Ribbonlength upper bounds for small crossing knots and links
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908600578867200 |
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| author | Chen, Zhicheng Denne, Elizabeth Patterson, Kyle Patterson, Timi |
| author_facet | Chen, Zhicheng Denne, Elizabeth Patterson, Kyle Patterson, Timi |
| contents | Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded ribbon knots, which is defined as the knot's length-to-width ratio. The {\em ribbonlength problem} asks to find the infimal folded ribbonlength of a knot or link type. By finding new methods of creating folded ribbon knots, we improve upon existing upper bounds for the folded ribbonlength of $(2,q)$-torus links, twist knots, and pretzel links. These give the best known bounds to date for small crossing knots in these families. For example, there is a folded ribbonlength twist knot $T_n$ with folded ribbonlength $\text{Rib}(T_n) = n +6$. Applying this to the figure-eight knot $T_2$ yields a folded ribbonlength $\text{Rib}(T_2)= 8$, which we conjecture is the infimum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ribbonlength upper bounds for small crossing knots and links Chen, Zhicheng Denne, Elizabeth Patterson, Kyle Patterson, Timi Geometric Topology Primary 57K10, Secondary 49Q10 Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded ribbon knots, which is defined as the knot's length-to-width ratio. The {\em ribbonlength problem} asks to find the infimal folded ribbonlength of a knot or link type. By finding new methods of creating folded ribbon knots, we improve upon existing upper bounds for the folded ribbonlength of $(2,q)$-torus links, twist knots, and pretzel links. These give the best known bounds to date for small crossing knots in these families. For example, there is a folded ribbonlength twist knot $T_n$ with folded ribbonlength $\text{Rib}(T_n) = n +6$. Applying this to the figure-eight knot $T_2$ yields a folded ribbonlength $\text{Rib}(T_2)= 8$, which we conjecture is the infimum. |
| title | Ribbonlength upper bounds for small crossing knots and links |
| topic | Geometric Topology Primary 57K10, Secondary 49Q10 |
| url | https://arxiv.org/abs/2510.16190 |