8 5
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 8 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
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8 5 is also known as the pretzel knot P(3,3,2). |
Knot presentations
| Planar diagram presentation | X6271 X8493 X2837 X14,10,15,9 X12,5,13,6 X4,13,5,14 X16,12,1,11 X10,16,11,15 |
| Gauss code | 1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7 |
| Dowker-Thistlethwaite code | 6 8 12 2 14 16 4 10 |
| Conway Notation | [3,3,2] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 8, width is 3, Braid index is 3 |
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 [{6, 11}, {1, 10}, {11, 9}, {10, 4}, {8, 3}, {9, 7}, {5, 8}, {4, 2}, {3, 6}, {2, 5}, {7, 1}] |
[edit Notes on presentations of 8 5]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["8 5"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X6271 X8493 X2837 X14,10,15,9 X12,5,13,6 X4,13,5,14 X16,12,1,11 X10,16,11,15 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7 |
In[6]:=
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DTCode[K]
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Out[6]=
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6 8 12 2 14 16 4 10 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[3,3,2] |
In[9]:=
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br = BR[K]
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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,-2,1,1,1,-2\}) }[/math] |
In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 3, 8, 3 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
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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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{6, 11}, {1, 10}, {11, 9}, {10, 4}, {8, 3}, {9, 7}, {5, 8}, {4, 2}, {3, 6}, {2, 5}, {7, 1}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-2} - t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ -z^6-3 z^4-z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 21, 4 } |
| Jones polynomial | [math]\displaystyle{ q^8-2 q^7+3 q^6-4 q^5+3 q^4-3 q^3+3 q^2-q+1 }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +z^4 a^{-6} +4 z^2 a^{-2} -8 z^2 a^{-4} +3 z^2 a^{-6} +4 a^{-2} -5 a^{-4} +2 a^{-6} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} +z^6 a^{-2} +4 z^6 a^{-4} +3 z^6 a^{-6} -3 z^5 a^{-3} +z^5 a^{-5} +4 z^5 a^{-7} -5 z^4 a^{-2} -15 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} -10 z^3 a^{-5} -8 z^3 a^{-7} +2 z^3 a^{-9} +8 z^2 a^{-2} +15 z^2 a^{-4} +4 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +3 z a^{-3} +7 z a^{-5} +4 z a^{-7} -4 a^{-2} -5 a^{-4} -2 a^{-6} }[/math] |
| The A2 invariant | [math]\displaystyle{ 1+ q^{-2} +2 q^{-4} +2 q^{-6} -3 q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-24} }[/math] |
