Prime Links with a Non-Prime Component

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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]:=

J31 = Jones[Knot[3, 1]]; J41 = Jones[Knot[4, 1]];

In[3]:=	CompositeJones = Expand[{J31[q] J31[q], J31[q] J31[1/q], J31[1/q] J31[1/q], J31[q] J41[q], J31[1/q] J41[q]}]
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 following program that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:

In[4]:=

SubLink[pd_PD, js_List] := Module[ {k, t0, t, t1, t2, S, P, T}, t0 = Flatten[List @@@ Skeleton[pd][[js]]]; t = pd /. x_X :> Select[x, MemberQ[t0, #] &]; t = DeleteCases[t, X[]]; k = 1; While[ k <= Length[t], If[ Length[t[[k]]] < 4, t = Delete[t, k] /. (Rule @@ t[[k]]), ++k]; ]; t1 = List @@ Union @@ t; t2 = Thread[(t1) -> Range[Length[t1]]]; S = t /. t2; P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S] ] SubLink[pd_PD, j_] := SubLink[pd, {j}]; SubLink[L_, js_] := SubLink[PD[L], js];

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