5 ms·

Generating SVG for the prime knots

230 points by prideout 3y ago

l0b0 3y ago
Love the double hearts in https://prideout.net/blog/svg_knots/knots/8_5.svg https://prideout.net/blog/svg_knots/knots/8_5.svg
- gilleain 3y ago
  Also known as the 'true lovers knot' I believe: https://en.wikipedia.org/wiki/True_lover%27s_knot https://en.wikipedia.org/wiki/True_lover%27s_knot (It isn't the smallest non-alternating knot. This page says 8_19 is the smallest. https://mathworld.wolfram.com/NonalternatingKnot.html https://mathworld.wolfram.com/NonalternatingKnot.html)
phkahler 3y ago
I've never studied knots, but understanding the algorithms used here seems like a really good starting point.
OscarCunningham 3y ago
The unknot isn't a prime knot for the same reason that 1 isn't a prime number.
- deleted 3y ago
  [deleted]
- Syzygies 3y ago
  Yes. Alas, mathematicians not in number theory tend to have p-ness envy. A more troubling example of imposing this world view is decomposing 3-manifolds, where people like to think of S^1 x S^2 as prime. This only makes sense as a form of respect for one's elders. One constructs 3-manifolds by gluing irreducible manifolds together after cutting out balls; one needs S^3 itself to express self-loops in this gluing graph, if there are no other pieces. S^1 x S^2 is S^3 with one self-loop. But the interest is in the pieces, not the graph (which can be freely reorganized), unless one aspires to be a number theorist.
deleted 3y ago
[deleted]
dekhn 3y ago
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.
- gilleain 3y ago
  > 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 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).
  - dekhn 3y ago
    Thanks! I was searching for this citation to include in my comment: https://www.cambridge.org/core/journals/protein-science/article/abs/protein-structural-topology-automated-analysis-and-diagrammatic-representation/6D6686CF1D6E16F2761FF46FADC87BFC 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)
- contingencies 3y ago
  A lot of knot books and websites provide property-based classifications.
  - dekhn 3y ago
    Yes. my goal would be to identify those properties from 3d models of knots, and a mechanism to generate plausible 3d models.
    - 3y ago
zokier 3y ago
Neat visualization. I noticed though that the images (when enlarged) have some awkward angles etc to them, they are not super rounded and smooth shapes... maybe the width modulation could use some easing or interpolation or something?
- mlyle 3y ago
  I feel like that would be nice. It also might be nice to have a slight change in hue over the length of the knot, so that there is more contrast on crossings and things are a little clearer.
dexwiz 3y ago
Very cool. I have been trying to do something similar with tilings, but there is more information to be encoded. I am targeting something similar to the ability to render the tessellation catalog [1] and perform searches. Turns out there are multiple notations, and most of them use some level of implied knowledge. Also seems like rendering of knots is more open ended, while tilings are essentially their rendering. But they all do use a similar approach of Notation > Half Edge Data Structure > Render. [1] https://zenorogue.github.io/tes-catalog/?c= https://zenorogue.github.io/tes-catalog/?c=
- lovegrenoble 3y ago
  Not sure I learned anything about "Knot Theory" from playing this, but that was fun. Knot theory game: https://knots.netlify.app https://knots.netlify.app
samstave 3y ago
As a knotable person myself, this is AMAZING. Some thoughts on application of this knowledge would be to look at the patterns as you have described as cross-sections of rope weaving with the circular looming as the individual bobbins/spinny-things in an industrial loom - so rather that just woven-sheeth and full spin style cabling, one might achieve some really incredible properties in the woven elements along a axis such as these represent. Especially if you further differentiate btwn material and woven state (Are you weaving in an already spun set of filiments? What are the materials for the various inputs, and even further - imagine you have a set of elements in the loom where youre certain threads are the static, more rigid scaffold - like woven titanium strands which then feed into another loom which is weaving in the kevlar or other materials including a core of optics which is protected by the outer woven sheath from these patterns of 2D knots stretched out along an axis - certain elements can be printed such like the articulating spine of a snake. It could make a machinable-high-tensile strength cable with an optical core with protected turn radii (titanium snake spine) See here for reference to advanced cabling: https://www.youtube.com/watch?v=AD5aAd8Oy84 https://www.youtube.com/watch?v=AD5aAd8Oy84
razster 3y ago
How is it feasible to have a .svg file that has links to other .SVG files? https://prideout.net/blog/svg_knots/knottable_v1.svg https://prideout.net/blog/svg_knots/knottable_v1.svg. Has this capacity been around for some time?
- Scaevolus 3y ago
  You can have <a href="..."> links inside an SVG. It's been there since the first SVG standard back in 2001: https://www.w3.org/TR/2001/REC-SVG-20010904/linking.html https://www.w3.org/TR/2001/REC-SVG-20010904/linking.html
  - razster 3y ago
    Thanks for the info, I honestly didn't know and didn't have time to search today.
WhitneyLand 3y ago
Very nice work Philip. It’s so satisfying, I wish I knew an easy way to explain why.

