(KnotPlot image)
|
See the full Rolfsen Knot Table.
Visit 10 124's page
at the Knot Server
(KnotPlot driven, includes 3D interactive images!)
Visit 10 124 at Knotilus!
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|
10_124 is also known as the torus knot T(5,3) or the pretzel knot P(5,3,-2). It is one of two knots which are both torus knots and pretzel knots, the other being 8_19 = T(4,3) = P(3,3,-2).
It seems like the prior statement is incorrect. I suspect what this should say is 10_124 and 8_19 are the only torus knots which are also almost alternating. See page 108 in the Encyclopedia of Knot Theory. Confirmation of this is that 3_1 is the pretzel knot P(1,1,1), i.e., the right-handed trefoil. It looks like 5_1 is a pretzel knot also, and so on, i.e. 7_1, 9_1, and should include the Hopf link and the Solomon link etc. These are torus knots/links also.
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If one takes the symmetric diagram for 10_123 and makes it doubly alternating one gets a diagram for 10_124. That's the torus knot view. There is then a nice representation of the quandle of 10_124 into the dodecahedral quandle [math]\displaystyle{ Q_{30} }[/math]. See [1].
10_124 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 X9,17,10,16 X5,15,6,14 X15,7,16,6 X11,19,12,18 X13,1,14,20 X17,11,18,10 X19,13,20,12 X2837
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| Gauss code
|
1, -10, 2, -1, -4, 5, 10, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
|
| Dowker-Thistlethwaite code
|
4 8 -14 2 -16 -18 -20 -6 -10 -12
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| Conway Notation
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[5,3,2-]
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| Minimum Braid Representative
|
A Morse Link Presentation
|
An Arc Presentation
|
Length is 10, width is 3,
Braid index is 3
|
|
[{4, 12}, {3, 5}, {1, 4}, {6, 11}, {5, 10}, {2, 6}, {12, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 2}, {7, 1}]
|
[edit Notes on presentations of 10 124]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["10 124"];
|
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
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X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X11,19,12,18 X13,1,14,20 X17,11,18,10 X19,13,20,12 X2837
|
Out[5]=
|
1, -10, 2, -1, -4, 5, 10, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
|
Out[6]=
|
4 8 -14 2 -16 -18 -20 -6 -10 -12
|
(The path below may be different on your system)
In[7]:=
|
AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
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ConwayNotation[K]
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|
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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[math]\displaystyle{ \textrm{BR}(3,\{1,1,1,1,1,2,1,1,1,2\}) }[/math]
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In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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|
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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In[11]:=
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Show[BraidPlot[br]]
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In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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|
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{4, 12}, {3, 5}, {1, 4}, {6, 11}, {5, 10}, {2, 6}, {12, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 2}, {7, 1}]
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Four dimensional invariants
Polynomial invariants
| Alexander polynomial |
[math]\displaystyle{ t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} }[/math] |
| Conway polynomial |
[math]\displaystyle{ z^8+7 z^6+14 z^4+8 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) |
[math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature |
{ 1, 8 } |
| Jones polynomial |
[math]\displaystyle{ -q^{10}+q^6+q^4 }[/math] |
| HOMFLY-PT polynomial (db, data sources) |
[math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -7 z^4 a^{-10} +21 z^2 a^{-8} -14 z^2 a^{-10} +z^2 a^{-12} +7 a^{-8} -8 a^{-10} +2 a^{-12} }[/math] |
| Kauffman polynomial (db, data sources) |
[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -8 z^6 a^{-10} -7 z^5 a^{-9} -7 z^5 a^{-11} +21 z^4 a^{-8} +21 z^4 a^{-10} +14 z^3 a^{-9} +14 z^3 a^{-11} -21 z^2 a^{-8} -22 z^2 a^{-10} -z^2 a^{-12} -8 z a^{-9} -8 z a^{-11} +7 a^{-8} +8 a^{-10} +2 a^{-12} }[/math] |
| The A2 invariant |
[math]\displaystyle{ q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-28} -2 q^{-30} -2 q^{-32} - q^{-34} + q^{-40} }[/math] |
| The G2 invariant |
[math]\displaystyle{ q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} +2 q^{-80} +2 q^{-82} + q^{-84} +2 q^{-86} +2 q^{-88} +2 q^{-90} +2 q^{-92} +2 q^{-94} + q^{-96} +2 q^{-98} + q^{-100} + q^{-104} + q^{-106} - q^{-112} - q^{-114} - q^{-118} -2 q^{-120} -2 q^{-122} - q^{-124} - q^{-126} -2 q^{-128} -2 q^{-130} -2 q^{-132} - q^{-134} - q^{-136} -2 q^{-138} -2 q^{-140} - q^{-142} - q^{-146} - q^{-148} + q^{-160} + q^{-162} + q^{-168} + q^{-170} + q^{-180} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 10_124.
