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The link volume of 3-manifolds is not multiplicative under coverings


Author: Jair Remigio–Juárez
Journal: Proc. Amer. Math. Soc. 144 (2016), 4075-4079
MSC (2010): Primary 57M10, 57M12, 57M25, 57M27
DOI: https://doi.org/10.1090/proc/13008
Published electronically: February 12, 2016
MathSciNet review: 3513562
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Abstract: We obtain an infinite family of $ 3-$manifolds $ \{{M}_n\}_{n\in \mathbb{N}}$ and an infinite family of coverings $ \{\varphi _n:\tilde {M}_n\to M_{n}\}_{n\in \mathbb{N}}$ with covering degrees unbounded and satisfying that $ \mbox {\rm LinkVol}[\tilde {M}]=\mbox {\rm LinkVol}[M].$ This shows that link volume of 3-manifolds is not well behaved under covering maps, in particular, it is not multiplicative, and gives a negative answer to a question posed in a work of Rieck and Yamashita, namely, how good is the bound $ \mbox {\rm LinkVol}[\tilde {M}]\leq q \mbox {\rm LinkVol}[M]$, when $ \tilde {M}$ is a $ q$-fold covering of $ M$?


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  • [1] James W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), no. 8, 370-372. MR 1560318, https://doi.org/10.1090/S0002-9904-1920-03319-7
  • [2] Mark E. Feighn, Branched covers according to J. W. Alexander, Collect. Math. 37 (1986), no. 1, 55-60. MR 882791 (88d:57002)
  • [3] Hugh M. Hilden, María Teresa Lozano, and José María Montesinos, The Whitehead link, the Borromean rings and the knot $ 9_{46}$ are universal, Collect. Math. 34 (1983), no. 1, 19-28. MR 747855 (86b:57001)
  • [4] Hugh M. Hilden, María Teresa Lozano, and José María Montesinos, On knots that are universal, Topology 24 (1985), no. 4, 499-504. MR 816529 (87a:57010), https://doi.org/10.1016/0040-9383(85)90019-9
  • [5] Peter Orlik and Frank Raymond, On $ 3$-manifolds with local $ {\rm SO}(2)$ action, Quart. J. Math. Oxford Ser. (2) 20 (1969), 143-160. MR 0266214 (42 #1121)
  • [6] Jair Remigio-Juárez,
    Branched coverings of Seifert manifolds,
    Ph.D. Thesis, 2008.
  • [7] Jair Remigio-Juárez and Yo'av Rieck, The link volumes of some prism manifolds, Algebr. Geom. Topol. 12 (2012), no. 3, 1649-1665. MR 2966698, https://doi.org/10.2140/agt.2012.12.1649
  • [8] Yo'av Rieck and Yasushi Yamashita, The link volume of 3-manifolds, Algebr. Geom. Topol. 13 (2013), no. 2, 927-958. MR 3044597, https://doi.org/10.2140/agt.2013.13.927
  • [9] William P. Thurston,
    Universal links,
    preprint.

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Additional Information

Jair Remigio–Juárez
Affiliation: División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, Km. 1 Carr. Cunduacán-Jalpa de Méndezm, Cunduacán, Tab. 86690, Mexico
Email: jair.remigio@ujat.mx

DOI: https://doi.org/10.1090/proc/13008
Received by editor(s): September 24, 2014
Received by editor(s) in revised form: September 8, 2015, and October 21, 2015
Published electronically: February 12, 2016
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2016 American Mathematical Society

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