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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2602.02642 |
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| _version_ | 1866911417656934400 |
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| author | Itzlinger, Paul Leon |
| author_facet | Itzlinger, Paul Leon |
| contents | Grid diagrams are special representations of knots in the three-sphere that are used to define a combinatorial version of knot Floer homology. Paolo Ghiggini and Yi Ni showed that knot Floer homology detects fibered knots. Their results imply, in particular, that grid diagrams with a unique grid state whose Alexander grading is maximal only exist for fibered knots. Whether every fibered knot admits such a diagram remains an open question. Here, we investigate the existence of such special grid diagrams for fibered knots. We develop an efficient method for deciding whether a given grid diagram meets the even stricter condition of having a unique grid state that realizes an upper bound for the Alexander function. By implementing this method in a Python package, we find suitable grid diagrams for 5385 of the 5397 fibered prime knots with crossing number at most 13. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02642 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Grid Diagrams of Fibered Knots Itzlinger, Paul Leon Geometric Topology Combinatorics Symplectic Geometry 57K18 (Primary) 05-08 (Secondary) Grid diagrams are special representations of knots in the three-sphere that are used to define a combinatorial version of knot Floer homology. Paolo Ghiggini and Yi Ni showed that knot Floer homology detects fibered knots. Their results imply, in particular, that grid diagrams with a unique grid state whose Alexander grading is maximal only exist for fibered knots. Whether every fibered knot admits such a diagram remains an open question. Here, we investigate the existence of such special grid diagrams for fibered knots. We develop an efficient method for deciding whether a given grid diagram meets the even stricter condition of having a unique grid state that realizes an upper bound for the Alexander function. By implementing this method in a Python package, we find suitable grid diagrams for 5385 of the 5397 fibered prime knots with crossing number at most 13. |
| title | Grid Diagrams of Fibered Knots |
| topic | Geometric Topology Combinatorics Symplectic Geometry 57K18 (Primary) 05-08 (Secondary) |
| url | https://arxiv.org/abs/2602.02642 |