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Journal of Combinatorial Theory Series A· 2026Q1

Triangulations of the 3-sphere with knotted edge

Dionne Ibarra, Daniel V. Mathews, Jessica S. Purcell, Jonathan Spreer

Short summary

Any knot K can be represented by a single vertex and edge within a tetrahedral decomposition of the 3-sphere.

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Abstract

We prove that for any knot K , there exists a decomposition of the 3-sphere into tetrahedra such that a representative of K is formed from one vertex and one edge of the decomposition. The proof is constructive, and based on fully augmented links. We use our method to produce a family of simplicial triangulations of the 3-sphere with a loop of four edges representing a t -fold connected sum of the trefoil, whose number of tetrahedra grows linearly in t . This is best possible up to a constant factor.

The authors' abstract, as published at the source. Journal of Combinatorial Theory Series A, 2026 · DOI ↗

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Field: Computer Graphics and Computer-Aided Design

Computer Graphics and Computer-Aided DesignComputer Science