A1 Invariants.
| Weight
|
Invariant
|
| 1
|
[math]\displaystyle{ q^{-7} + q^{-9} + q^{-11} + q^{-13} - q^{-19} - q^{-21} }[/math]
|
| 2
|
[math]\displaystyle{ q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} + q^{-24} + q^{-26} - q^{-36} - q^{-38} - q^{-40} - q^{-42} - q^{-44} + q^{-60} }[/math]
|
| 3
|
[math]\displaystyle{ q^{-21} + q^{-23} + q^{-25} + q^{-27} + q^{-29} + q^{-31} + q^{-33} + q^{-35} + q^{-37} + q^{-39} - q^{-53} - q^{-55} - q^{-57} - q^{-59} - q^{-61} - q^{-63} - q^{-65} - q^{-67} + q^{-97} + q^{-99} + q^{-101} + q^{-103} - q^{-109} - q^{-111} }[/math]
|
| 5
|
[math]\displaystyle{ q^{-35} + q^{-37} + q^{-39} + q^{-41} + q^{-43} + q^{-45} + q^{-47} + q^{-49} + q^{-51} + q^{-53} + q^{-55} + q^{-57} + q^{-59} + q^{-61} + q^{-63} + q^{-65} - q^{-87} - q^{-89} - q^{-91} - q^{-93} - q^{-95} - q^{-97} - q^{-99} - q^{-101} - q^{-103} - q^{-105} - q^{-107} - q^{-109} - q^{-111} - q^{-113} + q^{-171} + q^{-173} + q^{-175} + q^{-177} + q^{-179} + q^{-181} + q^{-183} + q^{-185} + q^{-187} + q^{-189} - q^{-203} - q^{-205} - q^{-207} - q^{-209} - q^{-211} - q^{-213} - q^{-215} - q^{-217} + q^{-247} + q^{-249} + q^{-251} + q^{-253} - q^{-259} - q^{-261} }[/math]
|
| 6
|
[math]\displaystyle{ q^{-42} + q^{-44} + q^{-46} + q^{-48} + q^{-50} + q^{-52} + q^{-54} + q^{-56} + q^{-58} + q^{-60} + q^{-62} + q^{-64} + q^{-66} + q^{-68} + q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} - q^{-104} - q^{-106} - q^{-108} - q^{-110} - q^{-112} - q^{-114} - q^{-116} - q^{-118} - q^{-120} - q^{-122} - q^{-124} - q^{-126} - q^{-128} - q^{-130} - q^{-132} - q^{-134} - q^{-136} + q^{-208} + q^{-210} + q^{-212} + q^{-214} + q^{-216} + q^{-218} + q^{-220} + q^{-222} + q^{-224} + q^{-226} + q^{-228} + q^{-230} + q^{-232} - q^{-250} - q^{-252} - q^{-254} - q^{-256} - q^{-258} - q^{-260} - q^{-262} - q^{-264} - q^{-266} - q^{-268} - q^{-270} + q^{-314} + q^{-316} + q^{-318} + q^{-320} + q^{-322} + q^{-324} + q^{-326} - q^{-336} - q^{-338} - q^{-340} - q^{-342} - q^{-344} + q^{-360} }[/math]
|
A2 Invariants.
