"Rubberband" Brunnian Links: Difference between revisions

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{{In|
{{In|
n = 2 |
n = 1 |
in = <nowiki>K0 =
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],
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{{In|
{{In|
n = 3 |
n = 2 |
in = <nowiki>RubberBandBrunnian[n_] :=
in = <nowiki>RubberBandBrunnian[n_] :=
Join @@ Table[K0 /. j_Integer :> j + 16 k, {k, 0, n - 1}] /. {16
Join @@ Table[K0 /. j_Integer :> j + 16 k, {k, 0, n - 1}] /. {16
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<!--$$DrawMorseLink[RBB3=RubberBandBrunnian[3]]$$-->
<!--$$DrawMorseLink[RBB3=RubberBandBrunnian[3]]$$-->
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{{In|
{{Graphics|
n = 4 |
n = 4 |
in = <nowiki>RBB3=RubberBandBrunnian[3];
in = <nowiki>DrawMorseLink[RBB3=RubberBandBrunnian[3]]</nowiki> |
img= "Rubberband" Brunnian Links_Out_3.gif |
RBB4=RubberBandBrunnian[4];
RBB5=RubberBandBrunnian[5];</nowiki>}}
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->
<!--END-->


<!--$$DrawMorseLink[RBB4=RubberBandBrunnian[4]]$$-->
<!--$$DrawMorseLink[RBB4=RubberBandBrunnian[4]]$$-->
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{{Graphics|
n = 6 |
in = <nowiki>DrawMorseLink[RBB4=RubberBandBrunnian[4]]</nowiki> |
img= "Rubberband" Brunnian Links_Out_5.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->
<!--END-->


<!--$$DrawMorseLink[RBB5=RubberBandBrunnian[5]]$$-->
<!--$$DrawMorseLink[RBB5=RubberBandBrunnian[5]]$$-->
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{{Graphics|
n = 8 |
in = <nowiki>DrawMorseLink[RBB5=RubberBandBrunnian[5]]</nowiki> |
img= "Rubberband" Brunnian Links_Out_7.gif |
out= <nowiki>-Graphics-</nowiki>}}
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{{InOut|
{{InOut|
n = 6 |
n = 9 |
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
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{{In|
{{In|
n = 7 |
n = 10 |
in = <nowiki>S = SubLink[RubberBandBrunnian[5], {1, 2, 3, 4}];</nowiki>}}
in = <nowiki>S = SubLink[RubberBandBrunnian[5], {1, 2, 3, 4}];</nowiki>}}
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{{InOut|
{{InOut|
n = 8 |
n = 11 |
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>}}
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{{In|
{{In|
n = 9 |
n = 12 |
in = <nowiki>BR /: Inverse[BR[n_, l_List]] := BR[n, -Reverse[l]];
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]];
BR /: BR[n1_, l1_] ** BR[n2_, l2_] := BR[Max[n1, n2], Join[l1, l2]];
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{{In|
{{In|
n = 10 |
n = 13 |
in = <nowiki>DeleteStrand[k_, BR[n_, l_List]] := BR[n - 1, DeleteStrand[k, l]];
in = <nowiki>DeleteStrand[k_, BR[n_, l_List]] := BR[n - 1, DeleteStrand[k, l]];
DeleteStrand[k_, {}] = {};
DeleteStrand[k_, {}] = {};
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{{Graphics|
{{Graphics|
n = 11 |
n = 15 |
in = <nowiki>(b = BrunnianBraid[4]) // BraidPlot </nowiki> |
in = <nowiki>(b = BrunnianBraid[4]) // BraidPlot </nowiki> |
img= User_IvaH_Examples_Out_11.gif |
img= "Rubberband" Brunnian Links_Out_14.gif |
out= <nowiki>-Graphics-</nowiki>}}
out= <nowiki>-Graphics-</nowiki>}}
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{{InOut|
{{InOut|
n = 12 |
n = 16 |
in = <nowiki>Jones[b][q]</nowiki> |
in = <nowiki>Jones[b][q]</nowiki> |
out= <nowiki> -(11/2) 4 6 5 5 1 3/2
out= <nowiki> -(11/2) 4 6 5 5 1 3/2
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{{Graphics|
{{Graphics|
n = 13 |
n = 18 |
in = <nowiki>(bb = DeleteStrand[4, b]) // BraidPlot</nowiki> |
in = <nowiki>(bb = DeleteStrand[4, b]) // BraidPlot</nowiki> |
img= User_IvaH_Examples_Out_13.gif |
img= "Rubberband" Brunnian Links_Out_17.gif |
out= <nowiki>-Graphics-</nowiki>}}
out= <nowiki>-Graphics-</nowiki>}}
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{{InOut|
{{InOut|
n = 14 |
n = 19 |
in = <nowiki>Jones[#][q] & /@ {bb, BR[3, {}]}</nowiki> |
in = <nowiki>Jones[#][q] & /@ {bb, BR[3, {}]}</nowiki> |
out= <nowiki> 1 1
out= <nowiki> 1 1

Revision as of 13:45, 12 October 2007


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

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[1]:= 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[2]:= 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]:= DrawMorseLink[RBB3=RubberBandBrunnian[3]]
File:"Rubberband" Brunnian Links Out 3.gif
Out[4]= -Graphics-
In[6]:= DrawMorseLink[RBB4=RubberBandBrunnian[4]]
File:"Rubberband" Brunnian Links Out 5.gif
Out[6]= -Graphics-
In[8]:= DrawMorseLink[RBB5=RubberBandBrunnian[5]]
File:"Rubberband" Brunnian Links Out 7.gif
Out[8]= -Graphics-
In[9]:= RBJones= Jones[#][q] & /@ {RBB3, RBB4, RBB5}
Out[9]= 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[10]:= S = SubLink[RubberBandBrunnian[5], {1, 2, 3, 4}];
In[11]:= J=Factor[Jones[S][q]]
Out[11]= 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[12]:= 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[13]:= 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[15]:= (b = BrunnianBraid[4]) // BraidPlot
File:"Rubberband" Brunnian Links Out 14.gif
Out[15]= -Graphics-
In[16]:= Jones[b][q]
Out[16]= -(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[18]:= (bb = DeleteStrand[4, b]) // BraidPlot
File:"Rubberband" Brunnian Links Out 17.gif
Out[18]= -Graphics-
In[19]:= Jones[#][q] & /@ {bb, BR[3, {}]}
Out[19]= 1 1 {2 + - + q, 2 + - + q} q q
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