Nielsen equivalence and trisections

Islambouli, Gabriel

doi:10.1007/s10711-021-00617-y

Cited by 7 publications

(4 citation statements)

References 22 publications

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“…Therefore, for each i, a spine of X i = B 4 \ νD i is isotopic to a spine of X i = B 4 \ νD i . As proven in [Isl21], this implies the two spines are related by edge slides and orientation reversals, and hence their induced Nielsen classes are equivalent.…”

Section: Fox Colorings and Quandle Coloringsmentioning

confidence: 55%

“…Here we show that bridge trisecting those same ribbon presentations yields non-isotopic bridge trisections. Nielsen equivalence was also used by Islambouli to find inequivalent trisections of a closed 4-manifold of the same parameters [Isl21].…”

Section: Fox Colorings and Quandle Coloringsmentioning

confidence: 99%

“…, x c i ), denoted N (X i ). This is well-defined because any two spines are related by Nielsen transformations [Isl21]. Note that one can arrange that these generators x i are meridians to the trivial disk system, one for each component.…”

Section: Fox Colorings and Quandle Coloringsmentioning

confidence: 99%

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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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Section: Fox Colorings and Quandle Coloringsmentioning

confidence: 55%

Section: Fox Colorings and Quandle Coloringsmentioning

confidence: 99%

See 1 more Smart Citation

Bridge trisections and classical knotted surface theory

Joseph,

Meier,

Miller

et al. 2021

Preprint

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show abstract

“…It would be interesting to see what the corresponding trisection diagrams of D 4 look like. In [20] spun lens spaces are used to construct non-diffeomorphic trisections on the same underlying 4-manifolds and one might wonder to which extent this might generalize to arbitrary open books. Moreover, it is claimed in [8] that trisection diagrams of some special open books can also be derived with the methods from [26].…”

Section: Previous Workmentioning

confidence: 99%

Trisecting a 4-dimensional book into three chapters

Kegel,

Schmäschke

2024

Geom Dedicata

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We describe an algorithm that takes as input an open book decomposition of a closed oriented 4-manifold and outputs an explicit trisection diagram of that 4-manifold. Moreover, a slight variation of this algorithm also works for open books on manifolds with non-empty boundary and for 3-manifold bundles over $$S^1$$ S 1 . We apply this algorithm to several simple open books, demonstrate that it is compatible with various topological constructions, and argue that it generalizes and unifies several previously known constructions.

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Trisections obtained by trivially regluing surface-knots

Isoshima

2024

Geom Dedicata

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Let S be a $$P^2$$ P 2 -knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted $$P^2$$ P 2 -knot with normal Euler number $${\pm }{2}$$ ± 2 in a closed 4-manifold X with trisection $$T_{X}$$ T X . Then, we show that the trisection of X obtained by the trivial gluing of relative trisections of $$\overline{\nu (S)}$$ ν ( S ) ¯ and $$X-\nu (S)$$ X - ν ( S ) is diffeomorphic to a stabilization of $$T_{X}$$ T X . It should be noted that this result is not obvious since boundary-stabilizations introduced by Kim and Miller are used to construct a relative trisection of $$X-\nu (S)$$ X - ν ( S ) . As a corollary, if $$X=S^4$$ X = S 4 and $$T_X$$ T X was the genus 0 trisection of $$S^4$$ S 4 , the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of $$S^4$$ S 4 . This result is related to the conjecture that is a 4-dimensional analogue of Waldhausen’s theorem on Heegaard splittings.

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Nielsen equivalence and trisections

Cited by 7 publications

References 22 publications

Bridge trisections and classical knotted surface theory

Bridge trisections and classical knotted surface theory

Trisecting a 4-dimensional book into three chapters

Trisections obtained by trivially regluing surface-knots

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