| Weight
|
Invariant
|
| 1,0
|
[math]\displaystyle{ q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-28} -2 q^{-30} -2 q^{-32} - q^{-34} + q^{-40} }[/math]
|
| 1,1
|
[math]\displaystyle{ q^{-28} +2 q^{-30} +4 q^{-32} +6 q^{-34} +7 q^{-36} +8 q^{-38} +6 q^{-40} +4 q^{-42} + q^{-44} -2 q^{-46} -6 q^{-48} -8 q^{-50} -9 q^{-52} -8 q^{-54} -6 q^{-56} -4 q^{-58} -2 q^{-60} +2 q^{-64} +2 q^{-66} +4 q^{-68} +4 q^{-70} +4 q^{-72} +2 q^{-74} + q^{-76} -2 q^{-78} -2 q^{-80} -2 q^{-82} - q^{-84} +2 q^{-90} }[/math]
|
| 2,0
|
[math]\displaystyle{ q^{-28} + q^{-30} +2 q^{-32} +2 q^{-34} +3 q^{-36} +3 q^{-38} +4 q^{-40} +3 q^{-42} +3 q^{-44} +2 q^{-46} +2 q^{-48} - q^{-52} -3 q^{-54} -4 q^{-56} -5 q^{-58} -5 q^{-60} -5 q^{-62} -4 q^{-64} -3 q^{-66} -2 q^{-68} - q^{-70} +2 q^{-74} +2 q^{-76} +4 q^{-78} +4 q^{-80} +4 q^{-82} +2 q^{-84} + q^{-86} -2 q^{-88} -2 q^{-90} -2 q^{-92} - q^{-94} + q^{-100} }[/math]
|
A3 Invariants.
| Weight
|
Invariant
|
| 0,1,0
|
[math]\displaystyle{ q^{-28} + q^{-30} +3 q^{-32} +4 q^{-34} +6 q^{-36} +6 q^{-38} +7 q^{-40} +3 q^{-42} + q^{-44} -4 q^{-46} -7 q^{-48} -9 q^{-50} -9 q^{-52} -6 q^{-54} -3 q^{-56} + q^{-58} +3 q^{-60} +4 q^{-62} +2 q^{-64} +2 q^{-66} }[/math]
|
| 1,0,0
|
[math]\displaystyle{ q^{-21} + q^{-23} +2 q^{-25} +3 q^{-27} +3 q^{-29} +3 q^{-31} +2 q^{-33} -2 q^{-37} -3 q^{-39} -4 q^{-41} -3 q^{-43} -2 q^{-45} + q^{-49} + q^{-51} + q^{-53} }[/math]
|
| 1,0,1
|
[math]\displaystyle{ q^{-42} +2 q^{-44} +5 q^{-46} +9 q^{-48} +14 q^{-50} +19 q^{-52} +23 q^{-54} +22 q^{-56} +19 q^{-58} +10 q^{-60} - q^{-62} -14 q^{-64} -26 q^{-66} -34 q^{-68} -38 q^{-70} -35 q^{-72} -26 q^{-74} -13 q^{-76} - q^{-78} +11 q^{-80} +18 q^{-82} +21 q^{-84} +19 q^{-86} +14 q^{-88} +8 q^{-90} +4 q^{-92} -3 q^{-96} -3 q^{-98} -4 q^{-100} -3 q^{-102} -3 q^{-104} - q^{-106} - q^{-108} +2 q^{-120} }[/math]
|
A4 Invariants.
