Re: [math-fun] [xbbn] Graduate Student Solves Decades-Old Conway Knot Problem | Quanta Magazine
Yup. A p-sphere and a q-sphere can link when they're embedded disjoint from each other in (p+q+1)-dimensional space. (Even when p = q = 0.) —Dan Cris Moore wrote: ----- Thanks. I wonder where I got this 2d+1 thing… maybe I’m misremembering some other embedding theorem. I do like the 5-dimensional construction of two linked 2-spheres, though, assuming I got that right. -----
huh! so it seems like d-surfaces can be knotted, linked, etc. in various ways in dimensions ranging from d+2 up to 2d+1.
On May 23, 2020, at 9:14 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yup. A p-sphere and a q-sphere can link when they're embedded disjoint from each other in (p+q+1)-dimensional space. (Even when p = q = 0.)
—Dan
Cris Moore wrote: ----- Thanks. I wonder where I got this 2d+1 thing… maybe I’m misremembering some other embedding theorem.
I do like the 5-dimensional construction of two linked 2-spheres, though, assuming I got that right. -----
Well now. First, I'm embarrassed that I misspelled Dr. Piccirillo's name. Second, I apologize if I gave the impression that I had read and understood the proof. I only read and roughly understood the outline of the summary in Quanta. Perhaps it was silly of me to think I could glean any sense of Piccirillo's style from the description of her work. I was mostly admiring the structure of the "trace sibling" she constructed I wouldn't expect to understand her paper in Annals right off the bat; knot theory is an old discipline with deep traditions. But if I wanted to, I imagine I'd start by consulting her references, and if I couldn't understand them, recurse. This approach has worked okay for me before (though it has also failed, notably in anything having to do with algebraic topology). On Sat, May 23, 2020 at 11:14 PM Dan Asimov <dasimov@earthlink.net> wrote:
Yup. A p-sphere and a q-sphere can link when they're embedded disjoint from each other in (p+q+1)-dimensional space. (Even when p = q = 0.)
—Dan
Cris Moore wrote: ----- Thanks. I wonder where I got this 2d+1 thing… maybe I’m misremembering some other embedding theorem.
I do like the 5-dimensional construction of two linked 2-spheres, though, assuming I got that right. -----
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From what the story seems to say, this construction was the main insight to completing the proof. So why not make the article an opportunity to talk more about similar constructions? (and give easier examples?)
All the article says is: <<Constructing trace siblings is a tricky business, but Piccirillo was an expert. “That’s just, like, a trade I’m in,” she said. “So I just went home and did it.”>> It could help to read all references and backtrack, but with trades come trade secrets. Not only that, Piccirillo must have some sort of special perspective, which isn't easily available to the public. Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read. --Brad On Sat, May 23, 2020 at 10:27 PM Allan Wechsler <acwacw@gmail.com> wrote:
I was mostly admiring the structure of the "trace sibling" she constructed
I don’t understand the animus here. A young person has solved a long-standing open problem with a clever construction, which deserves to be celebrated. Quanta articles rarely explain proof techniques: they play a specific journalistic role roughly at the old Scientific American level, for readers who want to feel they got a taste of the ideas. I know a little about knot and manifold invariants, and they can be introduced at a level accessible to high schoolers, but this isn’t what Quanta does… this would require an entire article of equal or greater length. How about learning about her proof and writing an accessible exposition?
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
She doesn’t have students yet, because she just graduated. Cris
On May 23, 2020, at 10:45 PM, Brad Klee <bradklee@gmail.com> wrote:
From what the story seems to say, this construction was the main insight to completing the proof. So why not make the article an opportunity to talk more about similar constructions? (and give easier examples?)
All the article says is: <<Constructing trace siblings is a tricky business, but Piccirillo was an expert. “That’s just, like, a trade I’m in,” she said. “So I just went home and did it.”>>
It could help to read all references and backtrack, but with trades come trade secrets. Not only that, Piccirillo must have some sort of special perspective, which isn't easily available to the public.