230 points by prideout 3y ago

l0b0 3y ago

Love the double hearts in https://prideout.net/blog/svg_knots/knots/8_5.svg https://prideout.net/blog/svg_knots/knots/8_5.svg

gilleain 3y ago
Also known as the 'true lovers knot' I believe: https://en.wikipedia.org/wiki/True_lover%27s_knot https://en.wikipedia.org/wiki/True_lover%27s_knot (It isn't the smallest non-alternating knot. This page says 8_19 is the smallest. https://mathworld.wolfram.com/NonalternatingKnot.html https://mathworld.wolfram.com/NonalternatingKnot.html)

phkahler 3y ago

I've never studied knots, but understanding the algorithms used here seems like a really good starting point.

OscarCunningham 3y ago

The unknot isn't a prime knot for the same reason that 1 isn't a prime number.

deleted 3y ago
[deleted]
Syzygies 3y ago
Yes. Alas, mathematicians not in number theory tend to have p-ness envy. A more troubling example of imposing this world view is decomposing 3-manifolds, where people like to think of S^1 x S^2 as prime. This only makes sense as a form of respect for one's elders. One constructs 3-manifolds by gluing irreducible manifolds together after cutting out balls; one needs S^3 itself to express self-loops in this gluing graph, if there are no other pieces. S^1 x S^2 is S^3 with one self-loop. But the interest is in the pieces, not the graph (which can be freely reorganized), unless one aspires to be a number theorist.

deleted 3y ago

[deleted]

dekhn 3y ago

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.

gilleain 3y ago
> 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 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).
- dekhn 3y ago
  Thanks! I was searching for this citation to include in my comment: https://www.cambridge.org/core/journals/protein-science/article/abs/protein-structural-topology-automated-analysis-and-diagrammatic-representation/6D6686CF1D6E16F2761FF46FADC87BFC 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)
contingencies 3y ago
A lot of knot books and websites provide property-based classifications.
- dekhn 3y ago
  Yes. my goal would be to identify those properties from 3d models of knots, and a mechanism to generate plausible 3d models.
  - 3y ago

zokier 3y ago

Neat visualization. I noticed though that the images (when enlarged) have some awkward angles etc to them, they are not super rounded and smooth shapes... maybe the width modulation could use some easing or interpolation or something?

mlyle 3y ago
I feel like that would be nice. It also might be nice to have a slight change in hue over the length of the knot, so that there is more contrast on crossings and things are a little clearer.

dexwiz 3y ago

Very cool. I have been trying to do something similar with tilings, but there is more information to be encoded. I am targeting something similar to the ability to render the tessellation catalog [1] and perform searches. Turns out there are multiple notations, and most of them use some level of implied knowledge. Also seems like rendering of knots is more open ended, while tilings are essentially their rendering. But they all do use a similar approach of Notation > Half Edge Data Structure > Render. [1] https://zenorogue.github.io/tes-catalog/?c= https://zenorogue.github.io/tes-catalog/?c=

lovegrenoble 3y ago
Not sure I learned anything about "Knot Theory" from playing this, but that was fun. Knot theory game: https://knots.netlify.app https://knots.netlify.app

samstave 3y ago

As a knotable person myself, this is AMAZING. Some thoughts on application of this knowledge would be to look at the patterns as you have described as cross-sections of rope weaving with the circular looming as the individual bobbins/spinny-things in an industrial loom - so rather that just woven-sheeth and full spin style cabling, one might achieve some really incredible properties in the woven elements along a axis such as these represent. Especially if you further differentiate btwn material and woven state (Are you weaving in an already spun set of filiments? What are the materials for the various inputs, and even further - imagine you have a set of elements in the loom where youre certain threads are the static, more rigid scaffold - like woven titanium strands which then feed into another loom which is weaving in the kevlar or other materials including a core of optics which is protected by the outer woven sheath from these patterns of 2D knots stretched out along an axis - certain elements can be printed such like the articulating spine of a snake. It could make a machinable-high-tensile strength cable with an optical core with protected turn radii (titanium snake spine) See here for reference to advanced cabling: https://www.youtube.com/watch?v=AD5aAd8Oy84 https://www.youtube.com/watch?v=AD5aAd8Oy84

razster 3y ago

How is it feasible to have a .svg file that has links to other .SVG files? https://prideout.net/blog/svg_knots/knottable_v1.svg https://prideout.net/blog/svg_knots/knottable_v1.svg. Has this capacity been around for some time?

Scaevolus 3y ago
You can have <a href="..."> links inside an SVG. It's been there since the first SVG standard back in 2001: https://www.w3.org/TR/2001/REC-SVG-20010904/linking.html https://www.w3.org/TR/2001/REC-SVG-20010904/linking.html
- razster 3y ago
  Thanks for the info, I honestly didn't know and didn't have time to search today.

WhitneyLand 3y ago

Very nice work Philip. It’s so satisfying, I wish I knew an easy way to explain why.