| Weight
|
Invariant
|
| 0,1,0,0
|
[math]\displaystyle{ q^{-42} + q^{-44} +3 q^{-46} +5 q^{-48} +8 q^{-50} +10 q^{-52} +14 q^{-54} +13 q^{-56} +13 q^{-58} +8 q^{-60} +2 q^{-62} -7 q^{-64} -14 q^{-66} -21 q^{-68} -23 q^{-70} -21 q^{-72} -16 q^{-74} -8 q^{-76} - q^{-78} +7 q^{-80} +9 q^{-82} +11 q^{-84} +9 q^{-86} +7 q^{-88} +3 q^{-90} +2 q^{-92} - q^{-96} - q^{-98} - q^{-100} - q^{-102} - q^{-104} }[/math]
|
| 1,0,0,0
|
[math]\displaystyle{ q^{-28} + q^{-30} +2 q^{-32} +3 q^{-34} +4 q^{-36} +4 q^{-38} +4 q^{-40} +2 q^{-42} + q^{-44} -2 q^{-46} -4 q^{-48} -5 q^{-50} -5 q^{-52} -4 q^{-54} -2 q^{-56} + q^{-60} +2 q^{-62} + q^{-64} + q^{-66} }[/math]
|
B2 Invariants.
| Weight
|
Invariant
|
| 0,1
|
[math]\displaystyle{ q^{-28} + q^{-30} + q^{-32} +2 q^{-34} +2 q^{-36} +2 q^{-38} + q^{-40} + q^{-42} + q^{-44} - q^{-48} - q^{-50} - q^{-52} -2 q^{-54} - q^{-56} - q^{-58} - q^{-60} }[/math]
|
| 1,0
|
[math]\displaystyle{ q^{-42} + q^{-46} + q^{-48} +2 q^{-50} + q^{-52} +3 q^{-54} +2 q^{-56} +3 q^{-58} +2 q^{-60} +3 q^{-62} +2 q^{-64} +2 q^{-66} - q^{-72} -2 q^{-74} -3 q^{-76} -3 q^{-78} -3 q^{-80} -4 q^{-82} -3 q^{-84} -3 q^{-86} -2 q^{-88} -2 q^{-90} + q^{-96} + q^{-98} +2 q^{-100} + q^{-102} + q^{-104} + q^{-106} + q^{-108} }[/math]
|
D4 Invariants.
| Weight
|
Invariant
|
| 1,0,0,0
|
[math]\displaystyle{ q^{-42} + q^{-44} +2 q^{-46} +4 q^{-48} +5 q^{-50} +7 q^{-52} +8 q^{-54} +8 q^{-56} +7 q^{-58} +5 q^{-60} + q^{-62} -3 q^{-64} -7 q^{-66} -10 q^{-68} -11 q^{-70} -11 q^{-72} -9 q^{-74} -6 q^{-76} -2 q^{-78} + q^{-80} +3 q^{-82} +4 q^{-84} +4 q^{-86} +3 q^{-88} +2 q^{-90} + q^{-92} }[/math]
|
G2 Invariants.
