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• W1979057472 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M Subscript k> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{M_k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-fold branched cyclic covering of a (tame) knot of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Superscript 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{S^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S cubed> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{S^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our main result is that the following statements are equivalent: (1) <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H 1 left-parenthesis upper M Subscript k Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{H_1}({M_k})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is periodic with period <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, i.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H 1 left-parenthesis upper M Subscript k Baseline right-parenthesis approximately-equals upper H 1 left-parenthesis upper M Subscript k plus n Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≅<!-- ≅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{H_1}({M_k}) cong {H_1}({M_{k + n}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, (2) <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H 1 left-parenthesis upper M Subscript k Baseline right-parenthesis approximately-equals upper H 1 left-parenthesis upper M Subscript left-parenthesis k comma n right-parenthesis Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≅<!-- ≅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{H_1}({M_k}) cong {H_1}({M_{(k,n)}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, (3) the first Alexander invariant of the knot, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda 1 left-parenthesis t right-parenthesis equals normal upper Delta 1 left-parenthesis t right-parenthesis slash normal upper Delta 2 left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{lambda _1}(t) = {Delta _1}(t)/{Delta _2}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, divides <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Superscript n Baseline minus 1> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>{t^n} - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1979057472 created "2016-06-24" @default.
• W1979057472 creator A5058356810 @default.
• W1979057472 date "1972-01-01" @default.
• W1979057472 modified "2023-10-16" @default.
• W1979057472 title "Knots whose branched cyclic coverings have periodic homology" @default.
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