Homological lifts of Arnold invariants J^- and J⁺

@misc{ito2026homological,
  title = {{Homological lifts of Arnold invariants $J^-$ and $J^+$}},
  author = {Noboru Ito},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11427},
  eprint = {2609.11427},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11427},
}

TY  - UNPB
TI  - Homological lifts of Arnold invariants J^- and J^+
AU  - Noboru Ito
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11427
UR  - https://arxiv.org/abs/2609.11427
AB  - Viro's Euler-integral polynomial P_C(q) and the Lanzat--Polyak quantized-curvature polynomial I_q(C) refine Arnold's invariants J^- and J^+ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is P_C(q). The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in I_q(C). Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full I_q(C) by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.
ER  -