10 76
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 76's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X4251 X14,10,15,9 X12,3,13,4 X2,13,3,14 X18,6,19,5 X20,8,1,7 X6,20,7,19 X8,18,9,17 X16,12,17,11 X10,16,11,15 |
| Gauss code | 1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6 |
| Dowker-Thistlethwaite code | 4 12 18 20 14 16 2 10 8 6 |
| Conway Notation | [3,3,2++] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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 [{5, 12}, {6, 3}, {1, 5}, {4, 2}, {3, 7}, {2, 6}, {10, 4}, {9, 11}, {8, 10}, {7, 9}, {12, 8}, {11, 1}] |
[edit Notes on presentations of 10 76]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 76"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X14,10,15,9 X12,3,13,4 X2,13,3,14 X18,6,19,5 X20,8,1,7 X6,20,7,19 X8,18,9,17 X16,12,17,11 X10,16,11,15 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 18 20 14 16 2 10 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{1,1,1,1,2,-1,-3,2,2,2,-3\})} |
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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{ 4, 11, 4 } |
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[{5, 12}, {6, 3}, {1, 5}, {4, 2}, {3, 7}, {2, 6}, {10, 4}, {9, 11}, {8, 10}, {7, 9}, {12, 8}, {11, 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 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -2 t^{-3} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^6-5 z^4-2 z^2+1} |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 57, 4 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-3 q^9+6 q^8-8 q^7+9 q^6-10 q^5+8 q^4-6 q^3+4 q^2-q+1} |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -4 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +4 z^2 a^{-2} -6 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +4 a^{-2} -4 a^{-4} + a^{-8} } |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} +z^7 a^{-3} -2 z^7 a^{-5} +2 z^7 a^{-7} +5 z^7 a^{-9} +z^6 a^{-2} -9 z^6 a^{-6} -3 z^6 a^{-8} +5 z^6 a^{-10} -2 z^5 a^{-3} +7 z^5 a^{-5} -2 z^5 a^{-7} -8 z^5 a^{-9} +3 z^5 a^{-11} -5 z^4 a^{-2} -7 z^4 a^{-4} +10 z^4 a^{-6} +4 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} -2 z^3 a^{-3} -15 z^3 a^{-5} -3 z^3 a^{-7} +7 z^3 a^{-9} -3 z^3 a^{-11} +8 z^2 a^{-2} +9 z^2 a^{-4} -7 z^2 a^{-6} -4 z^2 a^{-8} +3 z^2 a^{-10} -z^2 a^{-12} +4 z a^{-3} +8 z a^{-5} +2 z a^{-7} -2 z a^{-9} -4 a^{-2} -4 a^{-4} + a^{-8} } |
| The A2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+ q^{-2} +2 q^{-4} +3 q^{-6} - q^{-8} + q^{-10} -3 q^{-12} -2 q^{-14} -2 q^{-18} +2 q^{-20} - q^{-22} + q^{-24} + q^{-26} - q^{-28} + q^{-30} } |
| The G2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} +3 q^{-6} -2 q^{-8} +3 q^{-10} - q^{-12} +7 q^{-16} -9 q^{-18} +14 q^{-20} -12 q^{-22} +12 q^{-24} + q^{-26} -12 q^{-28} +31 q^{-30} -40 q^{-32} +46 q^{-34} -33 q^{-36} + q^{-38} +33 q^{-40} -65 q^{-42} +80 q^{-44} -67 q^{-46} +28 q^{-48} +17 q^{-50} -59 q^{-52} +71 q^{-54} -63 q^{-56} +24 q^{-58} +17 q^{-60} -50 q^{-62} +46 q^{-64} -24 q^{-66} -19 q^{-68} +58 q^{-70} -75 q^{-72} +60 q^{-74} -21 q^{-76} -35 q^{-78} +87 q^{-80} -115 q^{-82} +106 q^{-84} -59 q^{-86} -3 q^{-88} +66 q^{-90} -101 q^{-92} +104 q^{-94} -68 q^{-96} +17 q^{-98} +32 q^{-100} -60 q^{-102} +55 q^{-104} -22 q^{-106} -19 q^{-108} +51 q^{-110} -52 q^{-112} +28 q^{-114} +11 q^{-116} -53 q^{-118} +76 q^{-120} -74 q^{-122} +48 q^{-124} -8 q^{-126} -33 q^{-128} +60 q^{-130} -64 q^{-132} +54 q^{-134} -28 q^{-136} +3 q^{-138} +15 q^{-140} -29 q^{-142} +28 q^{-144} -21 q^{-146} +12 q^{-148} -2 q^{-150} -3 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} } |
Further Quantum Invariants
Further quantum knot invariants for 10_76.
