"Rubberband" Brunnian Links: Difference between revisions

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(New page: {{Manual TOC Sidebar}} =="Rubberband" Brunnian Links== A "Rubberband" Brunnian link is obtained by connecting unknots in a closed chain as illustrated in the diagram of the 10-component l...)
 
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{{In|
{{In|
n = 1 |
n = 2 |
in = <nowiki>Subscript[K, 0] =
in = <nowiki>K0 =
PD[X[1, 10, 5, 12], X[2, 12, 6, 14], X[5, 11, 8, 13],
PD[X[1, 10, 5, 12], X[2, 12, 6, 14], X[5, 11, 8, 13],
X[6, 13, 9, 15], X[10, 0, 16, 4], X[11, 4, 17, 8], X[14, 7, 19, 3],
X[6, 13, 9, 15], X[10, 0, 16, 4], X[11, 4, 17, 8], X[14, 7, 19, 3],
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{{In|
{{In|
n = 2 |
n = 3 |
in = <nowiki>RubberBandBrunnian[n_] :=
in = <nowiki>RubberBandBrunnian[n_] :=
Join @@ Table[
Join @@ Table[K0 /. j_Integer :> j + 16 k, {k, 0, n - 1}] /. {16
Subscript[K, 0] /. j_Integer :> j + 16 k, {k, 0, n - 1}] /. {16
n -> 0, 16 n + 1 -> 1, 16 n + 2 -> 2, 16 n + 3 -> 3}</nowiki>}}
n -> 0, 16 n + 1 -> 1, 16 n + 2 -> 2, 16 n + 3 -> 3}</nowiki>}}
<!--END-->
<!--END-->
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{{In|
{{In|
n = 3 |
n = 4 |
in = <nowiki>RBB3=RubberBandBrunnian[3];RBB4=RubberBandBrunnian[4];RBB5=RubberBandBrunnian[5];</nowiki>}}
in = <nowiki>RBB3=RubberBandBrunnian[3];
RBB4=RubberBandBrunnian[4];
RBB5=RubberBandBrunnian[5];</nowiki>}}
<!--END-->
<!--END-->