| The G2 invariant | [math]\displaystyle{ q^{-2} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +7 q^{-16} -5 q^{-18} +5 q^{-20} + q^{-22} -2 q^{-24} +8 q^{-26} -5 q^{-28} +5 q^{-30} +2 q^{-32} -2 q^{-34} +3 q^{-36} -3 q^{-38} +2 q^{-42} -4 q^{-44} +2 q^{-46} -4 q^{-48} -2 q^{-50} +2 q^{-52} -9 q^{-54} +4 q^{-56} -7 q^{-58} + q^{-62} -7 q^{-64} +7 q^{-66} -7 q^{-68} +4 q^{-70} +2 q^{-72} -5 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} + q^{-88} - q^{-90} +4 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-100} +3 q^{-102} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} -3 q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 8_5.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | [math]\displaystyle{ q+2 q^{-3} - q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} }[/math] |
| 2 | [math]\displaystyle{ q^6-q^2+2+2 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} - q^{-12} +2 q^{-14} - q^{-16} -2 q^{-18} + q^{-20} + q^{-22} - q^{-24} +2 q^{-28} -2 q^{-32} +2 q^{-34} + q^{-36} -3 q^{-38} + q^{-40} - q^{-44} + q^{-46} }[/math] |
| 3 | [math]\displaystyle{ q^{15}-q^{11}-q^9+2 q^7+3 q^5-4 q- q^{-1} +4 q^{-3} +4 q^{-5} -3 q^{-7} -4 q^{-9} +5 q^{-13} +2 q^{-15} -4 q^{-17} -2 q^{-19} +3 q^{-21} +4 q^{-23} -2 q^{-25} -4 q^{-27} +5 q^{-31} - q^{-33} -4 q^{-35} - q^{-37} +6 q^{-39} + q^{-41} -5 q^{-43} -3 q^{-45} +3 q^{-47} +3 q^{-49} - q^{-51} -3 q^{-53} -2 q^{-55} +4 q^{-57} +4 q^{-59} -2 q^{-61} -6 q^{-63} + q^{-65} +5 q^{-67} -3 q^{-71} + q^{-73} + q^{-75} - q^{-79} - q^{-85} + q^{-87} }[/math] |
| 4 | [math]\displaystyle{ q^{28}-q^{24}-q^{22}-q^{20}+3 q^{18}+3 q^{16}+q^{14}-2 q^{12}-7 q^{10}-q^8+4 q^6+8 q^4+5 q^2-7-8 q^{-2} -6 q^{-4} +5 q^{-6} +13 q^{-8} +4 q^{-10} -3 q^{-12} -13 q^{-14} -7 q^{-16} +9 q^{-18} +12 q^{-20} +10 q^{-22} -7 q^{-24} -14 q^{-26} -4 q^{-28} +5 q^{-30} +15 q^{-32} +4 q^{-34} -11 q^{-36} -12 q^{-38} -2 q^{-40} +12 q^{-42} +8 q^{-44} -5 q^{-46} -12 q^{-48} -5 q^{-50} +10 q^{-52} +9 q^{-54} -4 q^{-56} -12 q^{-58} -3 q^{-60} +11 q^{-62} +10 q^{-64} -4 q^{-66} -12 q^{-68} -3 q^{-70} +8 q^{-72} +13 q^{-74} +2 q^{-76} -9 q^{-78} -9 q^{-80} -6 q^{-82} +7 q^{-84} +13 q^{-86} +5 q^{-88} -7 q^{-90} -19 q^{-92} -6 q^{-94} +14 q^{-96} +17 q^{-98} +5 q^{-100} -19 q^{-102} -15 q^{-104} +4 q^{-106} +14 q^{-108} +10 q^{-110} -8 q^{-112} -8 q^{-114} - q^{-116} +3 q^{-118} +6 q^{-120} -3 q^{-122} -2 q^{-124} + q^{-126} +2 q^{-130} -2 q^{-132} - q^{-138} + q^{-140} }[/math] |
| 5 | [math]\displaystyle{ q^{45}-q^{41}-q^{39}-q^{37}+3 q^{33}+4 q^{31}+q^{29}-2 q^{27}-5 q^{25}-7 q^{23}-2 q^{21}+6 q^{19}+11 q^{17}+9 q^{15}+q^{13}-11 q^{11}-16 q^9-11 q^7+3 q^5+17 q^3+21 q+10 q^{-1} -8 q^{-3} -23 q^{-5} -25 q^{-7} -6 q^{-9} +18 q^{-11} +30 q^{-13} +25 q^{-15} + q^{-17} -27 q^{-19} -35 q^{-21} -16 q^{-23} +14 q^{-25} +36 q^{-27} +36 q^{-29} +7 q^{-31} -28 q^{-33} -43 q^{-35} -26 q^{-37} +10 q^{-39} +41 q^{-41} +40 q^{-43} +6 q^{-45} -34 q^{-47} -46 q^{-49} -24 q^{-51} +17 q^{-53} +45 q^{-55} +33 q^{-57} -7 q^{-59} -38 q^{-61} -39 q^{-63} -5 q^{-65} +33 q^{-67} +39 q^{-69} +10 q^{-71} -23 q^{-73} -32 q^{-75} -11 q^{-77} +23 q^{-79} +29 q^{-81} +6 q^{-83} -18 q^{-85} -24 q^{-87} -5 q^{-89} +23 q^{-91} +24 q^{-93} -2 q^{-95} -25 q^{-97} -25 q^{-99} -2 q^{-101} +26 q^{-103} +29 q^{-105} +6 q^{-107} -23 q^{-109} -33 q^{-111} -18 q^{-113} +8 q^{-115} +30 q^{-117} +33 q^{-119} +11 q^{-121} -20 q^{-123} -40 q^{-125} -34 q^{-127} +42 q^{-131} +55 q^{-133} +23 q^{-135} -28 q^{-137} -63 q^{-139} -45 q^{-141} +12 q^{-143} +58 q^{-145} +57 q^{-147} +6 q^{-149} -47 q^{-151} -54 q^{-153} -19 q^{-155} +28 q^{-157} +45 q^{-159} +22 q^{-161} -17 q^{-163} -31 q^{-165} -16 q^{-167} +6 q^{-169} +19 q^{-171} +12 q^{-173} -4 q^{-175} -11 q^{-177} -4 q^{-179} +3 q^{-181} +4 q^{-183} + q^{-185} - q^{-187} -2 q^{-189} +2 q^{-193} + q^{-195} -2 q^{-197} - q^{-203} + q^{-205} }[/math] |