A1 Invariants.
| Weight | Invariant |
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| 1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q+3 q^{-3} -2 q^{-5} +2 q^{-7} -2 q^{-9} - q^{-11} + q^{-13} -2 q^{-15} +3 q^{-17} -2 q^{-19} + q^{-21} } |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^6+3- q^{-2} -3 q^{-4} +8 q^{-6} -2 q^{-8} -11 q^{-10} +12 q^{-12} +3 q^{-14} -18 q^{-16} +9 q^{-18} +9 q^{-20} -14 q^{-22} +2 q^{-24} +10 q^{-26} -3 q^{-28} -7 q^{-30} +4 q^{-32} +10 q^{-34} -12 q^{-36} -4 q^{-38} +18 q^{-40} -11 q^{-42} -9 q^{-44} +15 q^{-46} -4 q^{-48} -7 q^{-50} +6 q^{-52} -2 q^{-56} + q^{-58} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{15}+2 q^7-q^5-q^3+q+6 q^{-1} -3 q^{-3} -10 q^{-5} +21 q^{-9} +3 q^{-11} -29 q^{-13} -18 q^{-15} +36 q^{-17} +37 q^{-19} -36 q^{-21} -54 q^{-23} +25 q^{-25} +73 q^{-27} -9 q^{-29} -78 q^{-31} -12 q^{-33} +79 q^{-35} +25 q^{-37} -67 q^{-39} -43 q^{-41} +58 q^{-43} +47 q^{-45} -37 q^{-47} -50 q^{-49} +17 q^{-51} +50 q^{-53} +4 q^{-55} -46 q^{-57} -31 q^{-59} +38 q^{-61} +52 q^{-63} -23 q^{-65} -74 q^{-67} +10 q^{-69} +83 q^{-71} +9 q^{-73} -82 q^{-75} -22 q^{-77} +70 q^{-79} +33 q^{-81} -53 q^{-83} -34 q^{-85} +34 q^{-87} +26 q^{-89} -17 q^{-91} -20 q^{-93} +9 q^{-95} +13 q^{-97} -5 q^{-99} -6 q^{-101} + q^{-103} +3 q^{-105} -2 q^{-109} + q^{-111} } |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-q^{20}+2 q^{18}-q^{16}+3 q^{12}-2 q^{10}+3 q^8-6 q^6-5 q^4+8 q^2+6+15 q^{-2} -17 q^{-4} -32 q^{-6} -5 q^{-8} +22 q^{-10} +71 q^{-12} +3 q^{-14} -76 q^{-16} -79 q^{-18} -15 q^{-20} +156 q^{-22} +118 q^{-24} -45 q^{-26} -187 q^{-28} -182 q^{-30} +141 q^{-32} +267 q^{-34} +135 q^{-36} -175 q^{-38} -382 q^{-40} -48 q^{-42} +278 q^{-44} +354 q^{-46} +10 q^{-48} -433 q^{-50} -266 q^{-52} +125 q^{-54} +433 q^{-56} +213 q^{-58} -319 q^{-60} -360 q^{-62} -48 q^{-64} +365 q^{-66} +301 q^{-68} -162 q^{-70} -333 q^{-72} -148 q^{-74} +237 q^{-76} +295 q^{-78} -9 q^{-80} -255 q^{-82} -214 q^{-84} +84 q^{-86} +257 q^{-88} +167 q^{-90} -131 q^{-92} -277 q^{-94} -132 q^{-96} +171 q^{-98} +360 q^{-100} +64 q^{-102} -282 q^{-104} -362 q^{-106} -6 q^{-108} +447 q^{-110} +278 q^{-112} -145 q^{-114} -456 q^{-116} -212 q^{-118} +341 q^{-120} +350 q^{-122} +53 q^{-124} -333 q^{-126} -287 q^{-128} +136 q^{-130} +234 q^{-132} +143 q^{-134} -135 q^{-136} -194 q^{-138} +18 q^{-140} +76 q^{-142} +98 q^{-144} -26 q^{-146} -78 q^{-148} + q^{-150} +8 q^{-152} +39 q^{-154} -5 q^{-156} -25 q^{-158} +5 q^{-160} -3 q^{-162} +12 q^{-164} -7 q^{-168} +2 q^{-170} -2 q^{-172} +3 q^{-174} -2 q^{-178} + q^{-180} } |
