Braid and Knot Theory in Dimension Four

Kamada, Seiichi

doi:10.1090/surv/095

Cited by 134 publications

(302 citation statements)

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“…A surface link diagram with the minimum number of triple points is described by a graph called a chart over the torus T : Recall that in the proof of Theorem 5.1 we took a surface link diagram using a specified projection [4,6,8]. Figure 5 presents an example of a chart that attains the minimum triple point number, which describes a surface link diagram of S 4 (X 1,1,(1,1,1) , ∆ 2 ).…”

Section: 2mentioning

confidence: 99%

On surface links whose link groups are abelian

Ito¹,

Nakamura²

2014

Math. Proc. Camb. Phil. Soc.

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Section: 2mentioning

confidence: 99%

On surface links whose link groups are abelian

Ito¹,

Nakamura²

2014

Math. Proc. Camb. Phil. Soc.

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“…The original notation of a chart, which is a graph in a 2-disk, was introduced for an "unclosed" surface braid and we can modify it to present a closed surface braid (cf. §23 in [12]). …”

Section: Chart Descriptionmentioning

confidence: 99%

“…S. Kamada proved that two m-charts are C-move equivalent if and only if their presented closed surface braids of degree m are equivalent (cf. [11,12]). …”

Section: Chart Descriptionmentioning

confidence: 99%

The lower bound of the 𝑤-indices of surface links via quandle cocycle invariants

Iwakiri

2009

Trans. Amer. Math. Soc.

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Abstract. The w-index of a surface link F is the minimal number of the triple points of surface braids representing F . In this paper, for a given 3-cocycle, we consider the minimal number of the w-indices of surface links whose quandle cocycle invariants associated with f are non-trivial, and denote it ω(f ). In particular, we show that ω(θ 3 ) = 6 and ω(θ p ) ≥ 7, where θ n is Mochizuki's 3-cocycle of the dihedral quandle of order n and p is an odd prime integer = 3. As a consequence, for a given non-negative integer g, there are surface knots with genus g with the w-index 6.

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“…Note that in the proof of Theorem 1.3, the following result due to Viro and Kamada [5] played an essential role: any orientable knotted surface in 4-space can be isotoped to a closed surface braid. However, it is known (see [8]) that a corresponding theorem does not hold for the non-orientable case.…”

Section: Theorem 14 Any Non-orientable Knotted Surface Can Be Isotomentioning

confidence: 99%

“…Let π 1 : R 4 → R 3 and π 2 : R 3 → R 2 be orthogonal projections. For a knotted surface F ⊂ R 4 , its projections π 1 | F into R 3 and π 2 • π 1 | F into R 2 are known to play essential roles in the theory of knotted surfaces (for example, see [2,3,4,5]). …”

Section: Introductionmentioning

confidence: 99%