10_153 is not [math]\displaystyle{ k }[/math]-colourable for any [math]\displaystyle{ k }[/math]. See The Determinant and the Signature.
Knot presentations
| Planar diagram presentation
|
X4251 X8493 X12,6,13,5 X13,18,14,19 X9,16,10,17 X17,10,18,11 X15,20,16,1 X19,14,20,15 X6,12,7,11 X2837
|
| Gauss code
|
1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7
|
| Dowker-Thistlethwaite code
|
4 8 12 2 -16 6 -18 -20 -10 -14
|
| Conway Notation
|
[(3,2)-(21,2)]
|
| Minimum Braid Representative
|
A Morse Link Presentation
|
An Arc Presentation
|
Length is 11, width is 4,
Braid index is 4
|
|
[{3, 9}, {2, 4}, {1, 3}, {10, 5}, {9, 2}, {11, 6}, {5, 7}, {4, 10}, {6, 8}, {7, 11}, {8, 1}]
|
[edit Notes on presentations of 10 153]
|
|
|
 A part of a knot and a part of a graph. |
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]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["10 153"];
|
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
X4251 X8493 X12,6,13,5 X13,18,14,19 X9,16,10,17 X17,10,18,11 X15,20,16,1 X19,14,20,15 X6,12,7,11 X2837
|
Out[5]=
|
1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7
|
Out[6]=
|
4 8 12 2 -16 6 -18 -20 -10 -14
|
(The path below may be different on your system)
In[7]:=
|
AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
|
ConwayNotation[K]
|
|
|
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
|
Out[9]=
|
[math]\displaystyle{ \textrm{BR}(4,\{-1,-1,-1,-2,-1,-1,3,2,2,2,3\}) }[/math]
|
In[10]:=
|
{First[br], Crossings[br], BraidIndex[K]}
|
|
|
KnotTheory::loading: Loading precomputed data in IndianaData`.
|
In[11]:=
|
Show[BraidPlot[br]]
|
In[12]:=
|
Show[DrawMorseLink[K]]
|
|
|
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
|
|
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
In[13]:=
|
ap = ArcPresentation[K]
|
Out[13]=
|
ArcPresentation[{3, 9}, {2, 4}, {1, 3}, {10, 5}, {9, 2}, {11, 6}, {5, 7}, {4, 10}, {6, 8}, {7, 11}, {8, 1}]
|
Four dimensional invariants
Polynomial invariants
| Alexander polynomial |
[math]\displaystyle{ t^3-t^2-t+3- t^{-1} - t^{-2} + t^{-3} }[/math] |
| Conway polynomial |
[math]\displaystyle{ z^6+5 z^4+4 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) |
[math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature |
{ 1, 0 } |
| Jones polynomial |
[math]\displaystyle{ -q^4+q^3-q^2+q+1+ q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math] |
| HOMFLY-PT polynomial (db, data sources) |
[math]\displaystyle{ z^6-z^4 a^{-2} +6 z^4-a^4 z^2-a^2 z^2-4 z^2 a^{-2} +10 z^2-a^4-a^2-3 a^{-2} +6 }[/math] |
| Kauffman polynomial (db, data sources) |
[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 z^6 a^{-2} -7 z^6+a^5 z^5+a^3 z^5-7 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} -4 a^4 z^4+10 z^4 a^{-2} +14 z^4-4 a^5 z^3-4 a^3 z^3+12 a z^3+22 z^3 a^{-1} +10 z^3 a^{-3} +3 a^4 z^2-2 a^2 z^2-7 z^2 a^{-2} -12 z^2+3 a^5 z+2 a^3 z-6 a z-10 z a^{-1} -5 z a^{-3} -a^4+a^2+3 a^{-2} +6 }[/math] |
| The A2 invariant |
[math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} - q^{-8} - q^{-10} - q^{-12} }[/math] |
| The G2 invariant |
[math]\displaystyle{ q^{80}+q^{76}-q^{74}+q^{70}-2 q^{68}+q^{64}-3 q^{62}+q^{60}-q^{58}-4 q^{56}+5 q^{54}-5 q^{52}+q^{50}+q^{48}-5 q^{46}+5 q^{44}-3 q^{42}-2 q^{40}+2 q^{38}-4 q^{36}+q^{34}+3 q^{32}-5 q^{30}+3 q^{28}-q^{24}+2 q^{22}-2 q^{20}+2 q^{18}+4 q^{14}-2 q^{12}+3 q^{10}+4 q^8-q^6+5 q^4-q^2+2+6 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +6 q^{-12} - q^{-14} -3 q^{-16} +4 q^{-18} -4 q^{-20} +2 q^{-22} -3 q^{-26} + q^{-28} - q^{-30} -3 q^{-32} -2 q^{-36} - q^{-38} -3 q^{-42} - q^{-48} - q^{-52} + q^{-56} + q^{-60} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 10_153.
