Here is a letter to Nicolaas Kuiper that describes the construction of the torus knots and their self-linking numbers.

Mathematics Department

Brown University

Providence RI 02912

December 29, 1991

Dear Nico,

Your question about the self-linking number of (p,q) knots on various flat tori in the 3-sphere has given me a number of ideas that I hope will lead to some further results. I am pleased that several ideas have come from studying images on the computer, although I would have to agree that someone with a careful pencil and paper could reconstruct most of the examples. Let me make some preliminary observations that already become clear with just a few cases.

1) The self-linking number of a curve on the 3-sphere is not given simply by the self-linking number of its image under stereographic projection from a point not on the curve to 3-space. The easiest way to see this is to consider a (3,2) knot, and observe that it projects stereographically to a (3,2) torus knot from (0,0,0,1) and to a (2,3) torus knot from (0,1,0,0). The former has self-linking number 3 and the latter has self-linking number 4.

2) It may be an interesting question to determine the average of the self-linking numbers of the stereographic images of a knot, but we cannot expect this to be the self-linking number of the curve, since in the above case this self-linking number of the (3,2) curve on the 3-sphere is 6. Observe that the involution (x,y,u,v)-->(u,v,x,y) on the 3-sphere is an isometry that preserves the self-linking of the curve, so we cannot distinguish between a (3,2) torus knot and a (2,3) knot on the 3-sphere. Both have self-linking number 6, and in general a (p,q) knot will have self-linking number pq.

3) To compute the self-linking number, we take the steregraphic projection not only of the curve but also of its push-off in the direction of the principal normal on the sphere. We can think of this as a thin strip formed by the segments from points of the curve to the corresponding points on the push-off. The linking number of these two curves will be independent of the point from which the stereographic projection is taken, and we may compute this number by projecting the curve into a plane in 3-space. When we do so, we find that in general the strip is immersed except at a number of points where "twisting" occurs, as well as a finite number of places where the planar projection of the original curve has a double point. In the case of the torus knots above, the planar projection of the original curve will be locally convex, so the self- linking number of the image of the curve in 3-space can be read off as the algebraic number of crossing points. But each twist contributes to the algebraic number of crossings of the curve in 3- space with its push-off in the strip. So for example we get 6 such twist points in the projection of the (3,2) curve, at three of which the original curve crosses over its push-off, with the same algebraic crossing numbers, giving the anticipated value 6. In the case of the (2,3) curve, we get 4 twist points, at two of which the original curves crosses over the push-off with the same algebraic crossing numbers, again yielding 6.

4) Consider the curve

X(t) = (cos(a) cos(pt), cos(a) sin(pt), sin(a) cos(qt), sin(a) sin(qt))

X ´ (t) = (-p cos(a) sin(pt), p cos(a) cos(pt),-q sin(a) sin(qt), q sin(a) cos(qt))

X ´´ (t) = (-p² cos(a) cos(pt), -p² cos(a) sin(pt), -q² sin(a) cos(qt), -q² sin(a) sin(qt))

Note that X(t) · X ´ (t) = 0 and X(t) · X ´´ (t) = -p² cos2(a) -q² sin2(a), so

X ´´ (t) - (X ´´ (t) · X(t)) X(t) =
(p² - q²) cos(a) sin(a) [-sin(a) cos(pt), -sin(a) sin(pt), cos(a) cos(qt), cos(a) sin(qt)].

The principal normal P(t) in the tangent space of the sphere is therefore

P(t) = (-sina)cos(pt), -sin(a)sin(pt), cos(a)cos(qt), cos(a)sin(qt)).

The push-off in the direction of P(t) at distance s is then

X(t) + sP(t) = [(cos(a)-s sin(a))cos(pt), (cos(a)-s sin(a))sin(pt), (sin(a)+s cos(a))cos(qt), (sin(a)+s cos(a))sin(qt)].

We may normalize to obtain the spherical push-off curve

[X(t) + sP(t)]/|X(t) + sP(t)| = [(cos(a)-s sin(a))cos(pt), (cos(a)-s sin(a))sin(pt), (sin(a)+s cos(a))cos(qt), (sin(a)+s cos(a))sin(qt)]/sqrt(1 + s²).

The stereographic projection of this curve from the point (0,0,0,1) will be

Y(t,s) = [(cos(a)-s sin(a))cos(pt), (cos(a)-s sin(a))sin(pt), (sin(a)+s cos(a))cos(qt)]/[sqrt(1 + s²) -(sin(a)+s cos(a))sin(qt)].

We wish to calculate the linking number of Y(t,0) with Y(t,s) for arbitrarily small s. If we project Y(t,0) to the x-y-plane, we obtain

(cos(a)cos(pt), cos(a)sin(pt))/[1 -sin(a)sin(qt)]. This is a locally convex curve with q(p-1) crossings.

We get a twisting if Y(t,0) = Y(t,s), at least in the limit as s tends to zero. This leads to the condition

[(cos(a)-s sin(a))cos(pt), (cos(a)-s sin(a))sin(pt)] [1 -sin(a)sin(qt)] =(cos(a)cos(pt), cos(a)sin(pt))[sqrt(1 + s²) -(sin(a)+s cos(a))sin(qt)],

or

(cos(a)-s sin(a))[1 -sin(a)sin(qt)] = cos(a)[sqrt(1 + s²) -(sin(a)+s cos(a))sin(qt)]

Thus

cos(a) -s sin(a) - cos(a)sin(a)sin(qt) + s sin2(a)sin(qt) =

cos(a)sqrt(1 + s²) - cos(a)sin(a)sin(qt) - s cos2(a)sin(qt)

so

s sin(qt) = cos(a)[sqrt(1 + s²) -1] - s sin(a).

Dividing by s and simplifying gives

sin(qt) = cos(a) s/[sqrt(1 + s²) +1] - sin(a).

As s approaches 0, we obtain the condition sin(qt) = -sin(a), which has 2q solutions for every value of a strictly between 0 and pi. At exactly half of these points, the image of original curve passes over the image of the perturbed curve, with the same algebraic index as at the q(p-1) crossings of the image of the original curve. Thus the linking number of these two curves in 3-space, and hence the self- linking number of the curve on the 3-sphere, is pq.

There, Nico, I hope that that is a correct answer to the question that you asked me a few days ago. I am sorry I could not get to it sooner, but I am glad to think about these things now as I get a bit more free time. Please give my regards to your family and to all of our mutual friends. And let me know what you think of my argument above. Gelukkig Nieuw Jaar, en tot ziens!

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