Minimal area of the spun trefoil knot on the canonical cubulation of ℝ⁴

2025/07/12 by Baray, Ana, Catalán, Juan José, Hinojosa, Gabriela +1
#57Q05 #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57Q15 #Secondary 57Q35

paper · doi:10.48550/arxiv.2507.09361

Abstract

We say that a cubical 2-knot K² is an embedding of the 2-sphere in the 2-skeleton of the canonical cubulation of ℝ⁴; in particular, K² is the union of m(K²) unit squares, hence m(K²) is its area. We define the minimal area of K² as the minimum over all the areas of cubical 2-knots isotopic to the given knot type. The minimal area of a cubical 2-knot is an invariant, and the following natural question arose: Given a knot type, what area is needed for a cubical 2-knot in the canonical cubulation of ℝ⁴ to realise that type with minimal area? In this paper, we answer this question for the spun trefoil knot in the weakly minimal case.

Citations

• Cubulated moves for 2-knots
• Topological surfaces as gridded surfaces in geometrical spaces
• Cubulated moves and discrete knots

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