Until this point, we have proven that all 118 links of this list are non-trivial and distinct from each other. In order to prove that they "deserve" a place in the list, we also have to show that they are "prime". The reader may recall that when discussing the problem of knots, we defined as prime knots they ones that could not be decomposed to a connected sum of two or more non-trivial knots. In the case of links, we also have to consider the case of unlinked unions. Therefore, a link is prime if it cannot be decomposed to either a connected sum, or an unlinked union of other links, whether the links are trivial or knot. This is because the unlinked union of a link with a trivial link is not equivalent to the original link, while the connected sum of a link to the trivial knot is equivalent to the original knot.
One may easily show that if two links have crossing numbers L(1) and L(2), the crossing number of their unlinked union is L(1)+L(2)+1. (In the "crossing number" we also include the imaginary crossings that were removed when the original knot was converted to the link). If a presentation of this union cannot be unlinked (without resorting to equivalence moves), its number of crossings is at least L(1)+L(2)+3. This fact is very helpful in eliminating almost all combinations without even having to consider any of their invariants.
Unfortunately there is no proven such theorem for connected sums. In principle, it could be possible for the connected sum of these two links to have any crossing number between 3 and L(1)+L(2). One may only show that it cannot be trivial. Therefore, for any non-trivial link, one would have to consider all (infinite!) combinations of connected sums. To avoid such a situation, we are going to use a conjecture, that is similar to the case of links. Their crossing number is equal to L(1)+L(2), while any of its prime presentation should have a crossing number of at least L(1)+L(2)+2. Therefore, for the 118 links of our list, one need only consider the following combinations. ("T/K" stands for "Trivial Knot")
In the following page we are going to prove that none of the 118 links is equivalent to any of the combinations above.
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