On the group of ring motions of an H-trivial link

Damiani, Celeste; Kamada, Seiichi

doi:10.1016/j.topol.2019.06.004

Cited by 5 publications

(2 citation statements)

References 11 publications

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“…Using this and Theorem B, we can compute the fundamental group of E(L) when L is a disjoint union of an n-component unlink and m Hopf links. Following [DK19], we call such a link H-trivial. Example 6.12 (H-trivial links).…”

Section: Now Consider the Homomorphismmentioning

confidence: 99%

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Embedding spaces of split links

Boyd¹,

Bregman²

2022

Preprint

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We study the homotopy type of the space E(L) of unparametrised embeddings of a split link L = L 1 ⊔ . . .⊔ Ln in R 3 . Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space of separating systems and show that this is homotopy equivalent to E(L). This combinatorial object provides a gateway to studying the homotopy type of E(L) via the homotopy type of the spaces E(L i ). We apply this tool to find a simple description of the fundamental group, or motion group, of E(L), and extend this to a description of the motion group of embeddings in S 3 .

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Section: Now Consider the Homomorphismmentioning

confidence: 99%

“…Goldsmith [Gol81], building on work of Dahm [Dah62], computed the motion group of U n (as noted above), and torus links [Gol82]. Damiani and Kamada computed the motion group for the round embedding space of the disjoint union of a Hopf link and an unknot [DK19].…”

Section: Introductionmentioning

confidence: 99%

Embedding spaces of split links

Boyd¹,

Bregman²

2022

Preprint

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Motion Groupoids and Mapping Class Groupoids

Martins

Martin

Torzewska

2023

Commun. Math. Phys.

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Here $${\underline{M}}$$ M ̲ denotes a pair (M, A) of a manifold and a subset (e.g. $$A=\partial M$$ A = ∂ M or $$A=\varnothing $$ A = ∅ ). We construct for each $${\underline{M}}$$ M ̲ its motion groupoid$$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ , whose object set is the power set $$ {{\mathcal {P}}}M$$ P M of M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M, that fix A, acting on $${{\mathcal {P}}}M$$ P M . These groupoids generalise the classical definition of a motion group associated to a manifold M and a submanifold N, which can be recovered by considering the automorphisms in $$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ of $$N\in {{\mathcal {P}}}M$$ N ∈ P M . We also construct the mapping class groupoid$$\textrm{MCG}_{{\underline{M}}}$$ MCG M ̲ associated to a pair $${\underline{M}}$$ M ̲ with the same object class, whose morphisms are now equivalence classes of homeomorphisms of M, that fix A. We recover the classical definition of the mapping class group of a pair by taking automorphisms at the appropriate object. For each pair $${\underline{M}}$$ M ̲ we explicitly construct a functor $${\textsf{F}}:\textrm{Mot}_{{\underline{M}}} \rightarrow \textrm{MCG}_{{\underline{M}}}$$ F : Mot M ̲ → MCG M ̲ , which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if $$\pi _0$$ π 0 and $$\pi _1$$ π 1 of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case $${\underline{M}}=([0,1]^n, \partial [0,1]^n)$$ M ̲ = ( [ 0 , 1 ] n , ∂ [ 0 , 1 ] n ) , for any $$n\in {\mathbb {N}}$$ n ∈ N . We show that the congruence relation used in the construction $$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows—worldlines (e.g. monotonic ‘tangles’). We examine several explicit examples of $$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ and $$\textrm{MCG}_{{\underline{M}}}$$ MCG M ̲ demonstrating the utility of the constructions.

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Examples of topologically unknotted tori

Juhász,

Powell

2024

Trans. Amer. Math. Soc. Ser. B

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We show that certain smooth tori with group Z \mathbb {Z} in S 4 S^4 have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all Z \mathbb {Z} -surfaces in S 4 S^4 are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of S 4 S^4 branched along any of these surfaces is a potentially exotic copy of S 2 × S 2 S^2 \times S^2 , and, in the case of turned twisted tori, we show they cannot be distinguished from S 2 × S 2 S^2 \times S^2 using Seiberg–Witten invariants.

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On the group of ring motions of an H-trivial link

Cited by 5 publications

References 11 publications

Embedding spaces of split links

Embedding spaces of split links

Motion Groupoids and Mapping Class Groupoids

Examples of topologically unknotted tori

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