Can I solve an integral (or other tough problem) by playing with knots?

I like your question. Unfortunately, I do not have a general answer, but I do have a specific example where it is possible. You can look more into this if you wish in Rolfsen, Knots and Links under linking numbers, which is my reference. A link is just a few disjoint knots, which may be "linked" together. There is something called a Gauss Integral $\ell k (J,K)$ defined for a link which has components $J$ and $K$ which is

$$ \frac{1}{4\pi} \int_J \int_K \frac{ (x'-x)(dxdz'-dzdy')+(y'-y)(dzdx'-dxdz')+(z'-z)(dxdy'-dydx') }{ [(x'-x)^2+(y'-y)^2+(z'-z)^2]^\frac{3}{2} } $$

where $(x,y,z)\in J$ and $(x',y',z')\in K.$ This is obviously an integral you do not want to have to work out. Luckily, Rolfsen gives seven equivalent ways of computing the linking number. The most common of which is the following: For an oriented link diagram of $J\cup K$, only look at the crossing where $K$ is the overarc (on top). Then rotate the picture until $K$ is "pointing" up at one of the crossing. If $J$ goes from right to left, assign a $+1$ to this crossing. If $J$ goes from left to right, a $-1$. The sum of these is your linking number, up to a sign convention. So, you never have to do the integral!

This is, as I said, very specific. But maybe it gives some intuition.