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: