10 92
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 92's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X18,7,19,8 X12,18,13,17 X6,19,7,20 X8,14,9,13 X2,10,3,9 |
| Gauss code | 1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4 |
| Dowker-Thistlethwaite code | 4 10 14 18 2 16 8 20 12 6 |
| Conway Notation | [.21.2.20] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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 [{3, 11}, {2, 6}, {1, 3}, {12, 8}, {10, 7}, {8, 5}, {6, 4}, {11, 9}, {5, 10}, {9, 2}, {4, 12}, {7, 1}] |
[edit Notes on presentations of 10 92]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 92"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X18,7,19,8 X12,18,13,17 X6,19,7,20 X8,14,9,13 X2,10,3,9 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 14 18 2 16 8 20 12 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[.21.2.20] |
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,\{1,1,1,2,2,-3,2,-1,2,-3,2\})} |
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[{3, 11}, {2, 6}, {1, 3}, {12, 8}, {10, 7}, {8, 5}, {6, 4}, {11, 9}, {5, 10}, {9, 2}, {4, 12}, {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 | 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+10 t^2-20 t+25-20 t^{-1} +10 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^6-2 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 | { 89, 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}-4 q^9+8 q^8-12 q^7+14 q^6-15 q^5+14 q^4-10 q^3+7 q^2-3 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} -2 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-2} -z^2 a^{-6} +z^2 a^{-8} + a^{-2} + a^{-4} - a^{-6} } |
| 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 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8 a^{-8} +3 z^7 a^{-3} +5 z^7 a^{-5} +12 z^7 a^{-7} +10 z^7 a^{-9} +z^6 a^{-2} -8 z^6 a^{-4} -22 z^6 a^{-6} -5 z^6 a^{-8} +8 z^6 a^{-10} -8 z^5 a^{-3} -22 z^5 a^{-5} -32 z^5 a^{-7} -14 z^5 a^{-9} +4 z^5 a^{-11} -3 z^4 a^{-2} +2 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +6 z^3 a^{-3} +18 z^3 a^{-5} +21 z^3 a^{-7} +7 z^3 a^{-9} -2 z^3 a^{-11} +3 z^2 a^{-2} +z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-3} -5 z a^{-5} -5 z a^{-7} -z a^{-9} - a^{-2} + a^{-4} + a^{-6} } |
| 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} + q^{-4} +2 q^{-6} -2 q^{-8} +4 q^{-10} - q^{-12} + q^{-14} + q^{-16} -3 q^{-18} +2 q^{-20} -3 q^{-22} + q^{-24} + q^{-26} -2 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} -2 q^{-4} +6 q^{-6} -10 q^{-8} +13 q^{-10} -12 q^{-12} +3 q^{-14} +19 q^{-16} -45 q^{-18} +75 q^{-20} -88 q^{-22} +65 q^{-24} -7 q^{-26} -85 q^{-28} +181 q^{-30} -229 q^{-32} +207 q^{-34} -97 q^{-36} -66 q^{-38} +227 q^{-40} -319 q^{-42} +300 q^{-44} -165 q^{-46} -29 q^{-48} +202 q^{-50} -276 q^{-52} +226 q^{-54} -72 q^{-56} -100 q^{-58} +223 q^{-60} -234 q^{-62} +120 q^{-64} +62 q^{-66} -250 q^{-68} +358 q^{-70} -329 q^{-72} +175 q^{-74} +56 q^{-76} -283 q^{-78} +422 q^{-80} -428 q^{-82} +287 q^{-84} -63 q^{-86} -176 q^{-88} +333 q^{-90} -352 q^{-92} +236 q^{-94} -38 q^{-96} -143 q^{-98} +230 q^{-100} -197 q^{-102} +55 q^{-104} +115 q^{-106} -235 q^{-108} +256 q^{-110} -158 q^{-112} -6 q^{-114} +173 q^{-116} -275 q^{-118} +278 q^{-120} -193 q^{-122} +61 q^{-124} +66 q^{-126} -157 q^{-128} +184 q^{-130} -156 q^{-132} +99 q^{-134} -28 q^{-136} -26 q^{-138} +54 q^{-140} -65 q^{-142} +54 q^{-144} -34 q^{-146} +16 q^{-148} + q^{-150} -8 q^{-152} +10 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} } |
Further Quantum Invariants
Further quantum knot invariants for 10_92.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 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-2 q^{-1} +4 q^{-3} -3 q^{-5} +4 q^{-7} - q^{-9} - q^{-11} +2 q^{-13} -4 q^{-15} +4 q^{-17} -3 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^6-2 q^4-q^2+9-6 q^{-2} -13 q^{-4} +24 q^{-6} + q^{-8} -34 q^{-10} +27 q^{-12} +21 q^{-14} -41 q^{-16} +11 q^{-18} +30 q^{-20} -26 q^{-22} -11 q^{-24} +22 q^{-26} +3 q^{-28} -25 q^{-30} +2 q^{-32} +33 q^{-34} -25 q^{-36} -21 q^{-38} +43 q^{-40} -12 q^{-42} -28 q^{-44} +28 q^{-46} + q^{-48} -15 q^{-50} +8 q^{-52} + q^{-54} -3 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^{13}-q^{11}+4 q^9+6 q^7-9 q^5-19 q^3+13 q+44 q^{-1} -3 q^{-3} -77 q^{-5} -33 q^{-7} +111 q^{-9} +92 q^{-11} -114 q^{-13} -172 q^{-15} +85 q^{-17} +247 q^{-19} -14 q^{-21} -291 q^{-23} -75 q^{-25} +293 q^{-27} +168 q^{-29} -258 q^{-31} -233 q^{-33} +192 q^{-35} +269 q^{-37} -116 q^{-39} -282 q^{-41} +44 q^{-43} +259 q^{-45} +34 q^{-47} -229 q^{-49} -106 q^{-51} +176 q^{-53} +181 q^{-55} -110 q^{-57} -243 q^{-59} +20 q^{-61} +288 q^{-63} +77 q^{-65} -295 q^{-67} -168 q^{-69} +262 q^{-71} +233 q^{-73} -192 q^{-75} -251 q^{-77} +108 q^{-79} +229 q^{-81} -42 q^{-83} -174 q^{-85} - q^{-87} +109 q^{-89} +19 q^{-91} -60 q^{-93} -16 q^{-95} +30 q^{-97} +5 q^{-99} -10 q^{-101} -3 q^{-103} +5 q^{-105} + q^{-107} -3 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}-2 q^{26}-q^{24}+4 q^{22}+q^{20}+3 q^{18}-15 q^{16}-13 q^{14}+21 q^{12}+30 q^{10}+38 q^8-61 q^6-117 q^4-19 q^2+116+269 q^{-2} +28 q^{-4} -330 q^{-6} -394 q^{-8} -75 q^{-10} +664 q^{-12} +656 q^{-14} -103 q^{-16} -949 q^{-18} -1039 