Here’s an equation for a a 3-torus I made:
How did I make it? Consider describing it as a function f(x,y,z)=0. This is called an implicit function. Since it’s the surface of zero values and zero times anything is zero, we superimpose functions by multiplying them. So if t1, t2, and t3 are tori, t1*t2*t3 is 3 tori. Now to smooth it out we subtract a number which causes the surface to move forward along something called a gradient.
Voila, a 3-trous!