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PreprintarXiv licencearXiv

Trisection invariants of 4-manifolds are uncomputable

  • Nathan M. Dunfield
  • Marc Kegel
  • Shana Yunsheng Li
  • Qiuyu Ren
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Abstract

We prove that two invariants of smooth 4-manifolds defined in terms of trisections are uncomputable: the trisection genus and the Kirby-Thompson L-invariant. That is, there does not exist an algorithm that takes as input a triangulated closed orientable 4-manifold and outputs one of these quantities. The result on the L-invariant resolves the second part of Problem 4.116 in the K3 problem list. In the same spirit, we show that the PL multisection genus of PL manifolds is uncomputable in dimensions at least four.

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Cite

BibTeXRIS
BibTeX
@misc{dunfield2026trisection,
  title = {{Trisection invariants of 4-manifolds are uncomputable}},
  author = {Nathan M. Dunfield and Marc Kegel and Shana Yunsheng Li and Qiuyu Ren},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27811},
  eprint = {2608.27811},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27811},
}
RIS
TY  - UNPB
TI  - Trisection invariants of 4-manifolds are uncomputable
AU  - Nathan M. Dunfield
AU  - Marc Kegel
AU  - Shana Yunsheng Li
AU  - Qiuyu Ren
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27811
UR  - https://arxiv.org/abs/2608.27811
AB  - We prove that two invariants of smooth 4-manifolds defined in terms of trisections are uncomputable: the trisection genus and the Kirby-Thompson L-invariant. That is, there does not exist an algorithm that takes as input a triangulated closed orientable 4-manifold and outputs one of these quantities. The result on the L-invariant resolves the second part of Problem 4.116 in the K3 problem list. In the same spirit, we show that the PL multisection genus of PL manifolds is uncomputable in dimensions at least four.
ER  -