What kind of knot is the square knot? I made an attempt to calculate the Connway notation of the square knot and it did not match with any of the prime knots. Either I did the calculation wrong or the prime knot is a composition of two primes.
Depending on which ends are fused, I have the following diagrams of a square knot.
The first diagram looks more like a link of two loops than a knot, but I am not sure.
This second diagram came from me tying a square knot and fusing the open end. This is from a single string so I don't think it is possible for it to be a loop.
The first diagram is actually equivalent to the unlink of two components. Notice how you can just slip one of them out of the other.
The second diagram represents a nontrivial knot, but it is not prime (this requires proof, but it's true) so you won't find it in the knot tables. A typical notation for it is $K_{3_1} \mathbin{\#} \overline{K}_{3_1}$ to indicate that it is the connected sum of the trefoil knot $K_{3_1}$ and its mirror image $\overline{K}_{3_1}$.