| 6 | [math]\displaystyle{ q^{66}-q^{62}-q^{60}-q^{58}+4 q^{52}+4 q^{50}+q^{48}-2 q^{46}-5 q^{44}-6 q^{42}-8 q^{40}+q^{38}+8 q^{36}+13 q^{34}+12 q^{32}+6 q^{30}-5 q^{28}-22 q^{26}-21 q^{24}-15 q^{22}+2 q^{20}+19 q^{18}+33 q^{16}+32 q^{14}+7 q^{12}-16 q^{10}-40 q^8-43 q^6-30 q^4+8 q^2+44+56 q^{-2} +50 q^{-4} +10 q^{-6} -34 q^{-8} -73 q^{-10} -67 q^{-12} -28 q^{-14} +23 q^{-16} +76 q^{-18} +88 q^{-20} +59 q^{-22} -13 q^{-24} -72 q^{-26} -98 q^{-28} -80 q^{-30} -11 q^{-32} +68 q^{-34} +117 q^{-36} +94 q^{-38} +29 q^{-40} -57 q^{-42} -123 q^{-44} -119 q^{-46} -44 q^{-48} +60 q^{-50} +121 q^{-52} +125 q^{-54} +56 q^{-56} -54 q^{-58} -137 q^{-60} -135 q^{-62} -52 q^{-64} +53 q^{-66} +136 q^{-68} +138 q^{-70} +53 q^{-72} -69 q^{-74} -141 q^{-76} -122 q^{-78} -36 q^{-80} +79 q^{-82} +144 q^{-84} +114 q^{-86} +8 q^{-88} -93 q^{-90} -129 q^{-92} -86 q^{-94} +16 q^{-96} +102 q^{-98} +114 q^{-100} +45 q^{-102} -42 q^{-104} -98 q^{-106} -85 q^{-108} -11 q^{-110} +63 q^{-112} +83 q^{-114} +36 q^{-116} -24 q^{-118} -64 q^{-120} -48 q^{-122} +5 q^{-124} +51 q^{-126} +52 q^{-128} - q^{-130} -47 q^{-132} -60 q^{-134} -20 q^{-136} +37 q^{-138} +74 q^{-140} +58 q^{-142} -11 q^{-144} -68 q^{-146} -84 q^{-148} -42 q^{-150} +22 q^{-152} +81 q^{-154} +95 q^{-156} +49 q^{-158} -20 q^{-160} -83 q^{-162} -101 q^{-164} -75 q^{-166} -2 q^{-168} +85 q^{-170} +131 q^{-172} +106 q^{-174} +15 q^{-176} -90 q^{-178} -161 q^{-180} -143 q^{-182} -29 q^{-184} +114 q^{-186} +188 q^{-188} +151 q^{-190} +26 q^{-192} -127 q^{-194} -205 q^{-196} -149 q^{-198} + q^{-200} +138 q^{-202} +186 q^{-204} +124 q^{-206} -20 q^{-208} -142 q^{-210} -156 q^{-212} -67 q^{-214} +47 q^{-216} +114 q^{-218} +109 q^{-220} +26 q^{-222} -61 q^{-224} -86 q^{-226} -50 q^{-228} +7 q^{-230} +44 q^{-232} +52 q^{-234} +16 q^{-236} -24 q^{-238} -30 q^{-240} -14 q^{-242} +3 q^{-244} +11 q^{-246} +15 q^{-248} +3 q^{-250} -9 q^{-252} -5 q^{-254} +2 q^{-258} +3 q^{-262} - q^{-264} -3 q^{-266} +2 q^{-268} + q^{-270} + q^{-272} -2 q^{-274} - q^{-280} + q^{-282} }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ 1+ q^{-2} +2 q^{-4} +2 q^{-6} -3 q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-24} }[/math] |
| 1,1 | [math]\displaystyle{ q^4+6-4 q^{-2} +12 q^{-4} -10 q^{-6} +16 q^{-8} -12 q^{-10} +10 q^{-12} -8 q^{-14} -2 q^{-16} +2 q^{-18} -14 q^{-20} +12 q^{-22} -22 q^{-24} +22 q^{-26} -20 q^{-28} +22 q^{-30} -12 q^{-32} +16 q^{-34} - q^{-36} +4 q^{-40} -10 q^{-42} +5 q^{-44} -10 q^{-46} +6 q^{-48} -6 q^{-50} +5 q^{-52} -2 q^{-54} +4 q^{-56} -4 q^{-58} +3 q^{-60} -2 q^{-62} +2 q^{-64} -2 q^{-66} + q^{-68} }[/math] |
| 2,0 | [math]\displaystyle{ q^4+q^2+1+ q^{-2} +3 q^{-4} +3 q^{-6} +2 q^{-8} + q^{-12} -2 q^{-14} -4 q^{-16} -4 q^{-18} -3 q^{-20} -3 q^{-22} - q^{-24} +2 q^{-26} +3 q^{-28} +3 q^{-30} +3 q^{-32} +3 q^{-34} - q^{-36} + q^{-40} -2 q^{-44} - q^{-46} - q^{-48} - q^{-50} - q^{-52} + q^{-56} + q^{-60} }[/math] |
