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

Valdez

et al. 2018

Self Cite

We say that a topological n-manifold N is a cubical n-manifold if it is contained in the n-skeleton of the canonical cubulation C of R n+k (k ≥ 1). In this paper, we prove that any closed, oriented cubical 2manifold has a transverse field of 2-planes in the sense of Whitehead and therefore it is smoothable by a small ambient isotopy. * 2010 Mathematics Subject Classification. Primary: 57R10. Secondary: 57R55, 57R25.

“…Observe that a cubical manifold can be subdivided into simplices to become a PL-manifold. M. Boege, G. Hinojosa and A. Verjovsky proved in [2] the following theorem. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 92%

“…Observe that by construction, v slkm(N ) is well-defined continuous transverse 2-field. By Theorem 3.5, we know that there exists a local transverse 2-vector field v Fm(N ) : F(N ) → G (2,4), such that it is a local transverse 2-field at y ∈ Int(F m (N )) and for y ∈ ∂(S 1 ∪ S 2 ), we have that v Fm(N ) (y) = v slkm(N ) (y).…”

Section: Remember Thatmentioning

confidence: 99%

Smoothing closed gridded surfaces embedded in ℝ4

Díaz

Valdez

et al. 2018

Self Cite

“…In [2] it was shown that any smooth knot K n : S n → R n+2 can be deformed isotopically into the n-skeleton of the canonical cubulation of R n+2 and this isotopic copy is called cubical n-knot. In particular, every smooth 1-knot S 1 ⊂ R 3 is isotopic to a cubical knot.…”

Section: Introductionmentioning

confidence: 99%

Cubulated moves for 2-knots

Díaz

Verjosvky

2019

Self Cite

“…In [5] it was shown that any smooth knot S n ∼ K n ⊂ R n+2 can be deformed isotopically into the n-skeleton of the canonical cubulation of R n+2 and this isotopic copy is called cubic knot. In particular, every classical smooth knot S 1 ⊂ R 3 is isotopic to a cubic knot.…”

Section: Introductionmentioning

confidence: 99%

Cubulated Moves and Discrete Knots

Verjovsky

Marcotte

2013

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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].