A1 Invariants.
| Weight
|
Invariant
|
| 1
|
[math]\displaystyle{ -q^{11}+q^3+q+2 q^{-1} - q^{-9} }[/math]
|
| 2
|
[math]\displaystyle{ q^{32}-q^{28}+q^{24}-3 q^{20}-q^{18}+2 q^{16}-2 q^{14}-q^{12}+2 q^{10}+q^6+q^4+2 q^2+1+ q^{-2} +2 q^{-4} -2 q^{-8} + q^{-10} + q^{-12} -2 q^{-14} - q^{-16} - q^{-24} + q^{-28} }[/math]
|
| 3
|
[math]\displaystyle{ -q^{63}+q^{59}+q^{57}-2 q^{53}-q^{51}+q^{49}+4 q^{47}+3 q^{45}-2 q^{43}-5 q^{41}-2 q^{39}+5 q^{37}+4 q^{35}-4 q^{33}-7 q^{31}-q^{29}+5 q^{27}+q^{25}-4 q^{23}-2 q^{21}+q^{19}+q^{17}+q^{15}+q^9+q^7-q^5+q^3+5 q+4 q^{-1} -2 q^{-3} -3 q^{-5} +6 q^{-7} +5 q^{-9} -3 q^{-11} -7 q^{-13} - q^{-15} +5 q^{-17} +2 q^{-19} -3 q^{-21} -5 q^{-23} -2 q^{-25} +3 q^{-27} +3 q^{-29} -3 q^{-33} -2 q^{-35} + q^{-37} + q^{-39} + q^{-41} - q^{-47} + q^{-51} + q^{-53} - q^{-57} }[/math]
|
A2 Invariants.
| Weight
|
Invariant
|
| 1,0
|
[math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} - q^{-8} - q^{-10} - q^{-12} }[/math]
|
| 1,1
|
[math]\displaystyle{ q^{44}+2 q^{40}-2 q^{38}+2 q^{34}-4 q^{32}+6 q^{30}-8 q^{28}+4 q^{26}-2 q^{24}-4 q^{22}+2 q^{20}-12 q^{18}+4 q^{16}-6 q^{14}+3 q^{12}+6 q^8+4 q^6+5 q^4+8 q^2+10 q^{-2} -2 q^{-4} +4 q^{-6} -4 q^{-10} +4 q^{-12} -6 q^{-14} -2 q^{-16} -2 q^{-18} -4 q^{-20} +2 q^{-22} -4 q^{-24} +2 q^{-32} + q^{-36} }[/math]
|
| 2,0
|
[math]\displaystyle{ q^{42}+q^{36}+q^{34}+q^{32}-q^{28}-2 q^{26}-5 q^{24}-2 q^{22}-3 q^{20}-3 q^{18}-q^{16}+q^{12}+3 q^8+3 q^6+6 q^4+5 q^2+8+4 q^{-2} +2 q^{-4} + q^{-6} -2 q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} -2 q^{-16} -3 q^{-18} -2 q^{-20} - q^{-22} - q^{-24} + q^{-30} + q^{-32} + q^{-34} }[/math]
|
A3 Invariants.