q^{-20} +406 q^{-22} +1467 q^{-24} +1122 q^{-26} -627 q^{-28} -2170 q^{-30} -1028 q^{-32} +1205 q^{-34} +2480 q^{-36} +964 q^{-38} -2076 q^{-40} -2550 q^{-42} -370 q^{-44} +2572 q^{-46} +2568 q^{-48} -683 q^{-50} -2837 q^{-52} -1928 q^{-54} +1465 q^{-56} +2993 q^{-58} +748 q^{-60} -2054 q^{-62} -2479 q^{-64} +263 q^{-66} +2457 q^{-68} +1487 q^{-70} -1081 q^{-72} -2308 q^{-74} -596 q^{-76} +1663 q^{-78} +1884 q^{-80} -133 q^{-82} -1952 q^{-84} -1480 q^{-86} +643 q^{-88} +2233 q^{-90} +1164 q^{-92} -1199 q^{-94} -2451 q^{-96} -951 q^{-98} +2029 q^{-100} +2574 q^{-102} +342 q^{-104} -2637 q^{-106} -2630 q^{-108} +727 q^{-110} +2948 q^{-112} +2022 q^{-114} -1453 q^{-116} -3072 q^{-118} -863 q^{-120} +1790 q^{-122} +2450 q^{-124} +104 q^{-126} -1962 q^{-128} -1354 q^{-130} +315 q^{-132} +1520 q^{-134} +670 q^{-136} -631 q^{-138} -778 q^{-140} -264 q^{-142} +507 q^{-144} +394 q^{-146} -67 q^{-148} -204 q^{-150} -174 q^{-152} +96 q^{-154} +98 q^{-156} - q^{-158} -15 q^{-160} -44 q^{-162} +17 q^{-164} +13 q^{-166} -6 q^{-168} +2 q^{-170} -6 q^{-172} +5 q^{-174} + q^{-176} -3 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}-2 q^{43}-q^{41}+4 q^{39}+q^{37}-2 q^{35}-3 q^{33}-9 q^{31}-5 q^{29}+24 q^{27}+36 q^{25}+8 q^{23}-40 q^{21}-97 q^{19}-89 q^{17}+41 q^{15}+232 q^{13}+284 q^{11}+74 q^9-338 q^7-670 q^5-512 q^3+244 q+1159 q^{-1} +1393 q^{-3} +420 q^{-5} -1360 q^{-7} -2638 q^{-9} -2020 q^{-11} +671 q^{-13} +3748 q^{-15} +4487 q^{-17} +1516 q^{-19} -3666 q^{-21} -7192 q^{-23} -5449 q^{-25} +1493 q^{-27} +8853 q^{-29} +10312 q^{-31} +3344 q^{-33} -7926 q^{-35} -14714 q^{-37} -10269 q^{-39} +3624 q^{-41} +16694 q^{-43} +17663 q^{-45} +3934 q^{-47} -14899 q^{-49} -23458 q^{-51} -13256 q^{-53} +9106 q^{-55} +25834 q^{-57} +22167 q^{-59} -386 q^{-61} -23990 q^{-63} -28639 q^{-65} -9350 q^{-67} +18542 q^{-69} +31428 q^{-71} +17884 q^{-73} -10903 q^{-75} -30419 q^{-77} -23837 q^{-79} +2987 q^{-81} +26597 q^{-83} +26569 q^{-85} +3629 q^{-87} -21302 q^{-89} -26442 q^{-91} -8267 q^{-93} +15967 q^{-95} +24539 q^{-97} +10786 q^{-99} -11497 q^{-101} -21849 q^{-103} -11951 q^{-105} +8041 q^{-107} +19517 q^{-109} +12632 q^{-111} -5369 q^{-113} -17777 q^{-115} -13776 q^{-117} +2565 q^{-119} +16606 q^{-121} +15936 q^{-123} +1127 q^{-125} -15198 q^{-127} -19017 q^{-129} -6419 q^{-131} +12642 q^{-133} +22214 q^{-135} +13326 q^{-137} -7954 q^{-139} -24321 q^{-141} -21034 q^{-143} +851 q^{-145} +23869 q^{-147} +28018 q^{-149} +8190 q^{-151} -20034 q^{-153} -32512 q^{-155} -17554 q^{-157} +12908 q^{-159} +32986 q^{-161} +25248 q^{-163} -3741 q^{-165} -29118 q^{-167} -29445 q^{-169} -5295 q^{-171} +21820 