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• W3113208124 abstract "Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a <italic>shifting chain map</italic> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=application/x-tex>sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on each quandle chain complex that lowers the dimensions by one. By using its pull-back <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Superscript normal ♯> <mml:semantics> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi mathvariant=normal>♯<!-- ♯ --></mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>sigma ^sharp</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cocycle <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi> <mml:semantics> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:annotation encoding=application/x-tex>phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives us the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cocycle <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Superscript normal ♯ Baseline phi> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi mathvariant=normal>♯<!-- ♯ --></mml:mi> </mml:msup> <mml:mi>ϕ<!-- ϕ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma ^sharp phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For oriented classical links in the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-space, we explore relation between their quandle <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cocycle invariants associated with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi> <mml:semantics> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:annotation encoding=application/x-tex>phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and their shadow <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cocycle invariants associated with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Superscript normal ♯ Baseline phi> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi mathvariant=normal>♯<!-- ♯ --></mml:mi> </mml:msup> <mml:mi>ϕ<!-- ϕ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma ^sharp phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For oriented surface links in the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=4> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=application/x-tex>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-space, we explore how powerful their quandle <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cocycle invariants associated with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Superscript normal ♯ Baseline phi> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi mathvariant=normal>♯<!-- ♯ --></mml:mi> </mml:msup> <mml:mi>ϕ<!-- ϕ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma ^sharp phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed." @default.
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• W3113208124 date "2022-07-29" @default.
• W3113208124 modified "2023-10-10" @default.
• W3113208124 title "Shifting chain maps in quandle homology and cocycle invariants" @default.
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