Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.27471 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914606453096448 |
|---|---|
| author | Shapiro, Boris |
| author_facet | Shapiro, Boris |
| contents | We study the minimum number of inflection points among generic immersed closed plane curves with a fixed embedded shadow. The word immersed is essential: a genuinely embedded Jordan curve has inflection minimum zero. For tree-like shadows, inflection criterion converts inflection-free realizability into a finite coorientation problem on the building polygons of the shadow. We sharpen this viewpoint into an exact finite formula for the minimum number of normalized inflections and record a dynamic-programming computation on the block tree. We then push the method beyond the tree-like case. For every embedded shadow the same coorientation model gives a universal lower bound. For a natural larger class, called tree--necklace shadows, in which the non-tree-like blocks are separated annular cycles, the lower bound is exact after imposing an explicit $\mathbb Z_2$ holonomy condition around each necklace. We also record the algorithmic status of the exact minimization problem and formulate a likely NP-hardness problem for unrestricted shadows. Finally, we introduce a related invariant: the minimum possible least multiplicity of the Gauss map, equivalently the smallest guaranteed number of oriented parallel tangencies. This ``parallel-tangent load'' is controlled by the same inflection folds but is not determined by their number alone. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27471 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Combinatorics of Inflection Points of Plane Curve Shadows Shapiro, Boris Geometric Topology Algebraic Topology Combinatorics 57R42, 53A04, 05C10 We study the minimum number of inflection points among generic immersed closed plane curves with a fixed embedded shadow. The word immersed is essential: a genuinely embedded Jordan curve has inflection minimum zero. For tree-like shadows, inflection criterion converts inflection-free realizability into a finite coorientation problem on the building polygons of the shadow. We sharpen this viewpoint into an exact finite formula for the minimum number of normalized inflections and record a dynamic-programming computation on the block tree. We then push the method beyond the tree-like case. For every embedded shadow the same coorientation model gives a universal lower bound. For a natural larger class, called tree--necklace shadows, in which the non-tree-like blocks are separated annular cycles, the lower bound is exact after imposing an explicit $\mathbb Z_2$ holonomy condition around each necklace. We also record the algorithmic status of the exact minimization problem and formulate a likely NP-hardness problem for unrestricted shadows. Finally, we introduce a related invariant: the minimum possible least multiplicity of the Gauss map, equivalently the smallest guaranteed number of oriented parallel tangencies. This ``parallel-tangent load'' is controlled by the same inflection folds but is not determined by their number alone. |
| title | Combinatorics of Inflection Points of Plane Curve Shadows |
| topic | Geometric Topology Algebraic Topology Combinatorics 57R42, 53A04, 05C10 |
| url | https://arxiv.org/abs/2605.27471 |