q^{-173} +29170 q^{-175} +12189 q^{-177} -13005 q^{-179} -25026 q^{-181} -15577 q^{-183} +4974 q^{-185} +18497 q^{-187} +15350 q^{-189} +806 q^{-191} -11613 q^{-193} -12610 q^{-195} -3736 q^{-197} +5992 q^{-199} +8795 q^{-201} +4272 q^{-203} -2239 q^{-205} -5256 q^{-207} -3493 q^{-209} +328 q^{-211} +2711 q^{-213} +2263 q^{-215} +317 q^{-217} -1146 q^{-219} -1230 q^{-221} -399 q^{-223} +416 q^{-225} +592 q^{-227} +226 q^{-229} -119 q^{-231} -217 q^{-233} -121 q^{-235} +23 q^{-237} +83 q^{-239} +48 q^{-241} -15 q^{-243} -24 q^{-245} -5 q^{-247} +2 q^{-251} +6 q^{-253} - q^{-255} -6 q^{-257} +5 q^{-259} + q^{-261} -3 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} + q^{-4} +2 q^{-6} -2 q^{-8} +4 q^{-10} - q^{-12} + q^{-14} + q^{-16} -3 q^{-18} +2 q^{-20} -3 q^{-22} + q^{-24} + q^{-26} -2 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^4-4 q^2+14-36 q^{-2} +82 q^{-4} -162 q^{-6} +298 q^{-8} -484 q^{-10} +726 q^{-12} -994 q^{-14} +1244 q^{-16} -1420 q^{-18} +1479 q^{-20} -1354 q^{-22} +1034 q^{-24} -528 q^{-26} -104 q^{-28} +814 q^{-30} -1524 q^{-32} +2132 q^{-34} -2575 q^{-36} +2796 q^{-38} -2780 q^{-40} +2510 q^{-42} -2033 q^{-44} +1402 q^{-46} -700 q^{-48} +16 q^{-50} +570 q^{-52} -998 q^{-54} +1256 q^{-56} -1340 q^{-58} +1276 q^{-60} -1116 q^{-62} +916 q^{-64} -706 q^{-66} +506 q^{-68} -344 q^{-70} +226 q^{-72} -136 q^{-74} +73 q^{-76} -38 q^{-78} +18 q^{-80} -6 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^2-2+3 q^{-2} +5 q^{-4} -2 q^{-6} -8 q^{-8} +4 q^{-10} +14 q^{-12} -7 q^{-14} -14 q^{-16} +10 q^{-18} +15 q^{-20} -10 q^{-22} -9 q^{-24} +14 q^{-26} +7 q^{-28} -11 q^{-30} + q^{-32} +10 q^{-34} -10 q^{-36} -6 q^{-38} +10 q^{-40} -8 q^{-42} -14 q^{-44} +8 q^{-46} +14 q^{-48} -12 q^{-50} -10 q^{-52} +16 q^{-54} +8 q^{-56} -13 q^{-58} -5 q^{-60} +11 q^{-62} +2 q^{-64} -5 q^{-66} -2 q^{-68} +3 q^{-70} -2 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 1-2 q^{-2} +2 q^{-4} +4 q^{-6} -9 q^{-8} +8 q^{-10} +9 q^{-12} -21 q^{-14} +18 q^{-16} +13 q^{-18} -28 q^{-20} +20 q^{-22} +14 q^{-24} -29 q^{-26} +7 q^{-28} +11 q^{-30} -14 q^{-32} -6 q^{-34} +5 q^{-36} +11 q^{-38} -13 q^{-40} -8 q^{-42} +29 q^{-44} -16 q^{-46} -17 q^{-48} +33 q^{-50} -13 q^{-52} -17 q^{-54} +22 q^{-56} -5 q^{-58} -10 q^{-60} +9 q^{-62} -3 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} +2 q^{-5} - q^{-7} +3 q^{-9} -2 q^{-11} +4 q^{-13} - q^{-15} +2 q^{-17} -3 q^{-25} +2 q^{-27} -3 q^{-29} +2 q^{-31} -2 q^{-33} +2 q^{-35} -2 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} +4 q^{-8} - q^{-10} -4 q^{-12} +6 q^{-14} +6 q^{-16} -8 q^{-18} -2 q^{-20} +18 q^{-22} +4 