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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.

enter image description here

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.

enter image description here

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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 $\overline{K}_{3_1}$.

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  • $\begingroup$ This is perfect. Thanks, I was thinking it must be a connected version of a trefoil. I did not realize I could take a mirror. $\endgroup$
    – Alex
    Commented Dec 29, 2022 at 1:27
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    $\begingroup$ Rumination, which of these diagrams would you say represents a square knot. The crossings are the same, but how one connects the end of the strings makes all the difference. $\endgroup$
    – Alex
    Commented Dec 29, 2022 at 1:28
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    $\begingroup$ Definitely, I would call the second one a square knot since it’s actually knotted and since it only consists of one component. $\endgroup$ Commented Dec 29, 2022 at 5:37
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    $\begingroup$ By the way, some knots are equivalent to their mirror image, but most, including the trefoil, are not. Knot tables only list one of each pair. $\endgroup$ Commented Dec 29, 2022 at 5:43

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