Prime Links with a Non-Prime Component: Difference between revisions

From Knot Atlas
Jump to navigationJump to search
No edit summary
No edit summary
Line 10: Line 10:
{{In|
{{In|
n = 2 |
n = 2 |
in = <nowiki>J31 = Jones[Knot[3, 1]]; J41 = Jones[Knot[4, 1]];</nowiki>}}
in = <nowiki>K31 = Knot[3, 1]; K41 = Knot[4, 1];</nowiki>}}
<!--END-->
<!--END-->


Line 22: Line 22:
n = 3 |
n = 3 |
in = <nowiki>CompositeJones =
in = <nowiki>CompositeJones =
Expand[{J31[q] J31[q], J31[q] J31[1/q], J31[1/q] J31[1/q],
Jones[#][q] & /@ {ConnectedSum[K31, K31],
ConnectedSum[K31, Mirror[K31]],
J31[q] J41[q], J31[1/q] J41[q]}]</nowiki> |
ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],
ConnectedSum[Mirror[K31], K41], ConnectedSum[K41, K41]}</nowiki> |
out= <nowiki> -8 2 -6 2 2 -2 -3 -2 1 2 3
out= <nowiki> -8 2 -6 2 2 -2 -3 -2 1 2 3
{q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q ,
{q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q ,
Line 38: Line 40:
1 2 3 4 5 6
1 2 3 4 5 6
-1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q }
-1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q ,
q</nowiki>}}
q
-4 2 3 4 2 3 4
5 + q - -- + -- - - - 4 q + 3 q - 2 q + q }
3 2 q
q q</nowiki>}}
<!--END-->
<!--END-->


Line 66: Line 73:
n = 4 |
n = 4 |
in = <nowiki>SubLink[pd_PD, js_List] := Module[
in = <nowiki>SubLink[pd_PD, js_List] := Module[
{k, t0, t, t1, t2, S, P, T},
{k, t0, t, t1, t2, S, P},
t0 = Flatten[List @@@ Skeleton[pd][[js]]];
t0 = Flatten[List @@@ Skeleton[pd][[js]]];
t = pd /. x_X :> Select[x, MemberQ[t0, #] &];
t = pd /. x_X :> Select[x, MemberQ[t0, #] &];

Revision as of 14:12, 3 July 2007



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], ConnectedSum[K41, 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 -4 2 3 4 2 3 4 5 + q - -- + -- - - - 4 q + 3 q - 2 q + q } 3 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}, 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.gif
L10a38
L10a39.gif
L10a39
L10a46.gif
L10a46
L10n35.gif
L10n35
L10n36.gif
L10n36
L10n37.gif
L10n37
L10n38.gif
L10n38
L10n39.gif
L10n39