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
--Brad
On Sat, May 23, 2020 at 10:27 PM Allan Wechsler <acwacw@gmail.com> wrote:
I was mostly admiring the structure of the "trace sibling" she constructed
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I think she hasn't just graduated, exactly -- I get the impression she did a post-doc and stuff, and the article says she's been offered a position at MIT. On Sun, May 24, 2020 at 12:47 AM Cris Moore via math-fun < math-fun@mailman.xmission.com> wrote:
I don’t understand the animus here. A young person has solved a long-standing open problem with a clever construction, which deserves to be celebrated. Quanta articles rarely explain proof techniques: they play a specific journalistic role roughly at the old Scientific American level, for readers who want to feel they got a taste of the ideas.
I know a little about knot and manifold invariants, and they can be introduced at a level accessible to high schoolers, but this isn’t what Quanta does… this would require an entire article of equal or greater length. How about learning about her proof and writing an accessible exposition?
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
She doesn’t have students yet, because she just graduated.
Cris
On May 23, 2020, at 10:45 PM, Brad Klee <bradklee@gmail.com> wrote:
From what the story seems to say, this construction was the main insight to completing the proof. So why not make the article an opportunity to talk more about similar constructions? (and give easier examples?)
All the article says is: <<Constructing trace siblings is a tricky business, but Piccirillo was an expert. “That’s just, like, a trade I’m in,” she said. “So I just went home and did it.”>>
It could help to read all references and backtrack, but with trades come trade secrets. Not only that, Piccirillo must have some sort of special perspective, which isn't easily available to the public.
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
--Brad
On Sat, May 23, 2020 at 10:27 PM Allan Wechsler <acwacw@gmail.com> wrote:
I was mostly admiring the structure of the "trace sibling" she constructed
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My mistake. But she’s not a faculty member yet, so she hasn’t had to think about recruiting students. When she does, she’ll face the same challenge all young faculty do, of going from the technical (and often incremental, though valuable) work of their theses to conveying a larger perspective about their field. But those of us outside her field can, if we want, try to build pedagogical bridges that make her work accessible, even if Quanta didn’t do so. I don’t have the bandwidth to do this myself, but I have the impression that it is in reach if we feel strongly about it. C
On May 23, 2020, at 10:49 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I think she hasn't just graduated, exactly -- I get the impression she did a post-doc and stuff, and the article says she's been offered a position at MIT.
On Sun, May 24, 2020 at 12:47 AM Cris Moore via math-fun < math-fun@mailman.xmission.com> wrote:
I don’t understand the animus here. A young person has solved a long-standing open problem with a clever construction, which deserves to be celebrated. Quanta articles rarely explain proof techniques: they play a specific journalistic role roughly at the old Scientific American level, for readers who want to feel they got a taste of the ideas.
I know a little about knot and manifold invariants, and they can be introduced at a level accessible to high schoolers, but this isn’t what Quanta does… this would require an entire article of equal or greater length. How about learning about her proof and writing an accessible exposition?
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
She doesn’t have students yet, because she just graduated.
Cris
On May 23, 2020, at 10:45 PM, Brad Klee <bradklee@gmail.com> wrote:
From what the story seems to say, this construction was the main insight to completing the proof. So why not make the article an opportunity to talk more about similar constructions? (and give easier examples?)
All the article says is: <<Constructing trace siblings is a tricky business, but Piccirillo was an expert. “That’s just, like, a trade I’m in,” she said. “So I just went home and did it.”>>
It could help to read all references and backtrack, but with trades come trade secrets. Not only that, Piccirillo must have some sort of special perspective, which isn't easily available to the public.
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
--Brad
On Sat, May 23, 2020 at 10:27 PM Allan Wechsler <acwacw@gmail.com> wrote:
I was mostly admiring the structure of the "trace sibling" she constructed
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The best way to celebrate a new result is to learn what it means for yourself. If there is animus here, it’s toward ignorance, including my own. Thanks Allan for the reference pointer, I will look into it and see if it says anything about knot isomorphism classes. I have a lot of catching up to do. Even if I can’t learn the new proof, maybe I can learn something. —Brad
On May 23, 2020, at 11:47 PM, Cris Moore via math-fun <math-fun@mailman.xmission.com> wrote:
I don’t understand the animus here. A young person has solved a long-standing open problem with a clever construction, which deserves to be celebrated. Quanta articles rarely explain proof techniques: they play a specific journalistic role roughly at the old Scientific American level, for readers who want to feel they got a taste of the ideas.
I know a little about knot and manifold invariants, and they can be introduced at a level accessible to high schoolers, but this isn’t what Quanta does… this would require an entire article of equal or greater length. How about learning about her proof and writing an accessible exposition?
