Is there a nice lower bound on the algebraic degree based on the number of crossings? For the trefoil, I can do it with degree 6 by first describing it parametrically and then using trig identities. But is it clear that 6 is the minimum? From projecting it into two dimensions, it seems clear that it must have degree at least 4… - Cris

On May 24, 2020, at 9:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:

I believe Veit's idea will always work for "classical" knots" — knotted topological circles S^1 in 3-space (or more commonly in knot theory, in the 3-sphere S^3). Classical knots are assumed "tame", i.e., equivalent to a knot that is a finite closed polygon.

That is because a continuous function can be approximated by a smooth one, and a smooth one by a real analytic one, and a real analytic one by a polynomial. And for a tame knot, any small enough approximation will produce an equivalent knot.

So Yes, all classical knots (up to knot equivalence) can be described as the intersection of the zero-loci of two real polynomials P(x,y,z) and Q(x,y,z) defined on R^3. Therefore it makes sense to ask what is the minimum (say) sum of degrees of all such polynomial pairs that define an equivalent knot.

There's an invariant called the "genus" of a knot K in R^3 defined as follows: A "Seifert surface" of a knot is a compact orientable surface in R^3 whose boundary is the knot.

The genus of a compact orientable surface with boundary = C_1 u ... u C_n (disjoint circles) is the genus of the surface *without* boundary obtained by glueing the boundary of a disk D_j to the circle C_j for each j: the result is an n-holed torus for some n, and the genus is defined as n.

A knot has many Seifert surfaces, but among them there is a minimum genus. That's defined as the genus of the knot. I would guess there's a simple relationship between the genus of a knot and the polynomial invariant suggested by Veit.

—Dan

Adam Goucher 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.

Veit Elser wrote: ----- 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? ----- -----

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Cris Moore moore@santafe.edu "For the most part they were men with well-defined and sound ideas on everything concerning exports, banking, the fruit or wine trade; men of proved ability in handling problems relating to insurance, the interpretation of ill-drawn contracts, and the like…. But as regards plague their competence was practically nil.” — Albert Camus, The Plague

Here’s an intuition that that the algebraic degree must be at least twice the number of crossings (which gives 6 for the trefoil). Draw the knot, and now imagine a plane (basically the page it’s drawn on) so that at each crossing, one strand is above this plane and the other is below it. Then if the knot has c crossings, it must intersect this plane in at least 2c places, and the variety has 2c roots in the xy plane. But this isn’t a proof, since the knot diagram with the minimum number of crossings doesn’t necessarily mean that there is a flat plane with this property — it could be a curved surface. - Cris

Is there a nice lower bound on the algebraic degree based on the number of crossings?

For the trefoil, I can do it with degree 6 by first describing it parametrically and then using trig identities. But is it clear that 6 is the minimum? From projecting it into two dimensions, it seems clear that it must have degree at least 4…

- Cris

On May 24, 2020, at 9:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:

I believe Veit's idea will always work for "classical" knots" — knotted topological circles S^1 in 3-space (or more commonly in knot theory, in the 3-sphere S^3). Classical knots are assumed "tame", i.e., equivalent to a knot that is a finite closed polygon.

That is because a continuous function can be approximated by a smooth one, and a smooth one by a real analytic one, and a real analytic one by a polynomial. And for a tame knot, any small enough approximation will produce an equivalent knot.

So Yes, all classical knots (up to knot equivalence) can be described as the intersection of the zero-loci of two real polynomials P(x,y,z) and Q(x,y,z) defined on R^3. Therefore it makes sense to ask what is the minimum (say) sum of degrees of all such polynomial pairs that define an equivalent knot.

There's an invariant called the "genus" of a knot K in R^3 defined as follows: A "Seifert surface" of a knot is a compact orientable surface in R^3 whose boundary is the knot.

The genus of a compact orientable surface with boundary = C_1 u ... u C_n (disjoint circles) is the genus of the surface *without* boundary obtained by glueing the boundary of a disk D_j to the circle C_j for each j: the result is an n-holed torus for some n, and the genus is defined as n.

A knot has many Seifert surfaces, but among them there is a minimum genus. That's defined as the genus of the knot. I would guess there's a simple relationship between the genus of a knot and the polynomial invariant suggested by Veit.

—Dan

Adam Goucher 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.

Veit Elser wrote: ----- 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? ----- -----

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