Computations of quandle cocycle invariants of surface-links using marked graph diagrams

Kamada, Seiichi; Kim, Ji-Eon; Lee, Sang Youl

doi:10.1142/s0218216515400106

Cited by 10 publications

(8 citation statements)

References 29 publications

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“…Associate the time parameter of a Reidemeister move with the height of a local broken sheet diagram. A translation of each Reidemeister move to a broken sheet diagram is done in [10]. A Reidemeister III move gives a triple point diagram seen in Figure 5, a Reidemeister I move corresponds to a branch point, and a Reidemeister II move corresponds to a maximum or minimum of a double point curve, Figures 5 & 6 of [10].…”

Section: Induced Broken Sheet Diagram Of a Motion Picturementioning

confidence: 99%

Surface-knot families with arbitrarily large triple point number

Cazet

2022

Preprint

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Analogous to a classical knot diagram, a surface-link can be generically projected to 3space and given crossing information to create a broken sheet diagram. The triple point number of a surface-link is the minimal number of triple points among all broken sheet diagrams that lift to that surface-link. This paper generalizes a family of Oshiro to show that there are non-split surface-links of arbitrarily many trivial components whose triple point number can be made arbitrarily large.

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Section: Induced Broken Sheet Diagram Of a Motion Picturementioning

confidence: 99%

Surface-knot families with arbitrarily large triple point number

Cazet

2022

Preprint

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“…To compute the 3-cocycle invariant, one performs these Reidemeister moves, keeping track of the quandle coloring and orientations on the surface. For a detailed analysis of the situation near a triple point, see [KKL15], who used movies corresponding to marked graph diagrams to compute this invariant.…”

Section: Fox Colorings and Quandle Coloringsmentioning

confidence: 99%

Bridge trisections and classical knotted surface theory

Joseph,

Meier,

Miller

et al. 2021

Preprint

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We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of knotted spheres that admit non-isotopic bridge trisections of minimal complexity. Third, we adapt Seifert's algorithm to the setting of tri-plane diagrams, which allows for the construction of a Seifert solid that is described by a Heegaard diagram. Finally, we give a classification result that says that a b-bridge trisection in which some sector contains at least b − 1 patches is completely decomposable, and thus the corresponding surface is unknotted. Corollary 3.9. Let D = (D 1 , D 2 , D 3 ) be a tri-plane diagram of a surface S ⊂ S 4 . Let w i be the writhe of the diagram D i ∪ D i+1 . Then e(S) = w 1 + w 2 + w 3 .As one application, we obtain the following well-known result.

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“…These invariants are defined as the state-sums over all quandle colorings of arcs and sheets and corresponding Boltzman weights that are the evaluations of a given quandle 2 and 3-cocycle at the crossings and triple points in a link diagram and broken surface diagram, respectively. In [10], S. Kamada, J. Kim and S. Y. Lee developed an interpretation of the quandle and shadow quandle cocycle invariants of surface-links in terms of marked graph diagram presentation of surface-links.…”

Section: Introductionmentioning

confidence: 99%