| Weight
|
Invariant
|
| 0,1,0
|
[math]\displaystyle{ q^{34}+q^{30}-q^{24}-3 q^{22}-q^{20}-3 q^{18}-3 q^{16}-q^{14}-2 q^{12}-2 q^{10}+q^8+3 q^6+5 q^4+8 q^2+10+7 q^{-2} +4 q^{-4} - q^{-6} -2 q^{-8} -7 q^{-10} -5 q^{-12} -3 q^{-14} -3 q^{-16} + q^{-20} + q^{-22} + q^{-26} }[/math]
|
| 1,0,0
|
[math]\displaystyle{ -q^{21}-q^{17}-q^{15}-q^{13}-q^{11}+2 q^5+3 q^3+4 q+3 q^{-1} +3 q^{-3} - q^{-7} - q^{-9} -2 q^{-11} - q^{-13} - q^{-15} }[/math]
|
A4 Invariants.
| Weight
|
Invariant
|
| 0,1,0,0
|
[math]\displaystyle{ q^{44}+q^{40}+2 q^{38}+q^{36}+q^{34}+q^{32}-q^{30}-3 q^{28}-3 q^{26}-5 q^{24}-6 q^{22}-7 q^{20}-6 q^{18}-7 q^{16}-8 q^{14}-2 q^{12}+q^{10}+4 q^8+13 q^6+19 q^4+20 q^2+20+17 q^{-2} +8 q^{-4} -8 q^{-8} -11 q^{-10} -15 q^{-12} -13 q^{-14} -8 q^{-16} -5 q^{-18} -2 q^{-20} +2 q^{-22} +3 q^{-24} +2 q^{-26} +2 q^{-28} + q^{-30} + q^{-32} }[/math]
|
| 1,0,0,0
|
[math]\displaystyle{ -q^{26}-q^{22}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}+2 q^6+3 q^4+5 q^2+4+4 q^{-2} +3 q^{-4} - q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} - q^{-16} - q^{-18} }[/math]
|
B2 Invariants.
| Weight
|
Invariant
|
| 0,1
|
[math]\displaystyle{ -q^{34}-q^{30}-q^{24}+q^{22}-q^{20}+q^{18}-q^{16}+q^{14}+q^8-q^6+3 q^4+2+ q^{-2} +2 q^{-4} + q^{-6} + q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-20} - q^{-22} - q^{-26} }[/math]
|
| 1,0
|
[math]\displaystyle{ q^{56}+q^{48}-q^{44}+q^{40}-q^{38}-2 q^{36}-q^{34}-q^{30}-3 q^{28}-2 q^{26}-2 q^{20}-q^{18}+q^{14}+q^{12}+q^{10}+2 q^8+4 q^6+3 q^4+4 q^2+3+4 q^{-2} +3 q^{-4} +3 q^{-6} - q^{-8} - q^{-14} -3 q^{-16} -3 q^{-18} - q^{-20} - q^{-22} -3 q^{-24} -2 q^{-26} + q^{-36} + q^{-44} }[/math]
|
D4 Invariants.
| Weight
|
Invariant
|
| 1,0,0,0
|
[math]\displaystyle{ q^{46}+q^{42}+q^{38}-2 q^{34}-q^{32}-3 q^{30}-q^{28}-3 q^{26}-q^{24}-3 q^{22}-q^{20}-q^{18}-2 q^{16}-2 q^{14}-2 q^{12}+q^{10}+q^8+6 q^6+6 q^4+10 q^2+10+10 q^{-2} +6 q^{-4} +4 q^{-6} -4 q^{-10} -5 q^{-12} -7 q^{-14} -5 q^{-16} -6 q^{-18} -2 q^{-20} -2 q^{-22} + q^{-26} + q^{-28} + q^{-30} + q^{-34} }[/math]
|
G2 Invariants.
