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Equivariant Grid Homology for Strongly Invertible Knots

  • Giovanni Framba
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Abstract

We study an equivariant version of grid homology for strongly invertible knots. We consider symmetric grid diagrams encoding the strong inversion, prove that every strongly invertible knot admits such a representation, and establish the corresponding analogue of Cromwell's theorem. A symmetric grid induces an involution on the associated grid complex. We study the mapping cone of the sum of the identity and this involution, proving that its homology is an invariant of strongly invertible knots. Finally, we define equivariant analogues of the tau invariant and of the Legendrian grid invariants, and show that the former provide lower bounds for equivariant unknotting numbers and for the genus of simple equivariant cobordisms.

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Cite

BibTeXRIS
BibTeX
@misc{framba2026equivariant,
  title = {{Equivariant Grid Homology for Strongly Invertible Knots}},
  author = {Giovanni Framba},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.04477},
  eprint = {2607.04477},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.04477},
}
RIS
TY  - UNPB
TI  - Equivariant Grid Homology for Strongly Invertible Knots
AU  - Giovanni Framba
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2607.04477
UR  - https://arxiv.org/abs/2607.04477
AB  - We study an equivariant version of grid homology for strongly invertible knots. We consider symmetric grid diagrams encoding the strong inversion, prove that every strongly invertible knot admits such a representation, and establish the corresponding analogue of Cromwell's theorem. A symmetric grid induces an involution on the associated grid complex. We study the mapping cone of the sum of the identity and this involution, proving that its homology is an invariant of strongly invertible knots. Finally, we define equivariant analogues of the tau invariant and of the Legendrian grid invariants, and show that the former provide lower bounds for equivariant unknotting numbers and for the genus of simple equivariant cobordisms.
ER  -