| 3,0 | [math]\displaystyle{ q^{12}+q^{10}+q^8+2 q^2+4+4 q^{-2} + q^{-4} - q^{-6} +3 q^{-10} +3 q^{-12} -2 q^{-14} -8 q^{-16} -6 q^{-18} -3 q^{-20} + q^{-22} -4 q^{-24} -5 q^{-26} -3 q^{-28} +5 q^{-30} +7 q^{-32} +6 q^{-34} +4 q^{-36} +6 q^{-38} +8 q^{-40} +5 q^{-42} + q^{-44} -3 q^{-46} -2 q^{-48} -6 q^{-50} -7 q^{-52} -8 q^{-54} - q^{-56} -2 q^{-58} -2 q^{-60} -3 q^{-62} +2 q^{-64} +6 q^{-66} +2 q^{-68} -2 q^{-70} -3 q^{-72} +4 q^{-74} +7 q^{-76} +3 q^{-78} -3 q^{-80} -3 q^{-82} + q^{-84} +4 q^{-86} + q^{-88} -2 q^{-90} -3 q^{-92} - q^{-96} - q^{-98} - q^{-100} + q^{-104} + q^{-108} }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ 1+3 q^{-4} +3 q^{-6} +3 q^{-8} +5 q^{-10} +3 q^{-12} - q^{-14} - q^{-16} -6 q^{-18} -8 q^{-20} -4 q^{-22} -3 q^{-24} - q^{-26} +3 q^{-28} +5 q^{-30} +4 q^{-32} +2 q^{-34} + q^{-36} + q^{-38} -3 q^{-40} - q^{-42} + q^{-44} -2 q^{-46} +2 q^{-50} - q^{-52} - q^{-54} + q^{-56} }[/math] |
| 1,0,0 | [math]\displaystyle{ q^{-1} + q^{-3} +3 q^{-5} +2 q^{-7} +3 q^{-9} - q^{-13} -3 q^{-15} -3 q^{-17} -2 q^{-19} - q^{-21} + q^{-23} +2 q^{-27} + q^{-31} }[/math] |
| 1,0,1 | [math]\displaystyle{ q^2+6 q^{-2} +2 q^{-4} +11 q^{-6} +8 q^{-8} +6 q^{-10} +12 q^{-12} -7 q^{-14} +7 q^{-16} -14 q^{-18} -8 q^{-20} -11 q^{-22} -19 q^{-24} -3 q^{-26} -13 q^{-28} +7 q^{-30} -3 q^{-32} +13 q^{-34} +7 q^{-36} +11 q^{-38} +9 q^{-40} + q^{-42} +10 q^{-44} -8 q^{-46} +7 q^{-48} -8 q^{-50} -2 q^{-52} - q^{-54} -9 q^{-56} +4 q^{-58} -6 q^{-60} + q^{-62} +2 q^{-64} -2 q^{-66} +2 q^{-68} + q^{-72} +2 q^{-74} - q^{-76} - q^{-78} +2 q^{-80} -2 q^{-82} + q^{-84} + q^{-86} -2 q^{-88} + q^{-90} }[/math] |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{-2} + q^{-4} +3 q^{-6} +5 q^{-8} +6 q^{-10} +7 q^{-12} +8 q^{-14} +5 q^{-16} +2 q^{-18} -2 q^{-20} -7 q^{-22} -13 q^{-24} -14 q^{-26} -12 q^{-28} -9 q^{-30} -6 q^{-32} +4 q^{-34} +10 q^{-36} +9 q^{-38} +11 q^{-40} +10 q^{-42} +4 q^{-44} - q^{-46} - q^{-48} -4 q^{-50} -5 q^{-52} -3 q^{-54} - q^{-58} - q^{-60} +2 q^{-62} + q^{-64} - q^{-66} + q^{-70} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^{-2} + q^{-4} +3 q^{-6} +3 q^{-8} +3 q^{-10} +3 q^{-12} - q^{-16} -4 q^{-18} -3 q^{-20} -4 q^{-22} -2 q^{-24} - q^{-26} + q^{-28} + q^{-30} + q^{-32} +2 q^{-34} + q^{-38} }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ 1+3 q^{-4} - q^{-6} +3 q^{-8} - q^{-10} +3 q^{-12} - q^{-14} + q^{-16} -2 q^{-20} +2 q^{-22} -5 q^{-24} +3 q^{-26} -5 q^{-28} +3 q^{-30} -4 q^{-32} +2 q^{-34} - q^{-36} + q^{-38} + q^{-40} - q^{-42} +3 q^{-44} -2 q^{-46} +2 q^{-48} -2 q^{-50} + q^{-52} - q^{-54} + q^{-56} }[/math] |
| 1,0 | [math]\displaystyle{ q^2+3 q^{-6} +2 q^{-8} +3 q^{-14} +3 q^{-16} +2 q^{-18} -2 q^{-20} - q^{-22} + q^{-24} + q^{-26} -3 q^{-28} -5 q^{-30} -3 q^{-32} - q^{-34} - q^{-36} -3 q^{-38} - q^{-40} + q^{-42} +2 q^{-44} + q^{-48} +2 q^{-50} +4 q^{-52} + q^{-54} - q^{-56} +2 q^{-60} + q^{-62} -2 q^{-64} -2 q^{-66} +2 q^{-70} - q^{-72} -2 q^{-74} - q^{-76} + q^{-78} +2 q^{-80} - q^{-84} - q^{-86} + q^{-90} }[/math] |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{-2} +3 q^{-6} + q^{-8} +6 q^{-10} +3 q^{-12} +6 