How can we express this sentence more concisely?

Question

Let $w_1,w_2,w_3,w_4$ be vertices of $K_4$.

The sentence I intend to express is the following:

Based on how the four vertices $w_i$'s are placed in the sphere, there are possibly two drawings of $K_4$.

Firstly, I am unsure if the statement is right because I have used the words $w_i$'s for the sake of brevity.

These similar expressions appear several times in my article. Of course, they may not all refer to the vertices of $K_4$, but instead consider the placement of $10$ vertices. To facilitate further discussion, it may be necessary to label these vertices as $w_1$, $w_2$, ..., $w_{10}$.

Answer 1

Based on how the four vertices $w_i$'s are placed in the sphere, there are possibly two drawings of $K_4$.

The phrase "possibly two" suggests uncertainty; otherwise, your sentence looks fine.

Since there are just four vertices, I would just list them.

Most concise:

• Based on the positions of the $21$ vertices, $w_i\,(i=1,2,\ldots,21),$ in the sphere, there are two possible drawings of $K_4.$

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Addendum

The rephrased version is much better. I have always felt that writing "$w_i$'s" like this is a bit problematic: I'm not sure if this notation can be used to represent "$w_1,$ $w_2,$ $w_3,$ $w_4$".

If we are requiring absolute perfectionistic clarity, then I'd firstly label the $i$'s (hehe, see what I've just done) so that they are no longer orphaned, then insert the two commas like I did, to leave without a doubt that that bit is not a restrictive clause, that is, that the immediate context has exactly four vertices and no others.