| Weight
|
Invariant
|
| 1,0
|
[math]\displaystyle{ q^{80}+q^{76}-q^{74}+q^{70}-2 q^{68}+q^{64}-3 q^{62}+q^{60}-q^{58}-4 q^{56}+5 q^{54}-5 q^{52}+q^{50}+q^{48}-5 q^{46}+5 q^{44}-3 q^{42}-2 q^{40}+2 q^{38}-4 q^{36}+q^{34}+3 q^{32}-5 q^{30}+3 q^{28}-q^{24}+2 q^{22}-2 q^{20}+2 q^{18}+4 q^{14}-2 q^{12}+3 q^{10}+4 q^8-q^6+5 q^4-q^2+2+6 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +6 q^{-12} - q^{-14} -3 q^{-16} +4 q^{-18} -4 q^{-20} +2 q^{-22} -3 q^{-26} + q^{-28} - q^{-30} -3 q^{-32} -2 q^{-36} - q^{-38} -3 q^{-42} - q^{-48} - q^{-52} + q^{-56} + q^{-60} }[/math]
|
.
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]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["10 153"];
|
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ t^3-t^2-t+3- t^{-1} - t^{-2} + t^{-3} }[/math]
|
Out[5]=
|
[math]\displaystyle{ z^6+5 z^4+4 z^2+1 }[/math]
|
In[6]:=
|
Alexander[K, 2][t]
|
|
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
[math]\displaystyle{ \{1\} }[/math]
|
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
[math]\displaystyle{ -q^4+q^3-q^2+q+1+ q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math]
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
|
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ z^6-z^4 a^{-2} +6 z^4-a^4 z^2-a^2 z^2-4 z^2 a^{-2} +10 z^2-a^4-a^2-3 a^{-2} +6 }[/math]
|
In[10]:=
|
Kauffman[K][a, z]
|
|
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 z^6 a^{-2} -7 z^6+a^5 z^5+a^3 z^5-7 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} -4 a^4 z^4+10 z^4 a^{-2} +14 z^4-4 a^5 z^3-4 a^3 z^3+12 a z^3+22 z^3 a^{-1} +10 z^3 a^{-3} +3 a^4 z^2-2 a^2 z^2-7 z^2 a^{-2} -12 z^2+3 a^5 z+2 a^3 z-6 a z-10 z a^{-1} -5 z a^{-3} -a^4+a^2+3 a^{-2} +6 }[/math]
|
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]):
{}
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["10 153"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ [math]\displaystyle{ t^3-t^2-t+3- t^{-1} - t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -q^4+q^3-q^2+q+1+ q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math] }
|
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
|
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
|
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
| V2,1 through V6,9:
|
| V2,1
|
V3,1
|
V4,1
|
V4,2
|
V4,3
|
V5,1
|
V5,2
|
V5,3
|
V5,4
|
V6,1
|
V6,2
|
V6,3
|
V6,4
|
V6,5
|
V6,6
|
V6,7
|
V6,8
|
V6,9
|
| [math]\displaystyle{ 16 }[/math]
|
[math]\displaystyle{ -8 }[/math]
|
[math]\displaystyle{ 128 }[/math]
|
[math]\displaystyle{ \frac{488}{3} }[/math]
|
[math]\displaystyle{ \frac{64}{3} }[/math]
|
[math]\displaystyle{ -128 }[/math]
