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• W2280972924 abstract "We obtain an infinite family of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3 minus> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>3-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>manifolds <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace upper M Subscript n Baseline right-brace Subscript n element-of double-struck upper N> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>M</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>N</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{{M}_n}_{nin mathbb {N}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an infinite family of coverings <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace phi Subscript n Baseline colon upper M overTilde Subscript n Baseline right-arrow upper M Subscript n Baseline right-brace Subscript n element-of double-struck upper N> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>φ<!-- φ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>M</mml:mi> <mml:mo stretchy=false>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>N</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{varphi _n:tilde {M}_nto M_{n}}_{nin mathbb {N}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with covering degrees unbounded and satisfying that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L i n k upper V o l left-bracket upper M overTilde right-bracket equals upper L i n k upper V o l left-bracket upper M right-bracket period> <mml:semantics> <mml:mrow> <mml:mi>LinkVol</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>M</mml:mi> <mml:mo stretchy=false>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>]</mml:mo> <mml:mo>=</mml:mo> <mml:mi>LinkVol</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {LinkVol}[tilde {M}]=operatorname {LinkVol}[M].</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L i n k upper V o l left-bracket upper M overTilde right-bracket less-than-or-equal-to q upper L i n k upper V o l left-bracket upper M right-bracket> <mml:semantics> <mml:mrow> <mml:mi>LinkVol</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>M</mml:mi> <mml:mo stretchy=false>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>]</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>q</mml:mi> <mml:mi>LinkVol</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {LinkVol}[tilde {M}]leq q operatorname {LinkVol}[M]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M overTilde> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>M</mml:mi> <mml:mo stretchy=false>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=application/x-tex>tilde {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-fold covering of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>?" @default.
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• W2280972924 date "2016-02-12" @default.
• W2280972924 modified "2023-09-27" @default.
• W2280972924 title "The link volume of 3-manifolds is not multiplicative under coverings" @default.
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