Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914389938929664 |
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| author | Klotz, Alexander R. |
| author_facet | Klotz, Alexander R. |
| contents | Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geqα_0 C^{3/4}$ for some constant $α_0$. There is currently an order-of-magnitude range for the value of $α_0$ between 1.105 and 10.76. In this work, we show that T(Q,Q) torus links can be constructed such that the upper bound is within a factor of 1.77 of the lower bound. We derive a stronger lower bound based on the convex hull around close-packed disks of approximately $α_{T_{QQ}}>\sqrt{8π\sqrt{3}}+(2π+\sqrt{2π+7\sqrt{3}-12}\ )Q^{-1/2}\approx6.60+7.61Q^{-1/2}$, significantly higher than the best universal lower bound of 1.105. We show that a link can be constructed without any free parameters or geometric optimization that, when $Q$ is large, has a coefficient $α_{T_{QQ}}<1.005\cdot 4π(5\sqrt{5}-8)/3\approx13.39$, and can be improved to to 11.68 by solving a helical no-overlap constraint equation that requires a conjectural approximation. For $Q$ up to 20 we construct links from smooth planar curves or toroidal helices minimized with respect to a small number of geometric parameters, that are between 6 and 60% greater in ropelength than the lower bound. Many such links can be annealed to within 10% of the lower bound using gradient descent. This represents significant progress towards developing sharp bounds on the ropelengths of specific classes of knots and links. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02416 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links Klotz, Alexander R. Geometric Topology Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geqα_0 C^{3/4}$ for some constant $α_0$. There is currently an order-of-magnitude range for the value of $α_0$ between 1.105 and 10.76. In this work, we show that T(Q,Q) torus links can be constructed such that the upper bound is within a factor of 1.77 of the lower bound. We derive a stronger lower bound based on the convex hull around close-packed disks of approximately $α_{T_{QQ}}>\sqrt{8π\sqrt{3}}+(2π+\sqrt{2π+7\sqrt{3}-12}\ )Q^{-1/2}\approx6.60+7.61Q^{-1/2}$, significantly higher than the best universal lower bound of 1.105. We show that a link can be constructed without any free parameters or geometric optimization that, when $Q$ is large, has a coefficient $α_{T_{QQ}}<1.005\cdot 4π(5\sqrt{5}-8)/3\approx13.39$, and can be improved to to 11.68 by solving a helical no-overlap constraint equation that requires a conjectural approximation. For $Q$ up to 20 we construct links from smooth planar curves or toroidal helices minimized with respect to a small number of geometric parameters, that are between 6 and 60% greater in ropelength than the lower bound. Many such links can be annealed to within 10% of the lower bound using gradient descent. This represents significant progress towards developing sharp bounds on the ropelengths of specific classes of knots and links. |
| title | Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2603.02416 |