| 5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{45}-q^{37}-q^{35}+2 q^{33}+3 q^{27}-5 q^{23}-2 q^{19}-q^{17}+10 q^{15}+10 q^{13}-4 q^{11}-9 q^9-20 q^7-18 q^5+15 q^3+43 q+39 q^{-1} +6 q^{-3} -60 q^{-5} -103 q^{-7} -50 q^{-9} +64 q^{-11} +165 q^{-13} +168 q^{-15} -4 q^{-17} -237 q^{-19} -315 q^{-21} -143 q^{-23} +213 q^{-25} +505 q^{-27} +420 q^{-29} -80 q^{-31} -615 q^{-33} -751 q^{-35} -264 q^{-37} +569 q^{-39} +1085 q^{-41} +751 q^{-43} -298 q^{-45} -1258 q^{-47} -1298 q^{-49} -243 q^{-51} +1179 q^{-53} +1780 q^{-55} +920 q^{-57} -819 q^{-59} -2018 q^{-61} -1629 q^{-63} +196 q^{-65} +1996 q^{-67} +2188 q^{-69} +519 q^{-71} -1668 q^{-73} -2508 q^{-75} -1203 q^{-77} +1184 q^{-79} +2542 q^{-81} +1713 q^{-83} -602 q^{-85} -2384 q^{-87} -2002 q^{-89} +130 q^{-91} +2039 q^{-93} +2073 q^{-95} +269 q^{-97} -1690 q^{-99} -2005 q^{-101} -479 q^{-103} +1337 q^{-105} +1828 q^{-107} +644 q^{-109} -1045 q^{-111} -1666 q^{-113} -744 q^{-115} +779 q^{-117} +1507 q^{-119} +879 q^{-121} -482 q^{-123} -1387 q^{-125} -1092 q^{-127} +127 q^{-129} +1251 q^{-131} +1354 q^{-133} +348 q^{-135} -1024 q^{-137} -1665 q^{-139} -924 q^{-141} +686 q^{-143} +1876 q^{-145} +1573 q^{-147} -168 q^{-149} -1950 q^{-151} -2164 q^{-153} -479 q^{-155} +1770 q^{-157} +2602 q^{-159} +1171 q^{-161} -1359 q^{-163} -2742 q^{-165} -1762 q^{-167} +739 q^{-169} +2579 q^{-171} +2132 q^{-173} -104 q^{-175} -2104 q^{-177} -2191 q^{-179} -454 q^{-181} +1477 q^{-183} +1971 q^{-185} +780 q^{-187} -854 q^{-189} -1530 q^{-191} -867 q^{-193} +345 q^{-195} +1034 q^{-197} +772 q^{-199} -47 q^{-201} -607 q^{-203} -545 q^{-205} -93 q^{-207} +295 q^{-209} +339 q^{-211} +114 q^{-213} -131 q^{-215} -179 q^{-217} -69 q^{-219} +45 q^{-221} +78 q^{-223} +38 q^{-225} -13 q^{-227} -40 q^{-229} -14 q^{-231} +13 q^{-233} +13 q^{-235} +3 q^{-237} -4 q^{-239} -4 q^{-241} -7 q^{-243} +3 q^{-245} +7 q^{-247} - q^{-249} -2 q^{-251} + q^{-253} - q^{-255} -2 q^{-257} +3 q^{-259} -2 q^{-263} + q^{-265} } |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+ q^{-2} +2 q^{-4} +3 q^{-6} - q^{-8} + q^{-10} -3 q^{-12} -2 q^{-14} -2 q^{-18} +2 q^{-20} - q^{-22} + q^{-24} + q^{-26} - q^{-28} + q^{-30} } |
