There's some fun stuff here (I'm reminded of the algorithms used to render planar drawings of proteins, which are similar to knots).
My real interest which I haven't seen much literature about is generating real-world knots that have good properties. For example if you look at the various knots, some knots have nice properties like "easy to untie" and "does not get tighter under load", which has huge impacts. These properties derive from the topology but also the physics of the knot. Would be nice to find a new hitch knot that worked better.
I'm reminded of the algorithms used to render planar drawings of proteins, which are similar to knots
Yes, like the PTGL - https://bio.tools/ptgl - or, er, TOPS diagrams. The main relationship to knot diagrams is really the chirality of beta-alpha-beta motifs (the majority of which are right handed).
Mathematics has probably already classified all knots that are human tieable. So from there you could iterate these knots in different physical positions, and perform different tests on them. The topological space has been investigated, now you need to decide between a teacup and a doughnut. This would be a mechanical engineering question, not a math question. Maybe look elsewhere?
math knots and real knots aren't the same thing and I don't think all possible human tieable knots have been enumerated and classified although I am happy to be pointed to evidence.
Math knots embed circles while real knots are typically made with free ended ropes (although some knots are not). Math knots ignore friction and the width of the rope; real knots can't ignore friction and the width (and other phyhsical properties) matter. See the comment in this article https://en.wikipedia.org/wiki/Overhand_knot
A knot with free ends can be turned into an embedded circle by connecting the ends far away from the knot. That's why they share a classification scheme.
The general properties of knot categories are already known. Real world factors such as the diameter of the line, its mechanical properties and environmental considerations (eg. presence of powder/dust, oil, rain, etc.) will significantly affect the actual deployment properties of a given knot. In addition, there are infinite places within a given knot that forces can be applied. Lines will also degrade over time, and factors such as complexity, time and fingers/hands/tools required-to-reliably tie/untie will also often be practical concerns in selection for deployment. Therefore, while your interest in the algorithmic exploration is interesting, if the goal is to generate novel practical results then there is a lot more complexity to add to the modeling before a useful result might be obtained, and any such result would have to be clearly based in assumptions around deployment scenario.
Sure. you're not telling me (a person who used to model knots in proteins using molecular dynamics, and works with FEA and other mechanical engineering tools) anything really novel.
Humans discovered hundreds of knots just playing around, and developed excellent knots in the past 400 years; new knots, never before tied, were invented some ~100 years ago. One imagines that a bit of searching with a computer might find a few cases that were overlooked.
new knots, never before tied, were invented some ~100 years ago.
No. I don't believe that. The archeological record has evidence of humans making rope for tens of thousands of years. I can believe that nobody alive had seen and named those particular interlocked pairs of overhand knots; that no written record of them existed until recently. But your claim that those knots had never been tied is quite presumptuous.
Comments
There's some fun stuff here (I'm reminded of the algorithms used to render planar drawings of proteins, which are similar to knots).
My real interest which I haven't seen much literature about is generating real-world knots that have good properties. For example if you look at the various knots, some knots have nice properties like "easy to untie" and "does not get tighter under load", which has huge impacts. These properties derive from the topology but also the physics of the knot. Would be nice to find a new hitch knot that worked better.
Yes, like the PTGL - https://bio.tools/ptgl - or, er, TOPS diagrams. The main relationship to knot diagrams is really the chirality of beta-alpha-beta motifs (the majority of which are right handed).
Thanks! I was searching for this citation to include in my comment: https://www.cambridge.org/core/journals/protein-science/arti...
(I have no idea how my brain can remember a paper from 20+ years ago but not enough to find it in the literature)
Mathematics has probably already classified all knots that are human tieable. So from there you could iterate these knots in different physical positions, and perform different tests on them. The topological space has been investigated, now you need to decide between a teacup and a doughnut. This would be a mechanical engineering question, not a math question. Maybe look elsewhere?
math knots and real knots aren't the same thing and I don't think all possible human tieable knots have been enumerated and classified although I am happy to be pointed to evidence.
Math knots embed circles while real knots are typically made with free ended ropes (although some knots are not). Math knots ignore friction and the width of the rope; real knots can't ignore friction and the width (and other phyhsical properties) matter. See the comment in this article https://en.wikipedia.org/wiki/Overhand_knot
While researching my answer I found out there's a term for what I wanted to do, https://en.wikipedia.org/wiki/Physical_knot_theory
I've had this discussion with mathematicians a few times now and they don't see the difference, so maybe I'm just missing something important.
A knot with free ends can be turned into an embedded circle by connecting the ends far away from the knot. That's why they share a classification scheme.
A lot of knot books and websites provide property-based classifications.
Yes. my goal would be to identify those properties from 3d models of knots, and a mechanism to generate plausible 3d models.
The general properties of knot categories are already known. Real world factors such as the diameter of the line, its mechanical properties and environmental considerations (eg. presence of powder/dust, oil, rain, etc.) will significantly affect the actual deployment properties of a given knot. In addition, there are infinite places within a given knot that forces can be applied. Lines will also degrade over time, and factors such as complexity, time and fingers/hands/tools required-to-reliably tie/untie will also often be practical concerns in selection for deployment. Therefore, while your interest in the algorithmic exploration is interesting, if the goal is to generate novel practical results then there is a lot more complexity to add to the modeling before a useful result might be obtained, and any such result would have to be clearly based in assumptions around deployment scenario.
Sure. you're not telling me (a person who used to model knots in proteins using molecular dynamics, and works with FEA and other mechanical engineering tools) anything really novel.
Humans discovered hundreds of knots just playing around, and developed excellent knots in the past 400 years; new knots, never before tied, were invented some ~100 years ago. One imagines that a bit of searching with a computer might find a few cases that were overlooked.
For example, take a look at https://en.wikipedia.org/wiki/Butterfly_loop and https://en.wikipedia.org/wiki/Butterfly_bend and https://en.wikipedia.org/wiki/Hunter%27s_bend and https://en.wikipedia.org/wiki/The_Ashley_Book_of_Knots
No. I don't believe that. The archeological record has evidence of humans making rope for tens of thousands of years. I can believe that nobody alive had seen and named those particular interlocked pairs of overhand knots; that no written record of them existed until recently. But your claim that those knots had never been tied is quite presumptuous.
OK, well go and do "a bit of searching with a computer" and let us know your novel results ignoring practical constraints, then.