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PreprintarXiv licencearXiv

Torus knots as loop-edges in three-sphere triangulations

  • Lezhi Lin
  • Jonathan Spreer
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Abstract

We present a procedure that, given a pair of coprime integers, produces a one-vertex triangulation of the three-sphere containing the corresponding torus knot as an edge. The construction is canonical, and produces a three-sphere triangulation from a pair of minimal layered triangulations of the solid torus. The size of the triangulation is, in the worst case, linear, but can be as small as logarithmic in the crossing number of the knot, motivating this triangulation size as an organising principle for knots, alternative to the crossing number.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026torus,
  title = {{Torus knots as loop-edges in three-sphere triangulations}},
  author = {Lezhi Lin and Jonathan Spreer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14200},
  eprint = {2609.14200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14200},
}
RIS
TY  - UNPB
TI  - Torus knots as loop-edges in three-sphere triangulations
AU  - Lezhi Lin
AU  - Jonathan Spreer
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14200
UR  - https://arxiv.org/abs/2609.14200
AB  - We present a procedure that, given a pair of coprime integers, produces a one-vertex triangulation of the three-sphere containing the corresponding torus knot as an edge. The construction is canonical, and produces a three-sphere triangulation from a pair of minimal layered triangulations of the solid torus. The size of the triangulation is, in the worst case, linear, but can be as small as logarithmic in the crossing number of the knot, motivating this triangulation size as an organising principle for knots, alternative to the crossing number.
ER  -