Bridge Presentations of Virtual Knots

Hirasawa, Mikami; Kamada, Naoko; Kamada, Seiichi

doi:10.1142/s0218216511009017

Cited by 10 publications

(15 citation statements)

References 4 publications

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“…A bridge of a virtual link diagram is an arc that contains one or more overcrossings (and any number of virtual crossings). The bridge number of a link L is the minimum number of bridges over all diagrams for L (for further details see [8]). It is a priori conceivable that a classical link may admit a virtual link diagram with fewer bridges than any of its classical diagrams: we use Theorem 5 to show that this cannot occur.…”

Section: Realising the Bridge And Ascending Numbers On Minimal Genus Diagramsmentioning

confidence: 99%

Minimal crossing number implies minimal supporting genus

Bodén

Rushworth

2021

Bull. London Math. Soc.

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A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. This extends corresponding results in the case of virtual knots due to Manturov and Chernov.

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Section: Realising the Bridge And Ascending Numbers On Minimal Genus Diagramsmentioning

confidence: 99%

Minimal crossing number implies minimal supporting genus

Bodén

Rushworth

2021

Bull. London Math. Soc.

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“…Bridge numbers of virtual and welded knots. Virtual and welded bridge numbers have been studied in [7,8,16,31,30] and the references therein. The definitions of virtual and welded bridge numbers that have appeared in the literature generalize perspectives from Sections 2.1.2, 2.1.3, and 2.1.4 to a virtual knot diagram by ignoring the virtual crossings.…”

Section: 32mentioning

confidence: 99%

Bridge numbers and meridional ranks of knotted surfaces and welded knots

Joseph¹,

Pongtanapaisan²

2021

Preprint

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The Meridional Rank Conjecture is an important open problem in the theory of knots and links in S 3 , and asks whether the bridge number of a knot is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the extent to which this is a good conjecture for other knotted objects, namely knotted surfaces in S 4 and virtual and welded knots. We develop criteria using tailored quotients of knot groups and the Wirtinger number to establish the equality of bridge number and meridional rank for several large families of examples. On the other hand, we show that the meridional rank of a connected sum of knotted spheres can achieve any value between the theoretical limits, so that either bridge number also collapses, or meridional rank is not equal to bridge number. We conclude with some applications of the Wirtinger number of virtual knots to the classical MRC and to hyperbolic volume of knots in S 3 .

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“…A bridge of a virtual link diagram is an arc that contains one or more overcrossings (and any number of virtual crossings). e bridge number of a link 𝐿 is the minimum number of bridges over all diagrams for 𝐿 (for further details see [7]). It is a priori conceivable that a classical link may admit a virtual link diagram with fewer bridges than any of its classical diagrams: we use eorem 10 to show that this cannot occur.…”

Section: Realising the Bridge And Ascending Numbers On Minimal Genus ...mentioning

confidence: 99%

Minimal crossing number implies minimal supporting genus

Boden,

Rushworth

2020

Preprint

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A virtual link may be de ned as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. is is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. is extends corresponding results in the case of virtual knots due to Manturov and Chernov.

show abstract

Bridge Presentations of Virtual Knots

Abstract: We study bridge presentations of virtual knots, and determine the virtual bridge numbers of pseudo-prime virtual knots with real crossing numbers less than 5, except two virtual knots.

Cited by 10 publications

References 4 publications

Minimal crossing number implies minimal supporting genus

Minimal crossing number implies minimal supporting genus

Bridge numbers and meridional ranks of knotted surfaces and welded knots

Minimal crossing number implies minimal supporting genus

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