|
[math]\displaystyle{ -\frac{560}{3} }[/math]
|
[math]\displaystyle{ -\frac{128}{3} }[/math]
|
[math]\displaystyle{ -8 }[/math]
|
[math]\displaystyle{ \frac{2048}{3} }[/math]
|
[math]\displaystyle{ 32 }[/math]
|
[math]\displaystyle{ \frac{7808}{3} }[/math]
|
[math]\displaystyle{ \frac{1024}{3} }[/math]
|
[math]\displaystyle{ \frac{41222}{15} }[/math]
|
[math]\displaystyle{ \frac{464}{5} }[/math]
|
[math]\displaystyle{ \frac{44528}{45} }[/math]
|
[math]\displaystyle{ \frac{442}{9} }[/math]
|
[math]\displaystyle{ \frac{1622}{15} }[/math]
|
|
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]0 is the signature of 10 153. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
|
|
|
-5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | χ |
| 9 | | | | | | | | | | | 1 | -1 |
| 7 | | | | | | | | | | | | 0 |
| 5 | | | | | | | | | 1 | 1 | | 0 |
| 3 | | | | | | | 1 | 1 | | | | 0 |
| 1 | | | | | | 1 | | 1 | | | | 2 |
| -1 | | | | | 1 | 3 | 1 | | | | | 1 |
| -3 | | | | 1 | | | | | | | | 1 |
| -5 | | | | 1 | 1 | | | | | | | 0 |
| -7 | | 1 | 1 | | | | | | | | | 0 |
| -9 | | | | | | | | | | | | 0 |
| -11 | 1 | | | | | | | | | | | -1 |
|
| Integral Khovanov Homology
(db, data source)
|
|
| [math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]
|
[math]\displaystyle{ i=-3 }[/math]
|
[math]\displaystyle{ i=-1 }[/math]
|
[math]\displaystyle{ i=1 }[/math]
|
| [math]\displaystyle{ r=-5 }[/math]
|
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
|
| [math]\displaystyle{ r=-4 }[/math]
|
|
[math]\displaystyle{ {\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
| [math]\displaystyle{ r=-3 }[/math]
|
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
|
| [math]\displaystyle{ r=-2 }[/math]
|
|
[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
| [math]\displaystyle{ r=-1 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
[math]\displaystyle{ {\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
| [math]\displaystyle{ r=0 }[/math]
|
[math]\displaystyle{ {\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
| [math]\displaystyle{ r=1 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
[math]\displaystyle{ {\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
| [math]\displaystyle{ r=2 }[/math]
|
[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
|
| [math]\displaystyle{ r=3 }[/math]
|
[math]\displaystyle{ {\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
|
| [math]\displaystyle{ r=4 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
|
|
| [math]\displaystyle{ r=5 }[/math]
|
[math]\displaystyle{ {\mathbb Z}_2 }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