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{{InOut|
{{InOut|
n = 4 |
n = 5 |
in = <nowiki>DrawMorseLink/@{RBB3,RBB4,RBB5}</nowiki> |
in = <nowiki>DrawMorseLink/@{RBB3,RBB4,RBB5}</nowiki> |
out= <nowiki>{-Graphics-, -Graphics-, -Graphics-}</nowiki>}}
out= <nowiki>{-Graphics-, -Graphics-, -Graphics-}</nowiki>}}
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{{InOut|
{{InOut|
n = 5 |
n = 6 |
in = <nowiki>RBJones= Jones[#][q] & /@ {RBB3, RBB4, RBB5}</nowiki> |
in = <nowiki>RBJones= Jones[#][q] & /@ {RBB3, RBB4, RBB5}</nowiki> |
out= <nowiki> 2 3 4 5 7 8 9 10
out= <nowiki> 2 3 4 5 7 8 9 10
Line 100: Line 101:
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{{In|
{{In|
n = 6 |
n = 7 |
in = <nowiki>S = SubLink[RubberBandBrunnian[5], {1, 2, 3, 4}];</nowiki>}}
in = <nowiki>S = SubLink[RubberBandBrunnian[5], {1, 2, 3, 4}];</nowiki>}}
<!--END-->
<!--END-->
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{{InOut|
{{InOut|
n = 7 |
n = 8 |
in = <nowiki>J=Factor[Jones[S][q]]</nowiki> |
in = <nowiki>J=Factor[Jones[S][q]]</nowiki> |
out= <nowiki> 6 3
out= <nowiki> 1/4 P[1, 10] P[5, 12]
-((Sqrt[q] SubLink[PD[q P[1, 12] P[5, 10] + -----------------,
-(q (1 + q) )</nowiki>}}
1/4
q
1/4 P[2, 12] P[6, 14]
q P[2, 14] P[6, 12] + -----------------,
1/4
q
1/4 P[5, 11] P[8, 13]
q P[5, 13] P[8, 11] + -----------------,
1/4
q
1/4 P[6, 13] P[9, 15]
q P[6, 15] P[9, 13] + -----------------,
1/4
q
1/4 P[0, 10] P[4, 16]
q P[0, 16] P[4, 10] + -----------------,
1/4
q
1/4 P[4, 11] P[8, 17]
q P[4, 17] P[8, 11] + -----------------,
1/4
q
P[3, 19] P[7, 14] 1/4
----------------- + q P[3, 14] P[7, 19],
1/4
q
P[7, 18] P[9, 15] 1/4
----------------- + q P[7, 15] P[9, 18],
1/4
q
1/4 P[17, 26] P[21, 28]
q P[17, 28] P[21, 26] + -------------------,
1/4
q
1/4 P[18, 28] P[22, 30]
q P[18, 30] P[22, 28] + -------------------,
1/4
q
1/4 P[21, 27] P[24, 29]
q P[21, 29] P[24, 27] + -------------------,
1/4
q
1/4 P[22, 29] P[25, 31]
q P[22, 31] P[25, 29] + -------------------,
1/4
q
1/4 P[16, 26] P[20, 32]
q P[16, 32] P[20, 26] + -------------------,
1/4
q
1/4 P[20, 27] P[24, 33]
q P[20, 33] P[24, 27] + -------------------,
1/4
q
P[19, 35] P[23, 30] 1/4
------------------- + q P[19, 30] P[23, 35],
1/4
q
P[23, 34] P[25, 31] 1/4
------------------- + q P[23, 31] P[25, 34],
1/4
q
1/4 P[33, 42] P[37, 44]
q P[33, 44] P[37, 42] + -------------------,
1/4
q
1/4 P[34, 44] P[38, 46]
q P[34, 46] P[38, 44] + -------------------,
1/4
q
1/4 P[37, 43] P[40, 45]
q P[37, 45] P[40, 43] + -------------------,
1/4
q
1/4 P[38, 45] P[41, 47]
q P[38, 47] P[41, 45] + -------------------,
1/4
q
1/4 P[32, 42] P[36, 48]
q P[32, 48] P[36, 42] + -------------------,
1/4
q
1/4 P[36, 43] P[40, 49]
q P[36, 49] P[40, 43] + -------------------,
1/4
q
P[35, 51] P[39, 46] 1/4
------------------- + q P[35, 46] P[39, 51],
1/4
q
P[39, 50] P[41, 47] 1/4
------------------- + q P[39, 47] P[41, 50],
1/4
q
1/4 P[49, 58] P[53, 60]
q P[49, 60] P[53, 58] + -------------------,
1/4
q
1/4 P[50, 60] P[54, 62]
q P[50, 62] P[54, 60] + -------------------,
1/4
q
1/4 P[53, 59] P[56, 61]
q P[53, 61] P[56, 59] + -------------------,
1/4
q
1/4 P[54, 61] P[57, 63]
q P[54, 63] P[57, 61] + -------------------,
1/4
q
1/4 P[48, 58] P[52, 64]
q P[48, 64] P[52, 58] + -------------------,
1/4
q
1/4 P[52, 59] P[56, 65]
q P[52, 65] P[56, 59] + -------------------,
1/4
q
P[51, 67] P[55, 62] 1/4
------------------- + q P[51, 62] P[55, 67],
1/4
q
P[55, 66] P[57, 63] 1/4
------------------- + q P[55, 63] P[57, 66],
1/4
q
1/4 P[65, 74] P[69, 76]
q P[65, 76] P[69, 74] + -------------------,
1/4
q
1/4 P[66, 76] P[70, 78]
q P[66, 78] P[70, 76] + -------------------,
1/4
q
1/4 P[69, 75] P[72, 77]
q P[69, 77] P[72, 75] + -------------------,
1/4
q
1/4 P[70, 77] P[73, 79]
q P[70, 79] P[73, 77] + -------------------,
1/4
q
P[0, 68] P[64, 74] 1/4
------------------ + q P[0, 64] P[68, 74],
1/4
q
P[1, 72] P[68, 75] 1/4
------------------ + q P[1, 68] P[72, 75],
1/4
q
1/4 P[3, 67] P[71, 78]
q P[3, 71] P[67, 78] + ------------------,
1/4
q
1/4 P[2, 71] P[73, 79]
q P[2, 73] P[71, 79] + ------------------], {1, 2, 3, 4}]) /
1/4
q
(1 + q))</nowiki>}}
<!--END-->
<!--END-->


