Edit T(8,3) Further Notes and Views
 Banco Internacional do Funchal [1] |
Knot presentations
| Planar diagram presentation
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X11,1,12,32 X22,2,23,1 X23,13,24,12 X2,14,3,13 X3,25,4,24 X14,26,15,25 X15,5,16,4 X26,6,27,5 X27,17,28,16 X6,18,7,17 X7,29,8,28 X18,30,19,29 X19,9,20,8 X30,10,31,9 X31,21,32,20 X10,22,11,21
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| Gauss code
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2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 1
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| Dowker-Thistlethwaite code
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22 -24 26 -28 30 -32 2 -4 6 -8 10 -12 14 -16 18 -20
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Polynomial invariants
| Alexander polynomial |
[math]\displaystyle{ t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} }[/math] |
| Conway polynomial |
[math]\displaystyle{ z^{14}+13 z^{12}+65 z^{10}+157 z^8+189 z^6+105 z^4+21 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) |
[math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature |
{ 3, 10 } |
| Jones polynomial |
[math]\displaystyle{ -q^{16}+q^9+q^7 }[/math] |
| HOMFLY-PT polynomial (db, data sources) |
[math]\displaystyle{ z^{14} a^{-14} +14 z^{12} a^{-14} -z^{12} a^{-16} +78 z^{10} a^{-14} -13 z^{10} a^{-16} +221 z^8 a^{-14} -65 z^8 a^{-16} +z^8 a^{-18} +338 z^6 a^{-14} -157 z^6 a^{-16} +8 z^6 a^{-18} +273 z^4 a^{-14} -189 z^4 a^{-16} +21 z^4 a^{-18} +105 z^2 a^{-14} -105 z^2 a^{-16} +21 z^2 a^{-18} +15 a^{-14} -21 a^{-16} +7 a^{-18} }[/math] |
| Kauffman polynomial (db, data sources) |
[math]\displaystyle{ z^{14} a^{-14} +z^{14} a^{-16} +z^{13} a^{-15} +z^{13} a^{-17} -14 z^{12} a^{-14} -14 z^{12} a^{-16} -13 z^{11} a^{-15} -13 z^{11} a^{-17} +78 z^{10} a^{-14} +78 z^{10} a^{-16} +65 z^9 a^{-15} +65 z^9 a^{-17} -221 z^8 a^{-14} -222 z^8 a^{-16} -z^8 a^{-18} -157 z^7 a^{-15} -157 z^7 a^{-17} +338 z^6 a^{-14} +346 z^6 a^{-16} +8 z^6 a^{-18} +189 z^5 a^{-15} +189 z^5 a^{-17} -273 z^4 a^{-14} -294 z^4 a^{-16} -21 z^4 a^{-18} -105 z^3 a^{-15} -105 z^3 a^{-17} +105 z^2 a^{-14} +126 z^2 a^{-16} +21 z^2 a^{-18} +21 z a^{-15} +21 z a^{-17} -15 a^{-14} -21 a^{-16} -7 a^{-18} }[/math] |
| The A2 invariant |
Data:T(8,3)/QuantumInvariant/A2/1,0 |
| The G2 invariant |
Data:T(8,3)/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot
5_2) as the notebook
PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["T(8,3)"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} }[/math]
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Out[5]=
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[math]\displaystyle{ z^{14}+13 z^{12}+65 z^{10}+157 z^8+189 z^6+105 z^4+21 z^2+1 }[/math]
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In[6]:=
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Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
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Out[6]=
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[math]\displaystyle{ \{1\} }[/math]
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ -q^{16}+q^9+q^7 }[/math]
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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[math]\displaystyle{ z^{14} a^{-14} +14 z^{12} a^{-14} -z^{12} a^{-16} +78 z^{10} a^{-14} -13 z^{10} a^{-16} +221 z^8 a^{-14} -65 z^8 a^{-16} +z^8 a^{-18} +338 z^6 a^{-14} -157 z^6 a^{-16} +8 z^6 a^{-18} +273 z^4 a^{-14} -189 z^4 a^{-16} +21 z^4 a^{-18} +105 z^2 a^{-14} -105 z^2 a^{-16} +21 z^2 a^{-18} +15 a^{-14} -21 a^{-16} +7 a^{-18} }[/math]
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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[math]\displaystyle{ z^{14} a^{-14} +z^{14} a^{-16} +z^{13} a^{-15} +z^{13} a^{-17} -14 z^{12} a^{-14} -14 z^{12} a^{-16} -13 z^{11} a^{-15} -13 z^{11} a^{-17} +78 z^{10} a^{-14} +78 z^{10} a^{-16} +65 z^9 a^{-15} +65 z^9 a^{-17} -221 z^8 a^{-14} -222 z^8 a^{-16} -z^8 a^{-18} -157 z^7 a^{-15} -157 z^7 a^{-17} +338 z^6 a^{-14} +346 z^6 a^{-16} +8 z^6 a^{-18} +189 z^5 a^{-15} +189 z^5 a^{-17} -273 z^4 a^{-14} -294 z^4 a^{-16} -21 z^4 a^{-18} -105 z^3 a^{-15} -105 z^3 a^{-17} +105 z^2 a^{-14} +126 z^2 a^{-16} +21 z^2 a^{-18} +21 z a^{-15} +21 z a^{-17} -15 a^{-14} -21 a^{-16} -7 a^{-18} }[/math]
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]):
{}
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["T(8,3)"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ [math]\displaystyle{ t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} }[/math], [math]\displaystyle{ -q^{16}+q^9+q^7 }[/math] }
