Wild knots in higher dimensions as limit sets of Kleinian groups

Boege, Margareta; Hinojosa, Gabriela; Verjovsky, Alberto

doi:10.1090/s1088-4173-09-00198-2

Cited by 4 publications

(5 citation statements)

References 35 publications

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“…Recent work on the subject can be found in the work of Louis Funar [8] and Rade T. Živaljević [15]. The results of the present paper were used in an essential way to construct higher dimensional wild knots as limit sets of geoemtrically-finite conformal groups in our article [1].…”

Section: Introductionmentioning

confidence: 82%

“…By the above,N m can be deformed to one connected component of ∂QN m by an isotopy in ∂Q + M m . By standard theorems (see [3,13]), this isotopy can be extended to a global isotopy {h t : R n+2 → R n+2 } t∈ [0,1] . Observe that this isotopic copy ofN m is contained in the n-dimensional skeleton of the cubulation C. To finish the proof we now rescale our construction back to its original size using the inverse homothetic transformation h 1 m .…”

Section: Corollary 25mentioning

confidence: 99%

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Any smooth knot $\mathbb{S}^{n}\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to a cubic knot contained in the canonical scaffolding of ℝ n+2

2010

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Any smooth knot S n → R n+2 is isotopic to a cubic knot contained in the canonical scaffolding of R n+2Abstract The n-skeleton of the canonical cubulation C of R n+2 into unit cubes is called the canonical scaffolding S. In this paper, we prove that any smooth, compact, closed, n-dimensional submanifold of R n+2 with trivial normal bundle can be continuously isotoped by an ambient isotopy to a cubic submanifold contained in S. In particular, any smooth knot S n → R n+2 can be continuously isotoped to a knot contained in S.

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Section: Introductionmentioning

confidence: 82%

Section: Corollary 25mentioning

confidence: 99%

Any smooth knot $\mathbb{S}^{n}\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to a cubic knot contained in the canonical scaffolding of ℝ n+2

2010

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“…Proof of Theorem 4. 6. Recall that there exist integer numbers m 1 and m 2 such that p −1 (t)∩J = K 1 for all t ≤ m 1 and p −1 (t)∩J = K 2 for all t ≥ m 2 .…”

Section: Ifmentioning

confidence: 99%

“…Another motivation for our theorem is to use it to construct dynamically defined wild knots as in [6].…”

Section: Introductionmentioning

confidence: 99%

“…The movement (M2) consists in replacing A by A . We conjecture that our theorem is also valid in this higher dimensional situation and the proof should be similar to ours.Another motivation for our theorem is to use it to construct dynamically defined wild knots as in [6].Remark 1.2. There are several mathematicians working with cubic knots and cube diagrams and their higher-dimensional analogous.…”

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confidence: 99%

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Cubulated Moves and Discrete Knots

Hinojosa

Verjovsky

Marcotte

2013

J. Knot Theory Ramifications

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In this paper, we prove that given two cubic knots K 1 , K 2 in R 3 , they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use this fact to describe a cubic knot in a discrete way, as a cyclic permutation of contiguous vertices of the Z 3 -lattice (with some restrictions); moreover, we describe a regular diagram of a cubic knot in terms of such cyclic permutations.This allows us to prove the following:Theorem 1. Given two cubic knots K 1 and K 2 in R 3 , they are isotopic if and only if K 1 is equivalent to K 2 by a finite sequence of cubulated moves; i.e., K 1Theorem 1 is analogous to the Reidemeister moves of classical tame knots for cubic knots.Since a cubic knot is given by a sequence of edges whose boundaries are in the canonical lattice of points with integer coefficients in R 3 , i.e., the abelian group Z 3 , each knot is determined by a cyclic permutation (a 1 , . . . , a n ) (with some restrictions), a i ∈ Z 3 . In section 5 we describe a regular diagram of a cubic knot in terms of such cyclic permutations by projecting onto a plane P , such that it is injective when restricted to the Z 3 -lattice and the image of the Z 3 -lattice, Λ P , is dense. More precisely, the projection of each knot is determined by a cyclic permutation (w 1 , . . . , w n ) (with some restrictions), w i ∈ Λ P . This fact allows us to develop algorithms to compute some invariants of cubic knots.Remark 1.1. Professor Scott Baldridge has brought to our attention the fact that our main theorem follows also from Theorem 1.1 of his paper with Adam Lowrance [3]. The methods we use are different from theirs and we remark that our moves have a natural higher dimensional counterpart. If K n ⊂ R n+2 is a cubic knot contained in the n-skeleton of the canonical cubulation C of R n+2 , then given a cube Q ∈ C, for a sufficiently fine subdivision of C, the union A of all n-dimensional faces of K contained in ∂Q ∼ = S n+1 , if nonempty, is homeomorphic to a cubulated n-disk, hence the closure A of ∂Q \ A, is also a cubulated disk. The movement (M2) consists in replacing A by A . We conjecture that our theorem is also valid in this higher dimensional situation and the proof should be similar to ours.Another motivation for our theorem is to use it to construct dynamically defined wild knots as in [6].Remark 1.2. There are several mathematicians working with cubic knots and cube diagrams and their higher-dimensional analogous. For example, Ben McCarty was the first to use cube knots to give the easiest known proof that the left-hand and right-hand trefoils are not isotopic. Also there are several papers which consider lattices and knots. See [16], [17], [4], [12], [13].

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Hausdorff Dimension Varies Continuously on Equivalent Dynamically Defined Wild Knots

Caetano

Hinojosa

Valdez

2021

Bull Braz Math Soc, New Series

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Wild knots in higher dimensions as limit sets of Kleinian groups

Abstract: Abstract. In this paper we construct infinitely many wild knots, S n → S n+2 , for n = 1, 2, 3, 4 and 5, each of which is a limit set of a geometrically finite Kleinian group. We also describe some of their properties.

Cited by 4 publications

References 35 publications

Any smooth knot $\mathbb{S}^{n}\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to a cubic knot contained in the canonical scaffolding of ℝ n+2

Any smooth knot $\mathbb{S}^{n}\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to a cubic knot contained in the canonical scaffolding of ℝ n+2

Cubulated Moves and Discrete Knots

Hausdorff Dimension Varies Continuously on Equivalent Dynamically Defined Wild Knots

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