q^{-14} +3 q^{-16} +5 q^{-18} - q^{-20} -2 q^{-22} -5 q^{-24} -7 q^{-26} -6 q^{-28} -9 q^{-30} -3 q^{-32} -6 q^{-34} +3 q^{-36} - q^{-38} +7 q^{-40} +2 q^{-42} +7 q^{-44} + q^{-46} +4 q^{-48} - q^{-54} -2 q^{-56} -2 q^{-60} +2 q^{-62} -2 q^{-64} + q^{-66} - q^{-68} +2 q^{-70} - q^{-72} - q^{-76} + q^{-78} }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{-2} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +7 q^{-16} -5 q^{-18} +5 q^{-20} + q^{-22} -2 q^{-24} +8 q^{-26} -5 q^{-28} +5 q^{-30} +2 q^{-32} -2 q^{-34} +3 q^{-36} -3 q^{-38} +2 q^{-42} -4 q^{-44} +2 q^{-46} -4 q^{-48} -2 q^{-50} +2 q^{-52} -9 q^{-54} +4 q^{-56} -7 q^{-58} + q^{-62} -7 q^{-64} +7 q^{-66} -7 q^{-68} +4 q^{-70} +2 q^{-72} -5 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} + q^{-88} - q^{-90} +4 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-100} +3 q^{-102} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} -3 q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134} }[/math] |
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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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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["8 5"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-2} - t^{-3} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -z^6-3 z^4-z^2+1 }[/math] |
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] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 21, 4 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ q^8-2 q^7+3 q^6-4 q^5+3 q^4-3 q^3+3 q^2-q+1 }[/math] |
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^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +z^4 a^{-6} +4 z^2 a^{-2} -8 z^2 a^{-4} +3 z^2 a^{-6} +4 a^{-2} -5 a^{-4} +2 a^{-6} }[/math] |
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^7 a^{-3} +z^7 a^{-5} +z^6 a^{-2} +4 z^6 a^{-4} +3 z^6 a^{-6} -3 z^5 a^{-3} +z^5 a^{-5} +4 z^5 a^{-7} -5 z^4 a^{-2} -15 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} -10 z^3 a^{-5} -8 z^3 a^{-7} +2 z^3 a^{-9} +8 z^2 a^{-2} +15 z^2 a^{-4} +4 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +3 z a^{-3} +7 z a^{-5} +4 z a^{-7} -4 a^{-2} -5 a^{-4} -2 a^{-6} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_141,}
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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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["8 5"];
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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^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^8-2 q^7+3 q^6-4 q^5+3 q^4-3 q^3+3 q^2-q+1 }[/math] } |
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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Out[5]=
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{10_141,} |
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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Out[6]=
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{} |
Vassiliev invariants
| V2 and V3: | (-1, -3) |
| V2,1 through V6,9: |
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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.