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In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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| V2,1 through V6,9:
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| V2,1
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V3,1
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V4,1
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V4,2
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V4,3
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V5,1
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V5,2
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V5,3
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V5,4
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V6,1
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V6,2
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V6,3
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V6,4
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V6,5
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V6,6
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V6,7
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V6,8
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V6,9
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| Data:T(8,3)/V 2,1
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Data:T(8,3)/V 3,1
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Data:T(8,3)/V 4,1
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Data:T(8,3)/V 4,2
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Data:T(8,3)/V 4,3
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Data:T(8,3)/V 5,1
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Data:T(8,3)/V 5,2
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Data:T(8,3)/V 5,3
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Data:T(8,3)/V 5,4
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Data:T(8,3)/V 6,1
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Data:T(8,3)/V 6,2
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Data:T(8,3)/V 6,3
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Data:T(8,3)/V 6,4
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Data:T(8,3)/V 6,5
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Data:T(8,3)/V 6,6
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Data:T(8,3)/V 6,7
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Data:T(8,3)/V 6,8
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Data:T(8,3)/V 6,9
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
| The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]10 is the signature of T(8,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | χ |
| 33 | | | | | | | | | | | | 1 | -1 |
| 31 | | | | | | | | | | 1 | | | -1 |
| 29 | | | | | | | | | | 1 | 1 | | 0 |
| 27 | | | | | | | | 1 | 1 | | | | 0 |
| 25 | | | | | | 1 | | | 1 | | | | 0 |
| 23 | | | | | | 1 | 1 | | | | | | 0 |
| 21 | | | | 1 | 1 | | | | | | | | 0 |
| 19 | | | | | 1 | | | | | | | | 1 |
| 17 | | | 1 | | | | | | | | | | 1 |
| 15 | 1 | | | | | | | | | | | | 1 |
| 13 | 1 | | | | | | | | | | | | 1 |
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| Integral Khovanov Homology
(db, data source)
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| [math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]
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[math]\displaystyle{ i=9 }[/math]
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[math]\displaystyle{ i=11 }[/math]
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[math]\displaystyle{ i=13 }[/math]
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[math]\displaystyle{ i=15 }[/math]
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| [math]\displaystyle{ r=0 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=1 }[/math]
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| [math]\displaystyle{ r=2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=3 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=4 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=5 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=6 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=7 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=8 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=9 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=10 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=11 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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