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((b0 ** BR[n, {n - 1, n - 1}]) ** Inverse[b0]) **
((b0 ** BR[n, {n - 1, n - 1}]) ** Inverse[b0]) **
BR[n, {1 - n, 1 - n}]
BR[n, {1 - n, 1 - n}]
]$$--> <!--END-->
]$$-->
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{{In|
n = 9 |
in = <nowiki>BR /: Inverse[BR[n_, l_List]] := BR[n, -Reverse[l]];
BR /: BR[n1_, l1_] ** BR[n2_, l2_] := BR[Max[n1, n2], Join[l1, l2]];
BrunnianBraid[2] = BR[2, {1, 1}];
BrunnianBraid[n_] /; n > 2 := Module[
{b0},
b0 = BrunnianBraid[n - 1];
((b0 ** BR[n, {n - 1, n - 1}]) ** Inverse[b0]) **
BR[n, {1 - n, 1 - n}]
]</nowiki>}}
<!--END-->


<!--$$DeleteStrand[k_, BR[n_, l_List]] := BR[n - 1, DeleteStrand[k, l]];
<!--$$DeleteStrand[k_, BR[n_, l_List]] := BR[n - 1, DeleteStrand[k, l]];
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k == Abs[j1] + 1, DeleteStrand[k - 1, {js}],
k == Abs[j1] + 1, DeleteStrand[k - 1, {js}],
k > Abs[j1] + 1, {j1}~Join~DeleteStrand[k, {js}]
k > Abs[j1] + 1, {j1}~Join~DeleteStrand[k, {js}]
]$$--> <!--END-->
]$$-->
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{{In|
n = 10 |
in = <nowiki>DeleteStrand[k_, BR[n_, l_List]] := BR[n - 1, DeleteStrand[k, l]];
DeleteStrand[k_, {}] = {};
DeleteStrand[k_, {j1_, js___}] := Which[
k < Abs[j1], {j1 - Sign[j1]}~Join~DeleteStrand[k, {js}],
k == Abs[j1], DeleteStrand[k + 1, {js}],
k == Abs[j1] + 1, DeleteStrand[k - 1, {js}],
k > Abs[j1] + 1, {j1}~Join~DeleteStrand[k, {js}]
]</nowiki>}}
<!--END-->


Testing for the Brunnian braid with four strands, we get:
Testing for the Brunnian braid with four strands, we get:


<!--$$(b = BrunnianBraid[4]) // BraidPlot $$--> <!--END-->
<!--$$(b = BrunnianBraid[4]) // BraidPlot $$-->
<!--Robot Land, no human edits to "END"-->
{{Graphics|
n = 11 |
in = <nowiki>(b = BrunnianBraid[4]) // BraidPlot </nowiki> |
img= User_IvaH/Examples_Out_11.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->


<!--$$Jones[b][q]$$--> <!--END-->
<!--$$Jones[b][q]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 12 |
in = <nowiki>Jones[b][q]</nowiki> |
out= <nowiki> -(11/2) 4 6 5 5 1 3/2
-q + ---- - ---- + ---- - ---- - ------- - Sqrt[q] - 5 q +
9/2 7/2 5/2 3/2 Sqrt[q]
q q q q
5/2 7/2 9/2 11/2
5 q - 6 q + 4 q - q</nowiki>}}
<!--END-->


<!--$$(bb = DeleteStrand[4, b]) // BraidPlot$$--> <!--END-->
<!--$$(bb = DeleteStrand[4, b]) // BraidPlot$$-->
<!--Robot Land, no human edits to "END"-->
{{Graphics|
n = 13 |
in = <nowiki>(bb = DeleteStrand[4, b]) // BraidPlot</nowiki> |
img= User_IvaH/Examples_Out_13.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->


<!--$$Jones[#][q] & /@ {bb, BR[3, {}]}$$--> <!--END-->
<!--$$Jones[#][q] & /@ {bb, BR[3, {}]}$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 14 |
in = <nowiki>Jones[#][q] & /@ {bb, BR[3, {}]}</nowiki> |
out= <nowiki> 1 1
{2 + - + q, 2 + - + q}
q q</nowiki>}}
<!--END-->

Revision as of 15:09, 22 September 2007


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

"Rubberband" Brunnian Links

A "Rubberband" Brunnian link is obtained by connecting unknots in a closed chain as illustrated in the diagram of the 10-component link, where the last knot gets connected to the first one.

