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| Format: | Preprint |
| Published: |
2007
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/0705.4348 |
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| _version_ | 1866914335237865472 |
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| author | Ozawa, Makoto |
| author_facet | Ozawa, Makoto |
| contents | We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its segment number is at least 3. Furthermore, we investigate the structural properties of the directed graph associated with a minimal segment number presentation. Specifically, we show that for any minimal presentation, the underlying graph is connected and cannot be a path. Finally, we discuss the relationship between the segment number and the bridge number, providing bounds and conjectures for future study. We also conjecture that the bridge number $b(K)$ provides a lower bound for the segment number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0705_4348 |
| institution | arXiv |
| publishDate | 2007 |
| record_format | arxiv |
| spellingShingle | Segment number of knots Ozawa, Makoto Geometric Topology Primary 57M25, Secondary 57Q35 We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its segment number is at least 3. Furthermore, we investigate the structural properties of the directed graph associated with a minimal segment number presentation. Specifically, we show that for any minimal presentation, the underlying graph is connected and cannot be a path. Finally, we discuss the relationship between the segment number and the bridge number, providing bounds and conjectures for future study. We also conjecture that the bridge number $b(K)$ provides a lower bound for the segment number. |
| title | Segment number of knots |
| topic | Geometric Topology Primary 57M25, Secondary 57Q35 |
| url | https://arxiv.org/abs/0705.4348 |