|
|
The Coloured Jones Polynomials
| [math]\displaystyle{ n }[/math]
|
[math]\displaystyle{ J_n }[/math]
|
| 2
|
[math]\displaystyle{ q^{13}-q^{12}-q^{11}+2 q^{10}-q^9-q^8+q^7-2 q^6+2 q^5+q^4-5 q^3+4 q^2+3 q-6+4 q^{-1} +4 q^{-2} -7 q^{-3} +4 q^{-4} +3 q^{-5} -5 q^{-6} + q^{-7} +2 q^{-8} - q^{-9} -2 q^{-10} +2 q^{-12} - q^{-13} - q^{-14} + q^{-15} }[/math]
|
| 3
|
[math]\displaystyle{ -q^{27}+q^{26}+q^{25}-2 q^{23}+2 q^{21}-q^{19}+2 q^{17}-3 q^{16}-2 q^{15}+3 q^{14}+5 q^{13}-3 q^{12}-7 q^{11}+7 q^9+2 q^8-4 q^7-6 q^6+q^5+6 q^4+4 q^3-5 q^2-8 q+7+10 q^{-1} -4 q^{-2} -12 q^{-3} +5 q^{-4} +12 q^{-5} -4 q^{-6} -13 q^{-7} +5 q^{-8} +13 q^{-9} -4 q^{-10} -13 q^{-11} +2 q^{-12} +11 q^{-13} + q^{-14} -9 q^{-15} -4 q^{-16} +5 q^{-17} +4 q^{-18} - q^{-19} -3 q^{-20} -2 q^{-21} + q^{-22} +2 q^{-23} +2 q^{-24} - q^{-25} -2 q^{-26} + q^{-28} + q^{-29} - q^{-30} }[/math]
|
| 4
|
[math]\displaystyle{ q^{46}-q^{45}-q^{44}+3 q^{41}-q^{40}-q^{39}-q^{38}-q^{37}+3 q^{36}-q^{35}-q^{33}+2 q^{32}+3 q^{31}-3 q^{30}-3 q^{29}-5 q^{28}+5 q^{27}+6 q^{26}+3 q^{25}-q^{24}-10 q^{23}-q^{22}+4 q^{20}+6 q^{19}+q^{18}+2 q^{17}-8 q^{16}-10 q^{15}-3 q^{14}+8 q^{13}+17 q^{12}+7 q^{11}-17 q^{10}-22 q^9-5 q^8+20 q^7+32 q^6-5 q^5-29 q^4-24 q^3+7 q^2+48 q+10-28 q^{-1} -34 q^{-2} -4 q^{-3} +54 q^{-4} +14 q^{-5} -26 q^{-6} -36 q^{-7} -7 q^{-8} +55 q^{-9} +14 q^{-10} -26 q^{-11} -35 q^{-12} -8 q^{-13} +53 q^{-14} +16 q^{-15} -21 q^{-16} -36 q^{-17} -15 q^{-18} +44 q^{-19} +21 q^{-20} -5 q^{-21} -29 q^{-22} -26 q^{-23} +19 q^{-24} +19 q^{-25} +16 q^{-26} -9 q^{-27} -23 q^{-28} -4 q^{-29} +2 q^{-30} +15 q^{-31} +7 q^{-32} -4 q^{-33} -5 q^{-34} -6 q^{-35} +3 q^{-37} +3 q^{-38} +2 q^{-39} + q^{-40} -3 q^{-41} -2 q^{-42} - q^{-43} +3 q^{-45} - q^{-48} - q^{-49} + q^{-50} }[/math]
|
| 5
|
[math]\displaystyle{ -q^{70}+q^{69}+q^{68}-q^{65}-2 q^{64}+2 q^{62}+q^{61}+q^{60}-q^{59}-q^{58}-q^{57}+q^{55}+q^{54}-3 q^{53}-q^{52}+2 q^{51}+2 q^{50}+4 q^{49}+2 q^{48}-5 q^{47}-7 q^{46}-3 q^{45}+5 q^{43}+8 q^{42}+4 q^{41}-q^{40}-4 q^{39}-5 q^{38}-6 q^{37}-3 q^{36}+4 q^{34}+11 q^{33}+11 q^{32}+6 q^{31}-8 q^{30}-19 q^{29}-20 q^{28}-6 q^{27}+13 q^{26}+31 q^{25}+26 q^{24}+2 q^{23}-23 q^{22}-41 q^{21}-29 q^{20}+5 q^{19}+37 q^{18}+49 q^{17}+25 q^{16}-20 q^{15}-55 q^{14}-53 q^{13}-14 q^{12}+49 q^{11}+74 q^{10}+40 q^9-23 q^8-80 q^7-75 q^6+4 q^5+83 q^4+87 q^3+22 q^2-72 q-108-33 q^{-1} +73 q^{-2} +106 q^{-3} +47 q^{-4} -66 q^{-5} -117 q^{-6} -46 q^{-7} +68 q^{-8} +111 q^{-9} +51 q^{-10} -65 q^{-11} -118 q^{-12} -47 q^{-13} +68 q^{-14} +112 q^{-15} +50 q^{-16} -65 q^{-17} -118 q^{-18} -48 q^{-19} +67 q^{-20} +113 q^{-21} +52 q^{-22} -59 q^{-23} -114 