Bounded ribbonlength for knot families and multi-twist Möbius bands
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| Format: | Preprint |
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2025
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| _version_ | 1866914117474844672 |
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| author | Denne, Elizabeth Patterson, Timi |
| author_facet | Denne, Elizabeth Patterson, Timi |
| contents | Take a thin, rectangular strip of paper, add in an odd number of half-twists, then join the ends together. This gives a multi-twist paper Möbius band. We prove that any multi-twist paper Möbius band can be constructed so the aspect ratio of the rectangle is $3\sqrt{3}+ε$ for any $ε>0$. We could also take the thin, rectangular strip of paper and tie a knot in it, then join the ends and fold flat in the plane. This creates a folded ribbon knot. We apply the techniques used to prove the multi-twist paper Möbius band result to $(2,q)$ torus knots and twist knots. We prove that any $(2,q)$-torus knot can be constructed so that the folded ribbonlength $\leq 13.86$. We prove that any twist knot can be constructed so that the folded ribbonlength is $\leq 17.59$. Both of these results give the lower bound for the ribbonlength crossing number problem which relates the infimal folded ribbonlength of a knot type $[K]$ to its crossing number $\text{Cr}(K)$. That is, we have shown $α=0$ in the equation $c\cdot \text{Cr}(K)^α\leq \text{Rib}([K])$, where $c$ is a constant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_18370 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded ribbonlength for knot families and multi-twist Möbius bands Denne, Elizabeth Patterson, Timi Geometric Topology Primary 57K10, Secondary 49Q10 Take a thin, rectangular strip of paper, add in an odd number of half-twists, then join the ends together. This gives a multi-twist paper Möbius band. We prove that any multi-twist paper Möbius band can be constructed so the aspect ratio of the rectangle is $3\sqrt{3}+ε$ for any $ε>0$. We could also take the thin, rectangular strip of paper and tie a knot in it, then join the ends and fold flat in the plane. This creates a folded ribbon knot. We apply the techniques used to prove the multi-twist paper Möbius band result to $(2,q)$ torus knots and twist knots. We prove that any $(2,q)$-torus knot can be constructed so that the folded ribbonlength $\leq 13.86$. We prove that any twist knot can be constructed so that the folded ribbonlength is $\leq 17.59$. Both of these results give the lower bound for the ribbonlength crossing number problem which relates the infimal folded ribbonlength of a knot type $[K]$ to its crossing number $\text{Cr}(K)$. That is, we have shown $α=0$ in the equation $c\cdot \text{Cr}(K)^α\leq \text{Rib}([K])$, where $c$ is a constant. |
| title | Bounded ribbonlength for knot families and multi-twist Möbius bands |
| topic | Geometric Topology Primary 57K10, Secondary 49Q10 |
| url | https://arxiv.org/abs/2509.18370 |