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
She doesn’t have students yet, because she just graduated.
Cris
On May 23, 2020, at 10:45 PM, Brad Klee <bradklee@gmail.com> wrote:
From what the story seems to say, this construction was the main insight to completing the proof. So why not make the article an opportunity to talk more about similar constructions? (and give easier examples?)
All the article says is: <<Constructing trace siblings is a tricky business, but Piccirillo was an expert. “That’s just, like, a trade I’m in,” she said. “So I just went home and did it.”>>
It could help to read all references and backtrack, but with trades come trade secrets. Not only that, Piccirillo must have some sort of special perspective, which isn't easily available to the public.
Another question is: what does Piccirillo teach to students whom she is interested in attracting to her research group? The answer to this question could lead to a better article for the general public to read.
--Brad
On Sat, May 23, 2020 at 10:27 PM Allan Wechsler <acwacw@gmail.com> wrote:
I was mostly admiring the structure of the "trace sibling" she constructed
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Here’s a graduate-student-level question about knots — the deepest I’m capable of: Is it possible to define a “real-algebraic degree” for knots, and if so, has it been tabulated? My idea is to first construct knots as non-singular algebraic varieties, i.e. as the solution set of a pair of polynomial equations in x,y,z, and I’m using non-singular to mean that locally, near every point of the solution set, the linear approximation of the variety is two planes intersecting as a line. For example, x^2+y^2=1 z=0 defines the unknot. The variety defined by the pair of equations will in general have several disconnected components. We only care if *one* of these components is our target knot, and if so, we take some suitable “minimal” combination of the degrees of the two polynomials as the “real-algebraic degree” of our knot, e.g. min(d1+d2), min(max(d1,d2)). By the second definition, the unknot would have degree 2. Two specific questions: 1) Can all knots be constructed in this way? 2) What’s the degree of the trefoil? -Veit
On May 24, 2020, at 1:07 AM, Brad Klee <bradklee@gmail.com> wrote:
The best way to celebrate a new result is to learn what it means for yourself. If there is animus here, it’s toward ignorance, including my own. Thanks Allan for the reference pointer, I will look into it and see if it says anything about knot isomorphism classes. I have a lot of catching up to do. Even if I can’t learn the new proof, maybe I can learn something. —Brad
For the trefoil, one way to construct the knot is to take the 2-variable *complex* parameterisation: a^2 = b^3, |a|^2 + |b|^2 = 1 and then 'compile' that into a 4-variable real parameterisation (where the second equation is just the equation for the 3-sphere), and then stereographically project that into a 3-variable real parameterisation. More generally, this idea will work for every torus knot.
Sent: Sunday, May 24, 2020 at 1:51 PM From: "Veit Elser" <ve10@cornell.edu> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] [xbbn] Graduate Student Solves Decades-Old Conway Knot Problem | Quanta Magazine
Here’s a graduate-student-level question about knots — the deepest I’m capable of:
Is it possible to define a “real-algebraic degree” for knots, and if so, has it been tabulated?
My idea is to first construct knots as non-singular algebraic varieties, i.e. as the solution set of a pair of polynomial equations in x,y,z, and I’m using non-singular to mean that locally, near every point of the solution set, the linear approximation of the variety is two planes intersecting as a line. For example,
x^2+y^2=1 z=0
defines the unknot.
The variety defined by the pair of equations will in general have several disconnected components. We only care if *one* of these components is our target knot, and if so, we take some suitable “minimal” combination of the degrees of the two polynomials as the “real-algebraic degree” of our knot, e.g. min(d1+d2), min(max(d1,d2)). By the second definition, the unknot would have degree 2.
Two specific questions:
1) Can all knots be constructed in this way? 2) What’s the degree of the trefoil?