Khovanov Homology
| 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]4 is the signature of 8 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ J_n }[/math] |
| 2 | [math]\displaystyle{ q^{22}-2 q^{21}+q^{20}+2 q^{19}-6 q^{18}+5 q^{17}+3 q^{16}-10 q^{15}+7 q^{14}+5 q^{13}-12 q^{12}+6 q^{11}+7 q^{10}-12 q^9+3 q^8+8 q^7-9 q^6+7 q^4-5 q^3-q^2+4 q-1- q^{-1} + q^{-2} }[/math] |
| 3 | [math]\displaystyle{ q^{42}-2 q^{41}+q^{40}-q^{37}+2 q^{36}-4 q^{34}+2 q^{33}+7 q^{32}-4 q^{31}-11 q^{30}+6 q^{29}+13 q^{28}-4 q^{27}-17 q^{26}+5 q^{25}+15 q^{24}-17 q^{22}-q^{21}+13 q^{20}+6 q^{19}-12 q^{18}-8 q^{17}+10 q^{16}+9 q^{15}-6 q^{14}-13 q^{13}+6 q^{12}+11 q^{11}-14 q^9+q^8+9 q^7+6 q^6-11 q^5-4 q^4+5 q^3+7 q^2-4 q-4+4 q^{-2} - q^{-4} - q^{-5} + q^{-6} }[/math] |
| 4 | [math]\displaystyle{ q^{68}-2 q^{67}+q^{66}-2 q^{64}+5 q^{63}-4 q^{62}+2 q^{61}-3 q^{60}-3 q^{59}+14 q^{58}-7 q^{57}-2 q^{56}-10 q^{55}-3 q^{54}+32 q^{53}-3 q^{52}-12 q^{51}-29 q^{50}-7 q^{49}+56 q^{48}+9 q^{47}-15 q^{46}-49 q^{45}-20 q^{44}+68 q^{43}+21 q^{42}-7 q^{41}-55 q^{40}-33 q^{39}+65 q^{38}+21 q^{37}+4 q^{36}-44 q^{35}-38 q^{34}+54 q^{33}+12 q^{32}+12 q^{31}-30 q^{30}-37 q^{29}+40 q^{28}+3 q^{27}+20 q^{26}-17 q^{25}-36 q^{24}+25 q^{23}-4 q^{22}+27 q^{21}-4 q^{20}-32 q^{19}+11 q^{18}-14 q^{17}+28 q^{16}+11 q^{15}-21 q^{14}+q^{13}-23 q^{12}+18 q^{11}+18 q^{10}-4 q^9+3 q^8-26 q^7+2 q^6+12 q^5+6 q^4+10 q^3-17 q^2-6 q+1+4 q^{-1} +11 q^{-2} -5 q^{-3} -3 q^{-4} -3 q^{-5} - q^{-6} +5 q^{-7} - q^{-10} - q^{-11} + q^{-12} }[/math] |
| 5 | [math]\displaystyle{ q^{100}-2 q^{99}+q^{98}-2 q^{96}+3 q^{95}+2 q^{94}-4 q^{93}-q^{92}+q^{91}+6 q^{89}+q^{88}-11 q^{87}-8 q^{86}+8 q^{85}+16 q^{84}+13 q^{83}-12 q^{82}-33 q^{81}-23 q^{80}+22 q^{79}+55 q^{78}+36 q^{77}-29 q^{76}-80 q^{75}-58 q^{74}+29 q^{73}+108 q^{72}+87 q^{71}-28 q^{70}-126 q^{69}-115 q^{68}+11 q^{67}+143 q^{66}+138 q^{65}+4 q^{64}-139 q^{63}-157 q^{62}-23 q^{61}+137 q^{60}+158 q^{59}+35 q^{58}-117 q^{57}-160 q^{56}-45 q^{55}+111 q^{54}+143 q^{53}+45 q^{52}-88 q^{51}-137 q^{50}-48 q^{49}+83 q^{48}+120 q^{47}+45 q^{46}-65 q^{45}-111 q^{44}-49 q^{43}+55 q^{42}+101 q^{41}+51 q^{40}-41 q^{39}-88 q^{38}-55 q^{37}+21 q^{36}+80 q^{35}+60 q^{34}-8 q^{33}-59 q^{32}-61 q^{31}-17 q^{30}+46 q^{29}+61 q^{28}+23 q^{27}-19 q^{26}-49 q^{25}-45 q^{24}+5 q^{23}+39 