q^{-24} -60 q^{-25} +48 q^{-26} +106 q^{-27} +70 q^{-28} -25 q^{-29} -92 q^{-30} -79 q^{-31} -5 q^{-32} +67 q^{-33} +78 q^{-34} +33 q^{-35} -32 q^{-36} -65 q^{-37} -49 q^{-38} -4 q^{-39} +39 q^{-40} +48 q^{-41} +29 q^{-42} -8 q^{-43} -33 q^{-44} -32 q^{-45} -16 q^{-46} +10 q^{-47} +25 q^{-48} +21 q^{-49} +6 q^{-50} -5 q^{-51} -17 q^{-52} -13 q^{-53} - q^{-54} +2 q^{-55} +7 q^{-56} +8 q^{-57} +2 q^{-58} -2 q^{-59} - q^{-60} -4 q^{-61} -4 q^{-62} + q^{-64} +2 q^{-65} +2 q^{-66} +2 q^{-67} - q^{-68} -2 q^{-69} - q^{-70} + q^{-73} + q^{-74} - q^{-75} }[/math]
|
| 6
|
[math]\displaystyle{ q^{99}-q^{98}-q^{97}+q^{94}+3 q^{92}-q^{91}-2 q^{90}-q^{89}-q^{88}+4 q^{85}-q^{84}-2 q^{80}+q^{79}+4 q^{78}-4 q^{77}-2 q^{76}-2 q^{75}-3 q^{73}+6 q^{72}+9 q^{71}+q^{70}-q^{69}-2 q^{68}-5 q^{67}-13 q^{66}+3 q^{64}+3 q^{63}+4 q^{62}+10 q^{61}+8 q^{60}-5 q^{59}-3 q^{57}-11 q^{56}-16 q^{55}-6 q^{54}+3 q^{53}+5 q^{52}+17 q^{51}+28 q^{50}+16 q^{49}-5 q^{48}-20 q^{47}-25 q^{46}-38 q^{45}-24 q^{44}+16 q^{43}+33 q^{42}+46 q^{41}+36 q^{40}+30 q^{39}-28 q^{38}-59 q^{37}-54 q^{36}-50 q^{35}-8 q^{34}+32 q^{33}+99 q^{32}+69 q^{31}+44 q^{30}-8 q^{29}-84 q^{28}-117 q^{27}-106 q^{26}+8 q^{25}+56 q^{24}+146 q^{23}+147 q^{22}+61 q^{21}-73 q^{20}-184 q^{19}-159 q^{18}-119 q^{17}+72 q^{16}+200 q^{15}+236 q^{14}+110 q^{13}-85 q^{12}-214 q^{11}-287 q^{10}-100 q^9+110 q^8+293 q^7+262 q^6+75 q^5-165 q^4-352 q^3-225 q^2-q+276+325 q^{-1} +175 q^{-2} -116 q^{-3} -361 q^{-4} -273 q^{-5} -54 q^{-6} +258 q^{-7} +341 q^{-8} +208 q^{-9} -102 q^{-10} -362 q^{-11} -281 q^{-12} -65 q^{-13} +255 q^{-14} +343 q^{-15} +213 q^{-16} -101 q^{-17} -363 q^{-18} -282 q^{-19} -65 q^{-20} +255 q^{-21} +343 q^{-22} +213 q^{-23} -102 q^{-24} -359 q^{-25} -285 q^{-26} -70 q^{-27} +250 q^{-28} +346 q^{-29} +226 q^{-30} -93 q^{-31} -343 q^{-32} -298 q^{-33} -106 q^{-34} +208 q^{-35} +337 q^{-36} +270 q^{-37} -19 q^{-38} -269 q^{-39} -305 q^{-40} -194 q^{-41} +69 q^{-42} +253 q^{-43} +300 q^{-44} +121 q^{-45} -82 q^{-46} -215 q^{-47} -241 q^{-48} -116 q^{-49} +55 q^{-50} +195 q^{-51} +173 q^{-52} +109 q^{-53} -15 q^{-54} -123 q^{-55} -155 q^{-56} -97 q^{-57} - q^{-58} +51 q^{-59} +109 q^{-60} +92 q^{-61} +39 q^{-62} -32 q^{-63} -54 q^{-64} -64 q^{-65} -58 q^{-66} -4 q^{-67} +28 q^{-68} +47 q^{-69} +33 q^{-70} +30 q^{-71} - q^{-72} -27 q^{-73} -28 q^{-74} -22 q^{-75} -8 q^{-76} +18 q^{-78} +16 q^{-79} +10 q^{-80} +3 q^{-81} -2 q^{-82} -7 q^{-83} -8 q^{-84} -5 q^{-85} -2 q^{-86} + q^{-87} +2 q^{-88} +4 q^{-89} +3 q^{-90} +3 q^{-91} - q^{-92} - q^{-93} -2 q^{-94} -2 q^{-95} -2 q^{-96} +3 q^{-98} + q^{-100} - q^{-103} - q^{-104} + q^{-105} }[/math]
|