| Weight
|
Invariant
|
| 1,0
|
[math]\displaystyle{ q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} +2 q^{-80} +2 q^{-82} + q^{-84} +2 q^{-86} +2 q^{-88} +2 q^{-90} +2 q^{-92} +2 q^{-94} + q^{-96} +2 q^{-98} + q^{-100} + q^{-104} + q^{-106} - q^{-112} - q^{-114} - q^{-118} -2 q^{-120} -2 q^{-122} - q^{-124} - q^{-126} -2 q^{-128} -2 q^{-130} -2 q^{-132} - q^{-134} - q^{-136} -2 q^{-138} -2 q^{-140} - q^{-142} - q^{-146} - q^{-148} + q^{-160} + q^{-162} + q^{-168} + q^{-170} + q^{-180} }[/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["10 124"];
|
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} }[/math]
|
Out[5]=
|
[math]\displaystyle{ z^8+7 z^6+14 z^4+8 z^2+1 }[/math]
|
In[6]:=
|
Alexander[K, 2][t]
|
|
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
[math]\displaystyle{ \{1\} }[/math]
|
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
[math]\displaystyle{ -q^{10}+q^6+q^4 }[/math]
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
|
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -7 z^4 a^{-10} +21 z^2 a^{-8} -14 z^2 a^{-10} +z^2 a^{-12} +7 a^{-8} -8 a^{-10} +2 a^{-12} }[/math]
|
In[10]:=
|
Kauffman[K][a, z]
|
|
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -8 z^6 a^{-10} -7 z^5 a^{-9} -7 z^5 a^{-11} +21 z^4 a^{-8} +21 z^4 a^{-10} +14 z^3 a^{-9} +14 z^3 a^{-11} -21 z^2 a^{-8} -22 z^2 a^{-10} -z^2 a^{-12} -8 z a^{-9} -8 z a^{-11} +7 a^{-8} +8 a^{-10} +2 a^{-12} }[/math]
|
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]):
{}
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["10 124"];
|
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^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^{10}+q^6+q^4 }[/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{ 32 }[/math]
|
[math]\displaystyle{ 160 }[/math]
|
[math]\displaystyle{ 512 }[/math]
|
[math]\displaystyle{ \frac{3184}{3} }[/math]
|
[math]\displaystyle{ \frac{416}{3} }[/math]
|
[math]\displaystyle{ 5120 }[/math]
|
[math]\displaystyle{ \frac{23680}{3} }[/math]
|
[math]\displaystyle{ \frac{4000}{3} }[/math]
|
[math]\displaystyle{ 832 }[/math]
|
[math]\displaystyle{ \frac{16384}{3} }[/math]
|
[math]\displaystyle{ 12800 }[/math]
|
[math]\displaystyle{ \frac{101888}{3} }[/math]
|
[math]\displaystyle{ \frac{13312}{3} }[/math]
|
[math]\displaystyle{ \frac{910684}{15} }[/math]
|
[math]\displaystyle{ \frac{59264}{15} }[/math]
|
[math]\displaystyle{ \frac{869536}{45} }[/math]
|
[math]\displaystyle{ \frac{2756}{9} }[/math]
|
[math]\displaystyle{ \frac{34684}{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]8 is the signature of 10 124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
|
|
|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | χ |
| 21 | | | | | | | | 1 | -1 |
| 19 | | | | | | 1 | | | -1 |
| 17 | | | | | | 1 | 1 | | 0 |
| 15 | | | | 1 | 1 | | | | 0 |
| 13 | | | | | 1 | | | | 1 |
| 11 | | | 1 | | | | | | 1 |
| 9 | 1 | | | | | | | | 1 |
| 7 | 1 | | | | | | | | 1 |
|
| Integral Khovanov Homology
(db, data source)
|
|
| [math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]
|
[math]\displaystyle{ i=5 }[/math]
|
[math]\displaystyle{ i=7 }[/math]
|
[math]\displaystyle{ i=9 }[/math]
|
| [math]\displaystyle{ r=0 }[/math]
|
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
| [math]\displaystyle{ r=1 }[/math]
