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PreprintCC BY 4.0arXiv

On Diagrammatic Categorification of Verma Modules I: Braiding

  • Pedro Guicardi
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Abstract

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of sl2. Our motivation is to construct a theory of Khovanov homology for knot complements in S3 (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of sl2, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in S1× D2, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

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Cite

BibTeXRIS
BibTeX
@misc{guicardi2026diagrammatic,
  title = {{On Diagrammatic Categorification of Verma Modules I: Braiding}},
  author = {Pedro Guicardi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10941},
  eprint = {2609.10941},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10941},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Diagrammatic Categorification of Verma Modules I: Braiding
AU  - Pedro Guicardi
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.10941
UR  - https://arxiv.org/abs/2609.10941
AB  - In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of sl_2. Our motivation is to construct a theory of Khovanov homology for knot complements in S^3 (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of sl_2, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in S^1× D^2, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.
ER  -