The Rubberband link with 10 components Brunnian Link Example.PNG

If we number the strands in one section of the link as shown and proceed with numbering each following section in the same manner, we can get its PD form. The PD of any "Rubberband" link can be generated in this way by varying the desired number of components:

(For In[1] see Setup)

In[2]:= K0 = PD[X[1, 10, 5, 12], X[2, 12, 6, 14], X[5, 11, 8, 13], X[6, 13, 9, 15], X[10, 0, 16, 4], X[11, 4, 17, 8], X[14, 7, 19, 3], X[15, 9, 18, 7]];
In[3]:= RubberBandBrunnian[n_] := Join @@ Table[K0 /. j_Integer :> j + 16 k, {k, 0, n - 1}] /. {16 n -> 0, 16 n + 1 -> 1, 16 n + 2 -> 2, 16 n + 3 -> 3}

For instance, let us draw the links with three, four, and five components and compute their Jones polynomials:

In[4]:= RBB3=RubberBandBrunnian[3]; RBB4=RubberBandBrunnian[4]; RBB5=RubberBandBrunnian[5];


In[5]:= DrawMorseLink/@{RBB3,RBB4,RBB5}
Out[5]= {-Graphics-, -Graphics-, -Graphics-}
In[6]:= RBJones= Jones[#][q] & /@ {RBB3, RBB4, RBB5}
Out[6]= 2 3 4 5 7 8 9 10 {-q + 5 q - 11 q + 14 q - 10 q + 11 q - 18 q + 24 q - 18 q + 11 13 14 15 16 17 11 q - 10 q + 14 q - 11 q + 5 q - q , 3/2 5/2 7/2 9/2 11/2 13/2 15/2 -q + 7 q - 24 q + 49 q - 56 q + 18 q + 51 q - 17/2 19/2 21/2 23/2 25/2 111 q + 131 q - 100 q + 32 q + 32 q - 27/2 29/2 31/2 33/2 35/2 37/2 100 q + 131 q - 111 q + 51 q + 18 q - 56 q + 39/2 41/2 43/2 45/2 49 q - 24 q + 7 q - q , 2 3 4 5 6 7 8 9 -q + 9 q - 40 q + 110 q - 189 q + 167 q + 57 q - 414 q + 10 11 12 13 14 15 660 q - 581 q + 189 q + 305 q - 672 q + 816 q - 16 17 18 19 20 21 22 672 q + 305 q + 189 q - 581 q + 660 q - 414 q + 57 q + 23 24 25 26 27 28 167 q - 189 q + 110 q - 40 q + 9 q - q }

We can also check that when one component is removed the remaining link is trivial:

In[7]:= S = SubLink[RubberBandBrunnian[5], {1, 2, 3, 4}];
In[8]:= J=Factor[Jones[S][q]]
Out[8]= 1/4 P[1, 10] P[5, 12] -((Sqrt[q] SubLink[PD[q P[1, 12] P[5, 10] + -----------------, 1/4 q 1/4 P[2, 12] P[6, 14] q P[2, 14] P[6, 12] + -----------------, 1/4 q 1/4 P[5, 11] P[8, 13] q P[5, 13] P[8, 11] + -----------------, 1/4 q 1/4 P[6, 13] P[9, 15] q P[6, 15] P[9, 13] + -----------------, 1/4 q 1/4 P[0, 10] P[4, 16] q P[0, 16] P[4, 10] + -----------------, 1/4 q 1/4 P[4, 11] P[8, 17] q P[4, 17] P[8, 11] + -----------------, 1/4 q P[3, 19] P[7, 14] 1/4 ----------------- + q P[3, 14] P[7, 19], 1/4 q P[7, 18] P[9, 15] 1/4 ----------------- + q P[7, 15] P[9, 18], 1/4 q 1/4 P[17, 26] P[21, 28] q P[17, 28] P[21, 26] + -------------------, 1/4 q 1/4 P[18, 28] P[22, 30] q P[18, 30] P[22, 28] + -------------------, 1/4 q 1/4 P[21, 27] P[24, 29] q P[21, 29] P[24, 27] + -------------------, 1/4 q 1/4 P[22, 29] P[25, 31] q P[22, 31] P[25, 29] + -------------------, 1/4 q 1/4 P[16, 26] P[20, 32] q P[16, 32] P[20, 26] + -------------------, 1/4 q 1/4 P[20, 27] P[24, 33] q P[20, 33] P[24, 27] + -------------------, 1/4 q P[19, 35] P[23, 30] 1/4 ------------------- + q P[19, 30] P[23, 35], 1/4 q P[23, 34] P[25, 31] 1/4 ------------------- + q P[23, 31] P[25, 34], 1/4 q 1/4 P[33, 42] P[37, 44] q P[33, 44] P[37, 42] + -------------------, 1/4 q 1/4 P[34, 44] P[38, 46] q P[34, 46] P[38, 44] + -------------------, 1/4 q 1/4 P[37, 43] P[40, 45] q P[37, 45] P[40, 43] + -------------------, 1/4 q 1/4 P[38, 45] P[41, 47] q P[38, 47] P[41, 45] + -------------------, 1/4 q 1/4 P[32, 42] P[36, 48] q P[32, 48] P[36, 42] + -------------------, 1/4 q 1/4 P[36, 43] P[40, 49] q P[36, 49] P[40, 43] + -------------------, 1/4 q P[35, 51] P[39, 46] 1/4 ------------------- + q P[35, 46] P[39, 51], 1/4 q P[39, 50] P[41, 47] 1/4 ------------------- + q P[39, 47] P[41, 50], 1/4 q 1/4 P[49, 58] P[53, 60] q P[49, 60] P[53, 58] + -------------------, 1/4 q 1/4 P[50, 60] P[54, 62] q P[50, 62] P[54, 60] + -------------------, 1/4 q 1/4 P[53, 59] P[56, 61] q P[53, 61] P[56, 59] + -------------------, 1/4 q 1/4 P[54, 61] P[57, 63] q P[54, 63] P[57, 61] + -------------------, 1/4 q 1/4 P[48, 58] P[52, 64] q P[48, 64] P[52, 58] + -------------------, 1/4 q 1/4 P[52, 59] P[56, 65] q P[52, 65] P[56, 59] + -------------------, 1/4 q P[51, 67] P[55, 62] 1/4 ------------------- + q P[51, 62] P[55, 67], 1/4 q P[55, 66] P[57, 63] 1/4 ------------------- + q P[55, 63] P[57, 66], 1/4 q 1/4 P[65, 74] P[69, 76] q P[65, 76] P[69, 74] + -------------------, 1/4 q 1/4 P[66, 76] P[70, 78] q P[66, 78] P[70, 76] + -------------------, 1/4 q 1/4 P[69, 75] P[72, 77] q P[69, 77] P[72, 75] + -------------------, 1/4 q 1/4 P[70, 77] P[73, 79] q P[70, 79] P[73, 77] + -------------------, 1/4 q P[0, 68] P[64, 74] 1/4 ------------------ + q P[0, 64] P[68, 74], 1/4 q P[1, 72] P[68, 75] 1/4 ------------------ + q P[1, 68] P[72, 75], 1/4 q 1/4 P[3, 67] P[71, 78] q P[3, 71] P[67, 78] + ------------------, 1/4 q 1/4 P[2, 71] P[73, 79] q P[2, 73] P[71, 79] + ------------------], {1, 2, 3, 4}]) / 1/4 q (1 + q))

