Merge pull request #46 from aklotzlb/main

Updated knot stuff

Author: teorth

README.md

@@ -44,7 +44,7 @@ We are arbitrarily numbering the constants as $C_{1}$, $C_{2}$, etc., mostly bas
 | [20b](https://teorth.github.io/optimizationproblems/constants/20b.html) | Isotropic constant of a log-concave probability measure | $1/e$ | $< \infty$ |
 | [20c](https://teorth.github.io/optimizationproblems/constants/20c.html) | KLS constant for log-concave probability measures | $\sqrt{\pi/2} \approx 1.25331$ | $\infty$ |
 | [21](https://teorth.github.io/optimizationproblems/constants/21a.html) | de Bruijn–Newman constant | 0 | 0.2 |
-| [22a](https://teorth.github.io/optimizationproblems/constants/22a.html) | Tight knot constant | 1.105 | 10.76 |
+| [22a](https://teorth.github.io/optimizationproblems/constants/22a.html) | Tight knot constant | 1.105 | 10.02 |
 | [22b](https://teorth.github.io/optimizationproblems/constants/22b.html) | Tight alternating knot constant | 0.017 | 7.31 |
 | [23a](https://teorth.github.io/optimizationproblems/constants/23a.html) | Smallest unsolved instance of the Hadamard conjecture | 668 | $\infty$ |
 | [23b](https://teorth.github.io/optimizationproblems/constants/23b.html) | Minimal condition number decay for sign matrices | $17/92$ | 1 |

constants/22a.md

@@ -17,6 +17,8 @@ Upper bounds are typically found by finding a tight instance of a specific knot
 | 12.81| [SDKP1998] | $8_{19}$ knot |
 | 12.63| [ACPR2011] | $10_{124}$ knot |
 | 10.76| [KM2021] | $T(25,26)$ knot |
+| 10.02| [Klotz2026] | $T(24,19)$ knot |
+
 
 ## Known lower bounds
 
@@ -43,6 +45,7 @@ Upper bounds are typically found by finding a tight instance of a specific knot
 - [BS1999] Buck, Gregory; Simon, Jonathan. Thickness and crossing number of knots. Topol. Appl. 91, No. 3, 245-257 (1999).
 - [Klotz2025] Klotz, Alexander. Geometric considerations for energy minimization of topological links and chainmail networks. [arXiv:2507.20903](https://arxiv.org/abs/2507.20903)
 - [Diao2003] Diao, Yuanan. The lower bounds of the lengths of thick knots. J. Knot Theory Ramifications 12, No. 01, 1-16 (2003).
+- [Klotz2026] Klotz, Alexander R. Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links. [arXiv:2603.02416](https://arxiv.org/abs/2603.02416).
 
 
 ## Contribution notes

constants/22b.md

@@ -4,7 +4,7 @@
 
 $C_{22b} = b_{o}$ is the largest constant for which one has an inequality
 $$ L \geq b_{o} C $$
-for all knots that admit an [alternating](https://en.wikipedia.org/wiki/Alternating_knot) diagram, where $L$ is the ropelength of a knot (or link) with [crossing number](https://en.wikipedia.org/wiki/Crossing_number_(graph_theory)) $C$.
+for all knots that admit an [alternating](https://en.wikipedia.org/wiki/Alternating_knot) diagram, where $L$ is the ropelength of a knot (or link) with [crossing number](https://en.wikipedia.org/wiki/Crossing_number_(knot_theory)) $C$.
 The _ropelength_ $L$ is the infimum over all embeddings of the knot (or link) of the ratio of the contour length of the knot to its thickness. The _thickness_ is defined as the radius of the smallest circle that passes through any three points on the knot (where collinear points yield an infinite radius). Colloquially, the ropelength is the least amount of rope required to tie a specific knot in a rope of unit radius. See [CKS2002] for the full definition.
 
 ## Known upper bounds
@@ -13,6 +13,7 @@ Upper bounds are typically found by constructing alternating torus knots or link
 
 | Bound | Reference | Comments |
 | ----- | --------- | -------- |
+|$2\pi+2\approx 8.28$ | Trivial | Hopf chain link of stadium curves
 | 8.50| [O2013] | Double helix |
 | 7.63| [Huh2018] | Four-strand superhelix |
 | $1+\pi\sqrt{4+\frac{1}{\pi^2}}\approx 7.36$| [Klotz2021] | Wrapped circle |