q^{22}+35 q^{21}+21 q^{20}-15 q^{19}-44 q^{18}-26 q^{17}+3 q^{16}+18 q^{15}+36 q^{14}+20 q^{13}-15 q^{12}-26 q^{11}-19 q^{10}-12 q^9+17 q^8+28 q^7+13 q^6-2 q^5-14 q^4-24 q^3-7 q^2+9 q+15+13 q^{-1} +4 q^{-2} -13 q^{-3} -11 q^{-4} -5 q^{-5} + q^{-6} +8 q^{-7} +9 q^{-8} - q^{-9} -3 q^{-10} -3 q^{-11} -4 q^{-12} +4 q^{-14} + q^{-15} - q^{-18} - q^{-19} + q^{-20} }[/math] |
| 6 | [math]\displaystyle{ q^{138}-2 q^{137}+q^{136}-2 q^{134}+3 q^{133}+2 q^{131}-7 q^{130}+3 q^{129}+4 q^{128}-5 q^{127}+5 q^{126}-2 q^{125}-3 q^{124}-11 q^{123}+15 q^{122}+16 q^{121}-9 q^{120}-3 q^{119}-19 q^{118}-19 q^{117}-5 q^{116}+55 q^{115}+52 q^{114}-17 q^{113}-40 q^{112}-76 q^{111}-55 q^{110}+20 q^{109}+142 q^{108}+135 q^{107}-12 q^{106}-107 q^{105}-190 q^{104}-144 q^{103}+34 q^{102}+264 q^{101}+279 q^{100}+50 q^{99}-155 q^{98}-327 q^{97}-294 q^{96}-22 q^{95}+342 q^{94}+432 q^{93}+175 q^{92}-118 q^{91}-401 q^{90}-437 q^{89}-136 q^{88}+324 q^{87}+503 q^{86}+280 q^{85}-27 q^{84}-376 q^{83}-483 q^{82}-223 q^{81}+251 q^{80}+477 q^{79}+298 q^{78}+36 q^{77}-307 q^{76}-437 q^{75}-237 q^{74}+192 q^{73}+413 q^{72}+256 q^{71}+52 q^{70}-250 q^{69}-368 q^{68}-221 q^{67}+155 q^{66}+356 q^{65}+216 q^{64}+65 q^{63}-204 q^{62}-315 q^{61}-222 q^{60}+109 q^{59}+303 q^{58}+200 q^{57}+105 q^{56}-144 q^{55}-268 q^{54}-242 q^{53}+35 q^{52}+229 q^{51}+187 q^{50}+161 q^{49}-57 q^{48}-199 q^{47}-254 q^{46}-51 q^{45}+127 q^{44}+144 q^{43}+197 q^{42}+44 q^{41}-93 q^{40}-224 q^{39}-116 q^{38}+12 q^{37}+58 q^{36}+178 q^{35}+116 q^{34}+29 q^{33}-139 q^{32}-118 q^{31}-71 q^{30}-47 q^{29}+95 q^{28}+114 q^{27}+112 q^{26}-29 q^{25}-49 q^{24}-75 q^{23}-108 q^{22}-9 q^{21}+39 q^{20}+108 q^{19}+37 q^{18}+37 q^{17}-10 q^{16}-85 q^{15}-58 q^{14}-40 q^{13}+39 q^{12}+19 q^{11}+63 q^{10}+49 q^9-13 q^8-29 q^7-52 q^6-14 q^5-32 q^4+24 q^3+43 q^2+26 q+15-12 q^{-1} -8 q^{-2} -44 q^{-3} -12 q^{-4} +5 q^{-5} +13 q^{-6} +18 q^{-7} +12 q^{-8} +15 q^{-9} -19 q^{-10} -11 q^{-11} -9 q^{-12} -4 q^{-13} + q^{-14} +6 q^{-15} +14 q^{-16} -2 q^{-17} -3 q^{-19} -3 q^{-20} -4 q^{-21} - q^{-22} +5 q^{-23} + q^{-25} - q^{-28} - q^{-29} + q^{-30} }[/math] |
