Prime Links with a Non-Prime Component

Let us find all (prime!) links in the Knot Atlas that have a non-prime component. Since the links listed in the Knot Atlas have at most 11 crossings, such a component may only be the sum of exactly two knots chosen among the trefoil, the figure eight knot, and their mirror images. The figure eight knot's mirror image is itself so we have five possibilities. Computing the Jones polynomial of each, we get:

(For In[1] see Setup)

 `In[2]:=` `K31 = Knot[3, 1]; K41 = Knot[4, 1];`
 `In[3]:=` ```CompositeJones = Jones[#][q] & /@ {ConnectedSum[K31, K31], ConnectedSum[K31, Mirror[K31]], ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41], ConnectedSum[Mirror[K31], K41]}``` `Out[3]=` ``` -8 2 -6 2 2 -2 -3 -2 1 2 3 {q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q , 7 5 4 q q q q 2 4 5 6 7 8 q + 2 q - 2 q + q - 2 q + q , -6 2 2 3 3 2 -1 - q + -- - -- + -- - -- + - + q, 5 4 3 2 q q q q q 1 2 3 4 5 6 -1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q } q```

Now, we can use the program `SubLink` that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:

 `In[4]:=` `Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];`

Using `SubLink` and the Jones polynomials of the five composite knots mentioned above, we can find all links that have one of these as a component:

 `In[5]:=` ```NonPrimeComponentQ[L_] := Or @@ (MemberQ[CompositeJones, Jones[SubLink[L, #]][q]] & /@ Range[Length[Skeleton[L]]])```
 `In[6]:=` `Exceptions= Select[AllLinks[], NonPrimeComponentQ]` `Out[6]=` ```{Link[10, Alternating, 38], Link[10, Alternating, 39], Link[10, Alternating, 46], Link[10, NonAlternating, 35], Link[10, NonAlternating, 36], Link[10, NonAlternating, 37], Link[10, NonAlternating, 38], Link[10, NonAlternating, 39], Link[11, Alternating, 91], Link[11, Alternating, 92], Link[11, Alternating, 93], Link[11, Alternating, 95], Link[11, Alternating, 121], Link[11, Alternating, 128], Link[11, Alternating, 130], Link[11, NonAlternating, 110], Link[11, NonAlternating, 111], Link[11, NonAlternating, 112], Link[11, NonAlternating, 113], Link[11, NonAlternating, 114], Link[11, NonAlternating, 115]}```

Thus, there are 21 links in the Knot Atlas that have a non-prime component. The first eight of those are:

 L10a38 L10a39 L10a46 L10n35
 L10n36 L10n37 L10n38 L10n39