-Veit
On May 24, 2020, at 1:07 AM, Brad Klee <bradklee@gmail.com> wrote:
The best way to celebrate a new result is to learn what it means for yourself. If there is animus here, it’s toward ignorance, including my own. Thanks Allan for the reference pointer, I will look into it and see if it says anything about knot isomorphism classes. I have a lot of catching up to do. Even if I can’t learn the new proof, maybe I can learn something. —Brad
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This is a nice problem and solution. There is a very fun connection with Riemann surfaces of elliptic curves once we identify C^2~R^4. So I spent a couple of hours drawing a (3,4) torus knot in Harold Edwards's normal form: https://0x0.st/ipSo.png https://0x0.st/ipSi.png https://0x0.st/ipS-.png When you have an algebraic definition, then you can integrate the knot to obtain a complex period. For the integral you don't necessarily need an algebraic definition of the knot itself. The dimensions of the period rectangle can be calculated on algebraic cycles. The knot takes a periodic trajectory through the complex, doubly-period plane of time, so its period can be counted as a pair of integers and then multiplied by the scale factor. The knot depicted above has period (3*4+4*2*i)*K(1/2), where K is the complete elliptic integral of the first kind. So, there it is, a congratulations to Lisa Piccirillo! I couldn't understand the proof, but it did, in some way, inspire me to integrate a knot period. --Brad PS. For more info on the calculation above, refer to Section IV of "An Alternative Theory of Simple Pendulum libration": https://github.com/bradklee/Dissertation/blob/master/SimplePendulum/SimplePe... On Sun, May 24, 2020 at 8:20 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
For the trefoil, one way to construct the knot is to take the 2-variable *complex* parameterisation:
a^2 = b^3, |a|^2 + |b|^2 = 1
and then 'compile' that into a 4-variable real parameterisation (where the second equation is just the equation for the 3-sphere), and then stereographically project that into a 3-variable real parameterisation.
More generally, this idea will work for every torus knot.
Sent: Sunday, May 24, 2020 at 1:51 PM From: "Veit Elser" <ve10@cornell.edu> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] [xbbn] Graduate Student Solves Decades-Old Conway Knot Problem | Quanta Magazine
Here’s a graduate-student-level question about knots — the deepest I’m capable of:
Is it possible to define a “real-algebraic degree” for knots, and if so, has it been tabulated?
My idea is to first construct knots as non-singular algebraic varieties, i.e. as the solution set of a pair of polynomial equations in x,y,z, and I’m using non-singular to mean that locally, near every point of the solution set, the linear approximation of the variety is two planes intersecting as a line. For example,
x^2+y^2=1 z=0
defines the unknot.
The variety defined by the pair of equations will in general have several disconnected components. We only care if *one* of these components is our target knot, and if so, we take some suitable “minimal” combination of the degrees of the two polynomials as the “real-algebraic degree” of our knot, e.g. min(d1+d2), min(max(d1,d2)). By the second definition, the unknot would have degree 2.
Two specific questions:
1) Can all knots be constructed in this way? 2) What’s the degree of the trefoil?
-Veit
On May 24, 2020, at 1:07 AM, Brad Klee <bradklee@gmail.com> wrote:
The best way to celebrate a new result is to learn what it means for yourself. If there is animus here, it’s toward ignorance, including my own. Thanks Allan for the reference pointer, I will look into it and see if it says anything about knot isomorphism classes. I have a lot of catching up to do. Even if I can’t learn the new proof, maybe I can learn something. —Brad
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On Sun, May 24, 2020 at 9:20 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
For the trefoil, one way to construct the knot is to take the 2-variable *complex* parameterisation:
a^2 = b^3, |a|^2 + |b|^2 = 1
I think you want |a|^2 = |b|^2 = 1/2 (a torus) rather than |a|^2 + |b|^2 = 1 (a 3-sphere). It makes sense that to define a 1-dimensional curve in C^2, which is R^4 with some extra structure, we need 3 real equations, not 2. But to get this to be an algebraic curve embedded in R^3, we need to specify a polynomial map from the torus |a|^2 = |b|^2 = 1 to a torus in 3-space. That's where you use sterographic projection to map the 3-sphere |a|^2 + |b|^2 = 1 to R^3, using the fact that the torus |a|^2 = |b|^2 = 1 lies in the 3-sphere |a|^2 + |b|^2 = 1.
and then 'compile' that into a 4-variable real parameterisation (where the second equation is just the equation for the 3-sphere), and then stereographically project that into a 3-variable real parameterisation.
More generally, this idea will work for every torus knot.
Sent: Sunday, May 24, 2020 at 1:51 PM From: "Veit Elser" <ve10@cornell.edu> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] [xbbn] Graduate Student Solves Decades-Old Conway Knot Problem | Quanta Magazine
Here’s a graduate-student-level question about knots — the deepest I’m capable of:
Is it possible to define a “real-algebraic degree” for knots, and if so, has it been tabulated?
