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

Remigio–Juárez, Jair

doi:10.1090/proc/13008

The link volume of 3-manifolds is not multiplicative under coverings
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Proc. Amer. Math. Soc. 144 (2016), 4075-4079 Request permission

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 $\operatorname {LinkVol}[\tilde {M}]=\operatorname {LinkVol}[M].$ 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 $\operatorname {LinkVol}[\tilde {M}]\leq q \operatorname {LinkVol}[M]$, when $\tilde {M}$ is a $q$-fold covering of $M$?

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