|
|
|
|
| [math]\displaystyle{ r=2 }[/math]
|
|
[math]\displaystyle{ {\mathbb Z} }[/math]
|
|
| [math]\displaystyle{ r=3 }[/math]
|
|
[math]\displaystyle{ {\mathbb Z}_2 }[/math]
|
[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]
|
|
[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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|
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The Coloured Jones Polynomials
| [math]\displaystyle{ n }[/math]
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[math]\displaystyle{ J_n }[/math]
|
| 2
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[math]\displaystyle{ q^{29}-q^{28}+q^{26}-q^{25}+q^{23}-q^{22}-q^{21}+q^{20}-q^{19}-q^{18}+q^{17}-q^{15}+q^{14}+q^{11}+q^8 }[/math]
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| 3
|
[math]\displaystyle{ -q^{54}+q^{52}+q^{48}-q^{46}+q^{44}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-q^{30}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+q^{12} }[/math]
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| 4
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[math]\displaystyle{ q^{88}-q^{87}+q^{83}-q^{82}-q^{80}+q^{78}-q^{77}+q^{73}-q^{72}+q^{71}+q^{68}-q^{67}+q^{66}-q^{64}+q^{63}-q^{62}+q^{61}-q^{59}+q^{58}-q^{57}+q^{56}-q^{54}+q^{53}-q^{52}+q^{51}-q^{49}+q^{48}-q^{47}+q^{46}-q^{44}-q^{42}+q^{41}-q^{39}-q^{37}+q^{36}-q^{34}+q^{31}-q^{29}+q^{26}+q^{21}+q^{16} }[/math]
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| 5
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[math]\displaystyle{ -q^{128}+q^{126}+q^{124}-q^{122}+q^{118}-q^{116}+q^{112}-q^{110}-q^{104}+q^{92}+q^{86}-q^{82}+q^{80}-q^{76}+q^{74}-q^{70}+q^{68}-q^{64}+q^{62}-q^{58}+q^{56}-q^{54}-q^{52}+q^{50}-q^{48}-q^{46}+q^{44}-q^{42}+q^{38}-q^{36}+q^{32}+q^{26}+q^{20} }[/math]
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| 6
|
[math]\displaystyle{ q^{177}-q^{176}+q^{170}-2 q^{169}+q^{164}+q^{163}-2 q^{162}+q^{160}+q^{157}+q^{156}-2 q^{155}+q^{150}+q^{149}-2 q^{148}+q^{143}+q^{142}-2 q^{141}+q^{136}+q^{135}-2 q^{134}-q^{132}+q^{129}+q^{128}-2 q^{127}-q^{125}+q^{122}+2 q^{121}-2 q^{120}-q^{118}+q^{115}+2 q^{114}-q^{113}-q^{111}+q^{108}+2 q^{107}-q^{106}-q^{104}+q^{101}+q^{100}-q^{99}-q^{97}+q^{94}+q^{93}-q^{92}-q^{90}+q^{87}+q^{86}-q^{85}-q^{83}+q^{80}+q^{79}-q^{78}-q^{76}+q^{73}+q^{72}-q^{71}-q^{69}+q^{66}-q^{64}-q^{62}+q^{59}-q^{57}-q^{55}+q^{52}-q^{50}+q^{45}-q^{43}+q^{38}+q^{31}+q^{24} }[/math]
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| 7
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[math]\displaystyle{ -q^{232}+q^{230}+q^{228}-2 q^{224}+q^{222}+q^{220}-2 q^{216}+q^{214}+q^{212}-q^{210}-2 q^{208}+q^{206}+q^{204}-2 q^{200}+q^{198}+2 q^{196}-2 q^{192}+q^{190}+2 q^{188}-q^{186}-2 q^{184}+q^{182}+2 q^{180}-q^{178}-2 q^{176}+q^{174}+2 q^{172}-q^{170}-2 q^{168}+q^{166}+2 q^{164}-q^{162}-2 q^{160}+2 q^{156}-q^{154}-2 q^{152}+2 q^{148}-q^{146}-q^{144}+2 q^{140}-q^{138}-q^{136}+q^{134}+2 q^{132}-q^{130}-q^{128}+q^{126}+2 q^{124}-q^{122}-q^{120}+2 q^{116}-q^{114}-q^{112}+2 q^{108}-q^{106}-q^{104}+2 q^{100}-q^{98}-q^{96}+2 q^{92}-q^{90}-q^{88}+2 q^{84}-q^{82}-q^{80}+q^{76}-q^{74}-q^{72}+q^{68}-q^{66}-q^{64}+q^{60}-q^{58}+q^{52}-q^{50}+q^{44}+q^{36}+q^{28} }[/math]
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