| 1,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4+6-4 q^{-2} +18 q^{-4} -20 q^{-6} +42 q^{-8} -62 q^{-10} +93 q^{-12} -140 q^{-14} +182 q^{-16} -240 q^{-18} +272 q^{-20} -302 q^{-22} +288 q^{-24} -244 q^{-26} +164 q^{-28} -46 q^{-30} -92 q^{-32} +246 q^{-34} -378 q^{-36} +502 q^{-38} -576 q^{-40} +612 q^{-42} -587 q^{-44} +512 q^{-46} -402 q^{-48} +254 q^{-50} -107 q^{-52} -40 q^{-54} +160 q^{-56} -244 q^{-58} +292 q^{-60} -300 q^{-62} +282 q^{-64} -246 q^{-66} +195 q^{-68} -144 q^{-70} +102 q^{-72} -66 q^{-74} +38 q^{-76} -20 q^{-78} +10 q^{-80} -4 q^{-82} + q^{-84} } |
| 2,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4+q^2+1+2 q^{-2} +5 q^{-4} +3 q^{-6} + q^{-10} +5 q^{-12} -4 q^{-14} -10 q^{-16} -2 q^{-18} -9 q^{-22} -6 q^{-24} +6 q^{-26} +5 q^{-28} -2 q^{-30} +4 q^{-32} +9 q^{-34} -3 q^{-36} - q^{-38} +7 q^{-40} -6 q^{-44} +2 q^{-46} +4 q^{-48} -6 q^{-50} -5 q^{-52} +5 q^{-54} +3 q^{-56} -6 q^{-58} - q^{-60} +4 q^{-62} - q^{-64} -2 q^{-66} +2 q^{-70} - q^{-74} + q^{-76} } |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+3 q^{-4} +4 q^{-6} +2 q^{-8} +6 q^{-10} +7 q^{-12} -7 q^{-14} +2 q^{-16} -20 q^{-20} +4 q^{-24} -14 q^{-26} +5 q^{-28} +12 q^{-30} -4 q^{-32} + q^{-34} +5 q^{-36} +3 q^{-38} -5 q^{-40} -3 q^{-42} +11 q^{-44} -5 q^{-46} -9 q^{-48} +14 q^{-50} -4 q^{-52} -11 q^{-54} +11 q^{-56} - q^{-58} -6 q^{-60} +5 q^{-62} -2 q^{-66} + q^{-68} } |
| 1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-1} + q^{-3} +3 q^{-5} +2 q^{-7} +4 q^{-9} - q^{-11} + q^{-13} -4 q^{-15} -2 q^{-17} -3 q^{-19} - q^{-21} - q^{-25} +2 q^{-27} - q^{-29} +2 q^{-31} - q^{-33} +2 q^{-35} - q^{-37} + q^{-39} } |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} + q^{-4} +3 q^{-6} +5 q^{-8} +7 q^{-10} +7 q^{-12} +9 q^{-14} +8 q^{-16} +4 q^{-18} -4 q^{-20} -4 q^{-22} -10 q^{-24} -21 q^{-26} -16 q^{-28} -4 q^{-30} -10 q^{-32} -9 q^{-34} +12 q^{-36} +15 q^{-38} +3 q^{-40} +4 q^{-42} +17 q^{-44} +3 q^{-46} -9 q^{-48} +4 q^{-50} +6 q^{-52} -11 q^{-54} -2 q^{-56} +10 q^{-58} -5 q^{-60} -8 q^{-62} +5 q^{-64} +4 q^{-66} -9 q^{-68} -3 q^{-70} +8 q^{-72} + q^{-74} -6 q^{-76} +2 q^{-78} +4 q^{-80} -2 q^{-82} - q^{-84} + q^{-86} } |
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} + q^{-4} +3 q^{-6} +3 q^{-8} +3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} -4 q^{-18} -3 q^{-20} -3 q^{-22} -3 q^{-24} - q^{-26} - q^{-28} + q^{-30} - q^{-32} +2 q^{-34} - q^{-36} +2 q^{-38} +2 q^{-44} - q^{-46} + q^{-48} } |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+3 q^{-4} -2 q^{-6} +6 q^{-8} -6 q^{-10} +11 q^{-12} -11 q^{-14} +14 q^{-16} -14 q^{-18} +12 q^{-20} -10 q^{-22} +2 q^{-24} +2 q^{-26} -11 q^{-28} +16 q^{-30} -24 q^{-32} +27 q^{-34} -29 q^{-36} +27 q^{-38} -23 q^{-40} +17 q^{-42} -9 q^{-44} +3 q^{-46} +5 q^{-48} -10 q^{-50} +14 q^{-52} -15 q^{-54} +15 q^{-56} -13 q^{-58} +10 q^{-60} -7 q^{-62} +4 q^{-64} -2 q^{-66} + q^{-68} } |
