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

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{{Manual TOC Sidebar}}
{{Manual TOC Sidebar}}


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:
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:


{{Startup Note}}
{{Startup Note}}


<!--$$J31 = Jones[Knot[3, 1]]; J41 = Jones[Knot[4, 1]];$$-->
<!--$$K31 = Knot[3, 1]; K41 = Knot[4, 1];$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{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-->


<!--$$CompositeJones =
<!--$$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]}]$$-->
ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],
ConnectedSum[Mirror[K31], K41]}$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
{{InOut|
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]}</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 40: Line 42:
<!--END-->
<!--END-->


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:
Now, we can use the program [[SubLink.m|<code>SubLink</code>]] that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:


<!--$$Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];$$-->
<!--$$SubLink[pd_PD, js_List] := Module[
{k, t0, t, t1, t2, S, P, T},
t0 = Flatten[List @@@ Skeleton[pd][[js]]];
t = pd /. x_X :> Select[x, MemberQ[t0, #] &];
t = DeleteCases[t, X[]];
k = 1;
While[
k <= Length[t],
If[ Length[t[[k]]] < 4,
t = Delete[t, k] /. (Rule @@ t[[k]]), ++k];
];
t1 = List @@ Union @@ t;
t2 = Thread[(t1) -> Range[Length[t1]]];
S = t /. t2;
P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S]
]
SubLink[pd_PD, j_] := SubLink[pd, {j}];
SubLink[L_, js_] := SubLink[PD[L], js];$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{In|
{{In|
n = 4 |
n = 4 |
in = <nowiki>Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];</nowiki>}}
in = <nowiki>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];</nowiki>}}
<!--END-->
<!--END-->


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


<!--$$NonPrimeComponentQ[L_] :=
<!--$$NonPrimeComponentQ[L_] :=

Latest revision as of 14:00, 20 October 2013


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

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.gif
L10a38 		L10a39.gif
L10a39 		L10a46.gif
L10a46 		L10n35.gif
L10n35
L10n36.gif
L10n36 		L10n37.gif
L10n37 		L10n38.gif
L10n38 		L10n39.gif
L10n39
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