My understanding is that when |p|≠ 1 ≠|q| and they are coprime for (p,q) torus knots, those knots are chiral and while rotation and translation in three dimensions cannot map a chiral knot to its obverse, the obverse of a (p,q) torus knot may be found in its (p,-q) form.

A parameterization of a (p,q) torus knot includes converting what would otherwise be z values into their additive inverses (z = -sin(qphi)) without which they would indicate a (p,-q) torus knot. What is the historical or practical rationale for requiring the additive inversion of z values in this context, i.e. why not invert the z axis to permit a (p,q) knot's z values to equal sin(qphi)?


I'm not exactly sure what the difference is between this question and Why does a a torus knot parameterization invert the sign of the z values?, but there is a little more to be said.

First, knots are usually studied in $S^3$, the unit sphere in $\mathbb{R}^4$, not $\mathbb{R}^3$. One can go back and forth between the two spaces using stereographic projection, and the parameterization Wikipedia gives for the torus knots are the result of such a stereographic projection.

Second, a torus knot is any knot confined to a torus $T\subset S^3$ such that the closure of each component of $S^3-T$ is a solid torus. As I described in my previous answer about this, given one of the two solid tori, there is a standard coordinate system on $T$ in terms of a longitude and a meridian. The classification of simple closed curves on $T$ is purely "homological," you only need to know how many longitudes and meridians the curve wrapped around $T$ to know the curve up to isotopy; the pair of numbers is commonly called the "slope." See Rolfsen's book Knots and Links for a comprehensive discussion of this. If we had chosen the other solid torus, the roles of longitude and meridian swap and we would have gotten $(q,p)$ instead of $(p,q)$. If we had chosen the opposite orientation for the longitude, we would have gotten $(-p,-q)$ instead. All of these are equivalent torus knots.

Third, there is a standard torus in $S^3$ called the Clifford torus, which has a flat metric and so $(p,q)$ can be thought of as an actual slope of a line (actually, a geodesic circle) on this torus. The isotopy between the $(p,q)$ and $(q,p)$ torus knots corresponds to a rotation of $S^3$. The $(p,q)$ and $(-p,-q)$ equivalence is also a rotation.

Fourth, this convention of having the meridian be a curve that has positive linking number with the core of the solid torus is related to the theory of complex hypersurfaces. If you take the zero set of the complex polynomial $z^p+w^q$ in $\mathbb{C}^2$ and see how it intersects with $S^3=\{(z,w)\in\mathbb{C}^2:z\overline{z}+w\overline{w}=1\}$, then you get the $(p,q)$ torus knot. You cannot get the $(p,-q)$ torus knot using polynomials.

For me, there are two takeaways:

  • The specific slope $(p,q)$ assigned to a torus knot comes a specific way of parameterizing a torus that bounds a solid torus on both sides. This is arbitrary, but there are good homological reasons for choosing this particular convention. Plus, it nicely matches what you get from links of complex hypersurfaces.

  • The parameterized curve on Wikipedia is just a way to give a representative torus knot. (Remember: any knot that is isotopic to such a curve is also a torus knot.) I have almost never contemplated this parameterization.

  • $\begingroup$ Thank you for elaborating on this line of questioning. I'm not sure about merge protocol on this site but I agree that the line of questioning is related. While all of it is relevant, your mention of the role of polynomials in complex hypersurfaces hits the spot regarding historical/practical basis for the convention. That and the Rolfsen recommendation should provide helpful context; thanks again. $\endgroup$ – bblohowiak Apr 1 at 20:40

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