Brunnian Braids

Similarly, in the case of Brunnian braids, removing one strand gives us a trivial braid. We can verify that using the following two programs. The first one constructs a Brunnian braid while the second one removes a selected strand:

In[9]:= BR /: Inverse[BR[n_, l_List]] := BR[n, -Reverse[l]]; BR /: BR[n1_, l1_] ** BR[n2_, l2_] := BR[Max[n1, n2], Join[l1, l2]]; BrunnianBraid[2] = BR[2, {1, 1}]; BrunnianBraid[n_] /; n > 2 := Module[ {b0}, b0 = BrunnianBraid[n - 1]; ((b0 ** BR[n, {n - 1, n - 1}]) ** Inverse[b0]) ** BR[n, {1 - n, 1 - n}] ]
In[10]:= DeleteStrand[k_, BR[n_, l_List]] := BR[n - 1, DeleteStrand[k, l]]; DeleteStrand[k_, {}] = {}; DeleteStrand[k_, {j1_, js___}] := Which[ k < Abs[j1], {j1 - Sign[j1]}~Join~DeleteStrand[k, {js}], k == Abs[j1], DeleteStrand[k + 1, {js}], k == Abs[j1] + 1, DeleteStrand[k - 1, {js}], k > Abs[j1] + 1, {j1}~Join~DeleteStrand[k, {js}] ]

Testing for the Brunnian braid with four strands, we get:

In[11]:= (b = BrunnianBraid[4]) // BraidPlot
File:User IvaH/Examples Out 11.gif
Out[11]= -Graphics-
In[12]:= Jones[b][q]
Out[12]= -(11/2) 4 6 5 5 1 3/2 -q + ---- - ---- + ---- - ---- - ------- - Sqrt[q] - 5 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 5 q - 6 q + 4 q - q
In[13]:= (bb = DeleteStrand[4, b]) // BraidPlot
File:User IvaH/Examples Out 13.gif
Out[13]= -Graphics-
In[14]:= Jones[#][q] & /@ {bb, BR[3, {}]}
Out[14]= 1 1 {2 + - + q, 2 + - + q} q q
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