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PreprintCC BY 4.0arXiv

Tangle Invariants Valued in Derived Categories of Sheaves and the Rouquier Complex

  • Jian Qiu
  • Alex Takeda
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Abstract

We develop effective calculation tools for constructing categorified invariants of tangles and knots, following the work of Cautis and Kamnitzer. These invariants, originally constructed by means of Fourier-Mukai transform functors between derived categories of coherent sheaves on certain iterated flag varieties, can be given an alternative algebraic description by using tilting bundles. We use this method and provide several examples of its application, in the hope of showing its utility to explicit pen-and-paper computations. This tilting-bundle method also allows us to lift the construction from derived to homotopy categories of complexes of modules, reducing the calculation of tangle invariants to a type of categorified matrix algebra, where the matrices in question have maps between their elements and calculations get turbocharged with chain homotopy.

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Cite

BibTeXRIS
BibTeX
@misc{qiu2026tangle,
  title = {{Tangle Invariants Valued in Derived Categories of Sheaves and the Rouquier Complex}},
  author = {Jian Qiu and Alex Takeda},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07046},
  eprint = {2611.07046},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07046},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Tangle Invariants Valued in Derived Categories of Sheaves and the Rouquier Complex
AU  - Jian Qiu
AU  - Alex Takeda
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.07046
UR  - https://arxiv.org/abs/2611.07046
AB  - We develop effective calculation tools for constructing categorified invariants of tangles and knots, following the work of Cautis and Kamnitzer. These invariants, originally constructed by means of Fourier-Mukai transform functors between derived categories of coherent sheaves on certain iterated flag varieties, can be given an alternative algebraic description by using tilting bundles. We use this method and provide several examples of its application, in the hope of showing its utility to explicit pen-and-paper computations. This tilting-bundle method also allows us to lift the construction from derived to homotopy categories of complexes of modules, reducing the calculation of tangle invariants to a type of categorified matrix algebra, where the matrices in question have maps between their elements and calculations get turbocharged with chain homotopy.
ER  -