Homological lifts of Arnold strangeness

@misc{ito2026homological,
  title = {{Homological lifts of Arnold strangeness}},
  author = {Noboru Ito},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36387},
  eprint = {2610.36387},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36387},
}

TY  - UNPB
TI  - Homological lifts of Arnold strangeness
AU  - Noboru Ito
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36387
UR  - https://arxiv.org/abs/2610.36387
AB  - We construct homological lifts of Arnold's strangeness St for long and closed plane curves. For long curves, we introduce a new q-variable quantization polynomial St_q, an edge state sum, designed for categorification and construct a triply graded homology that recovers St_q. We prove strictness by exhibiting long curves with the same number of crossings and the same St_q (in particular, the same rotation number and St) that are distinguished by the homology. For closed curves, we introduce a corresponding new polynomial St_q and construct an edge homology. In this case St_q is recovered from a signed q-character ch_q defined on the canonical cycles of the homology rather than directly from a graded rank. Again, we exhibit closed curves with the same number of crossings and the same St_q (in particular, the same rotation number and St) that are distinguished by the homology.
ER  -