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    Colourability is preserved under Reidemeister moves — Carmelics
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    Home/Modality & Possibility
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    Colourability is preserved under Reidemeister moves

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.By Reidemeister's theorem, knot equivalence can be characterised by sequences of Reidemeister moves
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    • 2.Each individual Reidemeister move preserves colourability
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.A diagrammatic proof that each Reidemeister move preserves colourability is not equivalent to a proof that arbitrary sequences of such moves preserve it.
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    • 2.Preservation under finite composition requires an explicit inductive argument over sequence length, which the case-by-case diagrammatic check does not supply.
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    • 3.Without the inductive closure argument, the inference from local move-preservation to global invariance commits a fallacy of composition.
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    Reason against 2 of 2
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    • 1.Colourability is defined relative to a modulus n, and the claim as stated suppresses this parameter, making it ambiguous across distinct colouring structures.
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    • 2.Reidemeister move preservation holds for Fox n-colourings only when n is fixed throughout; mixed or unspecified moduli can yield failures of preservation under type II moves.
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    Modality & Possibility

    Related

    A diagrammatic proof that each Reidemeister move preserves colourability is not ...By Reidemeister's theorem, knot equivalence can be characterised by sequences of...Colourability is defined relative to a modulus n, and the claim as stated suppre...Each individual Reidemeister move preserves colourability
    +3 moreShow less
    Preservation under finite composition requires an explicit inductive argument ov...Reidemeister move preservation holds for Fox n-colourings only when n is fixed t...Without the inductive closure argument, the inference from local move-preservati...

    Similar

    Feasibility is preserved under composition.83%If both Oscar and Oscar− survived, two distinct physical objects would...73%Each individual Reidemeister move preserves colourability72%L is properly contained in PSPACE72%

    Source

    AI-extracted1/3 agreementValid
    SEP: epistemology-visual-thinking
    View source passageHide passage
    The reference to colours here is inessential. Colourability is in fact a specific case of a kind of combinatorial property known as mod \(p\) labelling (for \(p\) an odd prime). Colourability is a knot invariant in the sense that if one diagram of a knot is colourable every diagram of that knot and of any equivalent knot is colourable. (By Reidemeister’s theorem this can be proved by showing that each Reidemeister move preserves colourability.) A standard diagram of an unknot, a diagram without
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit