Any smooth n-dimensional knot \mathbbSⁿ\hookrightarrowℝⁿ⁺² is isotopic to an n-knot contained in the Menger Mⁿ⁺²_n-continuum

2025/10/26 by Díaz, Juan Pablo, Hinojosa, Gabriela, Verjovsky, Alberto
#30F40 #54H20 #57M30 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.22794

Abstract

In this paper, we say that an n-dimensional closed submanifold N embedded in ℝⁿ⁺² is cubical if it is contained in the n-skeleton of the canonical cubulation of ℝⁿ⁺². It has been shown by the last two authors and M. Boege that any smooth n-dimensional closed submanifold of ℝⁿ⁺² can be deformed to a cubical n-manifold by a global continuous isotopy of ℝⁿ⁺². Here, we prove that there is an isotopic copy of Nⁿ contained in the Menger Mⁿ⁺²_n-continuum. In particular, any smooth knot \mathbbSⁿ\hookrightarrowℝⁿ⁺² is isotopic to a knot contained in the Menger Mⁿ⁺²_n-continuum.

Citations

• Knots Inside Fractals
• Topological surfaces as gridded surfaces in geometrical spaces

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