Ribbonlength bounds for pretzel links and knots with $\leq 9$ crossings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912764413345792 |
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| author | Denne, Elizabeth |
| author_facet | Denne, Elizabeth |
| 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. We prove that any $P(p,q,r)$ pretzel link can be constructed so that its infimal folded ribbonlength is $\leq \frac{55}{\sqrt{3}} \leq 31.755$. We prove that any $n$-strand pretzel link $P(p_1,p_2, \dots, p_n)$ can be constructed so that its infimal folded ribbonlength is $\leq \frac{18n+1}{\sqrt{3}}$. This means that there is an infinite link family with a uniform bound on infimal folded ribbonlength. That is, we have shown $α=0$ in the equation $c\cdot \text{Cr}(L)^α\leq \text{Rib}([L])$, where $L$ is any link and $c$ is a constant. This paper also contains a table showing the best known upper bounds on the infimal folded ribbonlength for all knots with $\leq 9$ crossings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ribbonlength bounds for pretzel links and knots with $\leq 9$ crossings Denne, Elizabeth Geometric Topology 57K10 (Primary), 49Q10 (Secondary) 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. We prove that any $P(p,q,r)$ pretzel link can be constructed so that its infimal folded ribbonlength is $\leq \frac{55}{\sqrt{3}} \leq 31.755$. We prove that any $n$-strand pretzel link $P(p_1,p_2, \dots, p_n)$ can be constructed so that its infimal folded ribbonlength is $\leq \frac{18n+1}{\sqrt{3}}$. This means that there is an infinite link family with a uniform bound on infimal folded ribbonlength. That is, we have shown $α=0$ in the equation $c\cdot \text{Cr}(L)^α\leq \text{Rib}([L])$, where $L$ is any link and $c$ is a constant. This paper also contains a table showing the best known upper bounds on the infimal folded ribbonlength for all knots with $\leq 9$ crossings. |
| title | Ribbonlength bounds for pretzel links and knots with $\leq 9$ crossings |
| topic | Geometric Topology 57K10 (Primary), 49Q10 (Secondary) |
| url | https://arxiv.org/abs/2512.12830 |