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^2+3 q^{-6} +3 q^{-8} + q^{-10} -2 q^{-12} + q^{-14} +6 q^{-16} +8 q^{-18} -3 q^{-20} -10 q^{-22} -3 q^{-24} +11 q^{-26} +7 q^{-28} -12 q^{-30} -18 q^{-32} -2 q^{-34} +14 q^{-36} +4 q^{-38} -13 q^{-40} -11 q^{-42} +7 q^{-44} +12 q^{-46} -9 q^{-50} +2 q^{-52} +11 q^{-54} +3 q^{-56} -9 q^{-58} -3 q^{-60} +9 q^{-62} +7 q^{-64} -8 q^{-66} -10 q^{-68} +6 q^{-70} +12 q^{-72} -2 q^{-74} -15 q^{-76} -4 q^{-78} +14 q^{-80} +11 q^{-82} -9 q^{-84} -15 q^{-86} +13 q^{-90} +6 q^{-92} -7 q^{-94} -8 q^{-96} + q^{-98} +6 q^{-100} +2 q^{-102} -2 q^{-104} -2 q^{-106} + q^{-110} } |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} +3 q^{-6} + q^{-8} +7 q^{-10} + q^{-12} +10 q^{-14} - q^{-16} +12 q^{-18} -9 q^{-20} +8 q^{-22} -15 q^{-24} +5 q^{-26} -18 q^{-28} -11 q^{-32} -6 q^{-38} +12 q^{-40} -8 q^{-42} +20 q^{-44} -19 q^{-46} +22 q^{-48} -20 q^{-50} +23 q^{-52} -20 q^{-54} +16 q^{-56} -14 q^{-58} +13 q^{-60} -4 q^{-62} + q^{-64} -5 q^{-68} +10 q^{-70} -11 q^{-72} +9 q^{-74} -13 q^{-76} +13 q^{-78} -9 q^{-80} +8 q^{-82} -8 q^{-84} +6 q^{-86} -3 q^{-88} +2 q^{-90} -2 q^{-92} + q^{-94} } |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} +3 q^{-6} -2 q^{-8} +3 q^{-10} - q^{-12} +7 q^{-16} -9 q^{-18} +14 q^{-20} -12 q^{-22} +12 q^{-24} + q^{-26} -12 q^{-28} +31 q^{-30} -40 q^{-32} +46 q^{-34} -33 q^{-36} + q^{-38} +33 q^{-40} -65 q^{-42} +80 q^{-44} -67 q^{-46} +28 q^{-48} +17 q^{-50} -59 q^{-52} +71 q^{-54} -63 q^{-56} +24 q^{-58} +17 q^{-60} -50 q^{-62} +46 q^{-64} -24 q^{-66} -19 q^{-68} +58 q^{-70} -75 q^{-72} +60 q^{-74} -21 q^{-76} -35 q^{-78} +87 q^{-80} -115 q^{-82} +106 q^{-84} -59 q^{-86} -3 q^{-88} +66 q^{-90} -101 q^{-92} +104 q^{-94} -68 q^{-96} +17 q^{-98} +32 q^{-100} -60 q^{-102} +55 q^{-104} -22 q^{-106} -19 q^{-108} +51 q^{-110} -52 q^{-112} +28 q^{-114} +11 q^{-116} -53 q^{-118} +76 q^{-120} -74 q^{-122} +48 q^{-124} -8 q^{-126} -33 q^{-128} +60 q^{-130} -64 q^{-132} +54 q^{-134} -28 q^{-136} +3 q^{-138} +15 q^{-140} -29 q^{-142} +28 q^{-144} -21 q^{-146} +12 q^{-148} -2 q^{-150} -3 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} } |
.