My idea is to first construct knots as non-singular algebraic varieties, i.e. as the solution set of a pair of polynomial equations in x,y,z, and I’m using non-singular to mean that locally, near every point of the solution set, the linear approximation of the variety is two planes intersecting as a line. For example,
x^2+y^2=1 z=0
defines the unknot.
The variety defined by the pair of equations will in general have several disconnected components. We only care if *one* of these components is our target knot, and if so, we take some suitable “minimal” combination of the degrees of the two polynomials as the “real-algebraic degree” of our knot, e.g. min(d1+d2), min(max(d1,d2)). By the second definition, the unknot would have degree 2.
Two specific questions:
1) Can all knots be constructed in this way? 2) What’s the degree of the trefoil?
-Veit
On May 24, 2020, at 1:07 AM, Brad Klee <bradklee@gmail.com> wrote:
The best way to celebrate a new result is to learn what it means for yourself. If there is animus here, it’s toward ignorance, including my own. Thanks Allan for the reference pointer, I will look into it and see if it says anything about knot isomorphism classes. I have a lot of catching up to do. Even if I can’t learn the new proof, maybe I can learn something. —Brad
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-- Andy.Latto@pobox.com
For the trefoil, one way to construct the knot is to take the 2-variable *complex* parameterisation:
a^2 = b^3, |a|^2 + |b|^2 = 1
I think you want |a|^2 = |b|^2 = 1/2 (a torus) rather than |a|^2 + |b|^2 = 1 (a 3-sphere). It makes sense that to define a 1-dimensional curve in C^2, which is R^4 with some extra structure, we need 3 real equations, not 2.
The first of these equations is a complex constraint (therefore two real constraints). Taking absolute values of the first equation gives: |a|^2 = |b|^3 which, together with the second equation, results in: |b|^3 + |b|^2 = 1 which implies that |b| is the unique positive real root of that cubic equation (and therefore |b| ~= 0.754878). This, in turn, forces |a| to be a constant, so (a, b) lies on a Clifford torus as you require. Best wishes, Adam P. Goucher
On May 25, 2020, at 9:38 AM, Andy Latto <andy.latto@pobox.com> wrote:
For the trefoil, one way to construct the knot is to take the 2-variable *complex* parameterisation:
a^2 = b^3, |a|^2 + |b|^2 = 1
I think you want |a|^2 = |b|^2 = 1/2 (a torus) rather than |a|^2 + |b|^2 = 1 (a 3-sphere). It makes sense that to define a 1-dimensional curve in C^2, which is R^4 with some extra structure, we need 3 real equations, not 2.
But to get this to be an algebraic curve embedded in R^3, we need to specify a polynomial map from the torus |a|^2 = |b|^2 = 1 to a torus in 3-space. That's where you use sterographic projection to map the 3-sphere |a|^2 + |b|^2 = 1 to R^3, using the fact that the torus |a|^2 = |b|^2 = 1 lies in the 3-sphere |a|^2 + |b|^2 = 1.
Andy, I think Adam’s embedding works and I like it. Let a=r exp(i p), b=s exp(i q), where r and s are positive. Then a^2=b^3 requires r=s (=1/sqrt(2) by the 3-sphere constraint), and 2p=3q mod 2pi. That’s the curve that winds around the torus trefoil-like. But Adam’s statement "More generally, this idea will work for every torus knot.” is intriguing. It suggests that f(a,b)=0, |a|^2+|b|^2=1 where f is a polynomial, might fall short of constructing all knots. Does the set of knots constructible by this route have a name? Is this subset easier to classify than general knots? Long ago I thought the best way to classify knots was to do so within each genus. After all, genus-1 knots (torus knots) are very straightforward. But it seems the complexity skyrockets already at genus-2! I like the algebraic approach, like Adam’s, because it opens the possibility of a natural taxonomy. Consider the space defined by the monomial coefficients, which is finite dimensional if we bound the degree. This space has singular loci where one knot/link combination transforms into another. Reidemeister moves also give you a taxonomy, but the algebraic scheme may be more “natural”. -Veit
participants (7)
-
Adam P. Goucher -
Allan Wechsler -
Andy Latto -
Brad Klee -
Cris Moore -
Dan Asimov -
Veit Elser