I am going through the paper, Physical Knots, by Jonathan Simon. Initially, on page 10, he describes the electrostatic energy of two charged stick, X and Y in space as follows.
$$ \int_{x \in X} \int_{y \in Y} \frac{dxdy}{||x-y||} $$
Now, he claims that it doesn't have the basic mathematical property which we want for knot energies. We want the energy to be infinite when the two charged stick cross one another. Here is the screenshot of the section.
I don't get this claim. When X intersects Y, $||x-y|| = 0$ which makes the energy infinite, isn't it?
You're mixing up the integrand and the integral. The knot's energy here is the integral of the energy density along the curve. Observing that the integrand blows up when $x=y$ is not sufficient to conclude that the integral is infinite; there are functions which blow up, yet can be integrated to a finite number.
An example of such a function: $$ \int_{-1}^1 \frac{1}{\sqrt{|x|}}\ dx = 4$$ The integrand is singular at $x=0$, but the integral converges nicely (it's a $p$-integral with $p < 1$ on a neighborhood of $0$).
In this case, consider the example of $X = [-1,1]\times \{0\}\subset\mathbb{R}^2$ and $Y = \{0\}\times[-1,1]\subset\mathbb{R}^2$. Then integrate: $$\int_{-1}^1\int_{-1}^1 \frac{dxdy}{\sqrt{x^2 + y^2}}$$ Mathematica tells me this is $8\sinh^{-1}(1)<\infty$. That is, the proposed knot energy of two intersecting segments is not infinite (and you want it to be infinite for intersections so that as the embedding flows along the energy gradient, it doesn't change knot type).