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["10 76"];
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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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -2 t^{-3} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^6-5 z^4-2 z^2+1} |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 57, 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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-3 q^9+6 q^8-8 q^7+9 q^6-10 q^5+8 q^4-6 q^3+4 q^2-q+1} |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -4 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +4 z^2 a^{-2} -6 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +4 a^{-2} -4 a^{-4} + a^{-8} } |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} +z^7 a^{-3} -2 z^7 a^{-5} +2 z^7 a^{-7} +5 z^7 a^{-9} +z^6 a^{-2} -9 z^6 a^{-6} -3 z^6 a^{-8} +5 z^6 a^{-10} -2 z^5 a^{-3} +7 z^5 a^{-5} -2 z^5 a^{-7} -8 z^5 a^{-9} +3 z^5 a^{-11} -5 z^4 a^{-2} -7 z^4 a^{-4} +10 z^4 a^{-6} +4 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} -2 z^3 a^{-3} -15 z^3 a^{-5} -3 z^3 a^{-7} +7 z^3 a^{-9} -3 z^3 a^{-11} +8 z^2 a^{-2} +9 z^2 a^{-4} -7 z^2 a^{-6} -4 z^2 a^{-8} +3 z^2 a^{-10} -z^2 a^{-12} +4 z a^{-3} +8 z a^{-5} +2 z a^{-7} -2 z a^{-9} -4 a^{-2} -4 a^{-4} + a^{-8} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}
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["10 76"];
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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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{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -2 t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-3 q^9+6 q^8-8 q^7+9 q^6-10 q^5+8 q^4-6 q^3+4 q^2-q+1} } |
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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{} |
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: | (-2, -6) |
| 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of 10 76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
|
The Coloured Jones Polynomials
The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1}
-dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)}
)
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-3 q^{27}+2 q^{26}+7 q^{25}-16 q^{24}+5 q^{23}+26 q^{22}-40 q^{21}+3 q^{20}+55 q^{19}-62 q^{18}-5 q^{17}+77 q^{16}-68 q^{15}-16 q^{14}+81 q^{13}-55 q^{12}-24 q^{11}+65 q^{10}-32 q^9-24 q^8+38 q^7-11 q^6-15 q^5+15 q^4-2 q^3-5 q^2+4 q- q^{-1} + q^{-2} } |
| 3 | |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{88}-3 q^{87}+2 q^{86}+3 q^{85}-5 q^{84}+5 q^{83}-12 q^{82}+9 q^{81}+15 q^{80}-20 q^{79}+13 q^{78}-42 q^{77}+29 q^{76}+59 q^{75}-51 q^{74}+6 q^{73}-121 q^{72}+81 q^{71}+183 q^{70}-73 q^{69}-52 q^{68}-333 q^{67}+140 q^{66}+461 q^{65}+18 q^{64}-150 q^{63}-756 q^{62}+94 q^{61}+847 q^{60}+315 q^{59}-159 q^{58}-1309 q^{57}-150 q^{56}+1158 q^{55}+738 q^{54}+10 q^{53}-1762 q^{52}-506 q^{51}+1238 q^{50}+1084 q^{49}+306 q^{48}-1951 q^{47}-809 q^{46}+1093 q^{45}+1230 q^{44}+604 q^{43}-1861 q^{42}-982 q^{41}+795 q^{40}+1189 q^{39}+850 q^{38}-1557 q^{37}-1040 q^{36}+410 q^{35}+1004 q^{34}+1021 q^{33}-1094 q^{32}-976 q^{31}-3 q^{30}+692 q^{29}+1062 q^{28}-562 q^{27}-756 q^{26}-311 q^{25}+301 q^{24}+895 q^{23}-119 q^{22}-412 q^{21}-387 q^{20}-25 q^{19}+561 q^{18}+88 q^{17}-102 q^{16}-255 q^{15}-151 q^{14}+238 q^{13}+83 q^{12}+40 q^{11}-92 q^{10}-113 q^9+67 q^8+19 q^7+43 q^6-13 q^5-45 q^4+18 q^3-8 q^2+16 q+2-13 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} + q^{-5} -4 q^{-6} +4 q^{-7} - q^{-8} - q^{-11} + q^{-12} } |
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. |
|