| 7 | [math]\displaystyle{ q^{182}-2 q^{181}+q^{180}-2 q^{178}+3 q^{177}-q^{174}-3 q^{173}+6 q^{172}-q^{171}-6 q^{170}+5 q^{169}-3 q^{168}+2 q^{166}+q^{165}+13 q^{164}-4 q^{163}-19 q^{162}-7 q^{161}-9 q^{160}+14 q^{159}+29 q^{158}+18 q^{157}+17 q^{156}-31 q^{155}-66 q^{154}-47 q^{153}-11 q^{152}+77 q^{151}+119 q^{150}+77 q^{149}+4 q^{148}-131 q^{147}-203 q^{146}-146 q^{145}+226 q^{143}+332 q^{142}+233 q^{141}+2 q^{140}-328 q^{139}-488 q^{138}-374 q^{137}-40 q^{136}+427 q^{135}+679 q^{134}+562 q^{133}+119 q^{132}-503 q^{131}-868 q^{130}-766 q^{129}-256 q^{128}+507 q^{127}+1020 q^{126}+996 q^{125}+439 q^{124}-466 q^{123}-1113 q^{122}-1177 q^{121}-631 q^{120}+349 q^{119}+1131 q^{118}+1308 q^{117}+807 q^{116}-211 q^{115}-1082 q^{114}-1363 q^{113}-937 q^{112}+85 q^{111}+999 q^{110}+1339 q^{109}+988 q^{108}+24 q^{107}-888 q^{106}-1286 q^{105}-999 q^{104}-69 q^{103}+815 q^{102}+1191 q^{101}+942 q^{100}+91 q^{99}-736 q^{98}-1108 q^{97}-900 q^{96}-79 q^{95}+707 q^{94}+1031 q^{93}+828 q^{92}+68 q^{91}-655 q^{90}-969 q^{89}-803 q^{88}-71 q^{87}+627 q^{86}+920 q^{85}+770 q^{84}+94 q^{83}-564 q^{82}-865 q^{81}-772 q^{80}-144 q^{79}+495 q^{78}+808 q^{77}+772 q^{76}+207 q^{75}-399 q^{74}-734 q^{73}-770 q^{72}-288 q^{71}+280 q^{70}+648 q^{69}+767 q^{68}+367 q^{67}-163 q^{66}-533 q^{65}-728 q^{64}-445 q^{63}+16 q^{62}+406 q^{61}+688 q^{60}+494 q^{59}+106 q^{58}-252 q^{57}-590 q^{56}-527 q^{55}-239 q^{54}+100 q^{53}+483 q^{52}+507 q^{51}+320 q^{50}+65 q^{49}-326 q^{48}-454 q^{47}-384 q^{46}-200 q^{45}+179 q^{44}+346 q^{43}+371 q^{42}+304 q^{41}-3 q^{40}-214 q^{39}-332 q^{38}-351 q^{37}-109 q^{36}+66 q^{35}+209 q^{34}+336 q^{33}+212 q^{32}+71 q^{31}-101 q^{30}-271 q^{29}-213 q^{28}-158 q^{27}-43 q^{26}+153 q^{25}+191 q^{24}+202 q^{23}+126 q^{22}-49 q^{21}-99 q^{20}-168 q^{19}-176 q^{18}-54 q^{17}+10 q^{16}+112 q^{15}+162 q^{14}+91 q^{13}+66 q^{12}-23 q^{11}-108 q^{10}-94 q^9-104 q^8-39 q^7+43 q^6+52 q^5+98 q^4+73 q^3+13 q^2-5 q-60-70 q^{-1} -41 q^{-2} -36 q^{-3} +22 q^{-4} +46 q^{-5} +35 q^{-6} +47 q^{-7} +15 q^{-8} -11 q^{-9} -21 q^{-10} -46 q^{-11} -23 q^{-12} -4 q^{-13} -2 q^{-14} +21 q^{-15} +22 q^{-16} +18 q^{-17} +14 q^{-18} -13 q^{-19} -12 q^{-20} -8 q^{-21} -13 q^{-22} -4 q^{-23} + q^{-24} +7 q^{-25} +13 q^{-26} + q^{-27} + q^{-29} -4 q^{-30} -3 q^{-31} -4 q^{-32} - q^{-33} +4 q^{-34} + q^{-35} + q^{-37} - q^{-40} - q^{-41} + q^{-42} }[/math] |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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