Ribbon Torus Knots Presented by Virtual Knots With Up to Four Crossings

Ichimori, Atsushi; Kanenobu, Taizo

doi:10.1142/s0218216512400056

Cited by 11 publications

(6 citation statements)

References 27 publications

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“…However the Tube map is not injective and the lack of injectivity is not fully understood yet: the Tube map is known to be invariant under a global reversal move (see Prop. 3.4 in [IK12] or Prop. 2.7 in the present paper) but it is not known whether the Tube map quotiented by this move is injective.…”

Section: Correspondance Between Oriented and Signed Pairs And Crossingsmentioning

confidence: 99%

On the Welded Tube Map

Audoux¹

2016

Contemporary Mathematics

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This paper investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in S 4 . It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with welded knots before quotient under classical Reidemeister moves and reformulates these moves and the known source of non-injectivity of the Tube map in terms of filling changes.

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Section: Correspondance Between Oriented and Signed Pairs And Crossingsmentioning

confidence: 99%

On the Welded Tube Map

Audoux¹

2016

Contemporary Mathematics

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“…Consider an ext. welded knot diagram K. Let us denote by sK the sign reversal of K. It is easy to see that K † has the same Gauss data as sK (also shown in [IK12]). By Lemma 3.6, sK and K † are equivalent.…”

Section: Damianimentioning

confidence: 99%

A Markov's theorem for extended welded braids and links

Damiani

2017

Preprint

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“…This question is in general open, see [5,4]. It is true when restricting to welded braids by [7] and to string links up to self-virtualization [2], but fails in the case of welded knots [17].…”

Section: Some Open Questionsmentioning

confidence: 99%

On usual, virtual and welded knotted objects up to homotopy

Audoux¹,

Bellingeri²,

Meilhan³

et al. 2017

J. Math. Soc. Japan

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We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or selfvirtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.• P n and SL n stand for the (usual) sets of pure braids and string links on n strands, and the prefix v and w refer to their virtual and welded counterpart, respectively; • the superscripts sv and sc refer respectively to the equivalence relations generated by self-virtualization and self-crossing change.

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Ribbon Torus Knots Presented by Virtual Knots With Up to Four Crossings

Cited by 11 publications

References 27 publications

On the Welded Tube Map

On the Welded Tube Map

A Markov's theorem for extended welded braids and links

On usual, virtual and welded knotted objects up to homotopy

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