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Discrete Representation of Cubical 2-Knots

In: Handbook of Visual, Experimental and Computational Mathematics

Author

Listed:
  • Gabriela Hinojosa

    (Universidad Autónoma del Estado de Morelos, Centro de Investigación en Ciencias)

  • Ana Baray

    (Universidad Autónoma del Estado de Morelos, Centro de Investigación en Ciencias)

  • Juan Pablo Díaz

    (Universidad Autónoma del Estado de Morelos, Centro de Investigación en Ciencias)

Abstract

A cubical 2-knot K 2 ⊂ ℝ 4 $$K^2\subset \mathbb {R}^4$$ is an embedding of the 2-sphere in the 2-skeleton of the canonical cubulation of ℝ 4 $$\mathbb {R}^4$$ . In this paper, we describe cubical 2-knots in a discrete way, i.e., as a sequence of a finite number of points; in particular, we prove that there exists a generic projection p : ℝ 4 → P $$p:\mathbb {R}^4\rightarrow P$$ onto a suitable hyperplane P such that p ( K ) $$p(K)$$ is a knot diagram and using this fact, we develop an algorithm to compute its fundamental group.

Suggested Citation

  • Gabriela Hinojosa & Ana Baray & Juan Pablo Díaz, 2026. "Discrete Representation of Cubical 2-Knots," Springer Books, in: Bharath Sriraman (ed.), Handbook of Visual, Experimental and Computational Mathematics, pages 55-76, Springer.
    • Handle: RePEc:spr:sprchp:978-3-032-16368-4_7
    DOI: 10.1007/978-3-032-16368-4_7
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