
Article copyright: © Copyright 2019 American Mathematical Society