q^{-24} -15 q^{-26} +11 q^{-28} +26 q^{-30} -17 q^{-32} -19 q^{-34} +20 q^{-36} +3 q^{-38} -32 q^{-40} -3 q^{-42} +20 q^{-44} -14 q^{-46} -15 q^{-48} +23 q^{-50} +9 q^{-52} -24 q^{-54} +9 q^{-56} +23 q^{-58} -17 q^{-60} -15 q^{-62} +20 q^{-64} +7 q^{-66} -20 q^{-68} -2 q^{-70} +16 q^{-72} -2 q^{-74} -12 q^{-76} +4 q^{-78} +7 q^{-80} -3 q^{-82} -2 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} +2 q^{-6} +3 q^{-12} -2 q^{-14} +4 q^{-16} - q^{-18} +2 q^{-20} + q^{-22} - q^{-28} -3 q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} - q^{-42} +2 q^{-44} -2 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 1-2 q^{-2} +6 q^{-4} -10 q^{-6} +17 q^{-8} -24 q^{-10} +31 q^{-12} -35 q^{-14} +38 q^{-16} -33 q^{-18} +26 q^{-20} -12 q^{-22} -4 q^{-24} +23 q^{-26} -41 q^{-28} +57 q^{-30} -68 q^{-32} +72 q^{-34} -69 q^{-36} +59 q^{-38} -45 q^{-40} +26 q^{-42} -7 q^{-44} -10 q^{-46} +23 q^{-48} -33 q^{-50} +37 q^{-52} -37 q^{-54} +32 q^{-56} -25 q^{-58} +18 q^{-60} -11 q^{-62} +6 q^{-64} -3 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^2-2 q^{-2} -2 q^{-4} +4 q^{-6} +7 q^{-8} -2 q^{-10} -13 q^{-12} -5 q^{-14} +18 q^{-16} +19 q^{-18} -13 q^{-20} -30 q^{-22} - q^{-24} +38 q^{-26} +21 q^{-28} -28 q^{-30} -34 q^{-32} +13 q^{-34} +40 q^{-36} +6 q^{-38} -34 q^{-40} -17 q^{-42} +22 q^{-44} +20 q^{-46} -16 q^{-48} -23 q^{-50} +9 q^{-52} +23 q^{-54} -6 q^{-56} -27 q^{-58} +28 q^{-62} +9 q^{-64} -28 q^{-66} -19 q^{-68} +25 q^{-70} +30 q^{-72} -15 q^{-74} -38 q^{-76} - q^{-78} +38 q^{-80} +19 q^{-82} -25 q^{-84} -30 q^{-86} +7 q^{-88} +27 q^{-90} +8 q^{-92} -15 q^{-94} -14 q^{-96} +3 q^{-98} +10 q^{-100} +3 q^{-102} -3 q^{-104} -3 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} -2 q^{-4} +4 q^{-6} -5 q^{-8} +10 q^{-10} -13 q^{-12} +18 q^{-14} -20 q^{-16} +27 q^{-18} -28 q^{-20} +30 q^{-22} -26 q^{-24} +30 q^{-26} -19 q^{-28} +14 q^{-30} -3 q^{-32} -2 q^{-34} +14 q^{-36} -31 q^{-38} +33 q^{-40} -44 q^{-42} +48 q^{-44} -59 q^{-46} +53 q^{-48} -52 q^{-50} +52 q^{-52} -42 q^{-54} +31 q^{-56} -23 q^{-58} +17 q^{-60} -10 q^{-64} +13 q^{-66} -21 q^{-68} +30 q^{-70} -29 q^{-72} +25 q^{-74} -29 q^{-76} +27 q^{-78} -18 q^{-80} +14 q^{-82} -14 q^{-84} +10 q^{-86} -4 q^{-88} +3 q^{-90} -3 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} -2 q^{-4} +6 q^{-6} -10 q^{-8} +13 q^{-10} -12 q^{-12} +3 q^{-14} +19 q^{-16} -45 q^{-18} +75 q^{-20} -88 q^{-22} +65 q^{-24} -7 q^{-26} -85 q^{-28} +181 q^{-30} -229 q^{-32} +207 q^{-34} -97 q^{-36} -66 q^{-38} +227 q^{-40} -319 q^{-42} +300 q^{-44} -165 q^{-46} -29 q^{-48} +202 q^{-50} -276 q^{-52} +226 q^{-54} -72 q^{-56} -100 q^{-58} +223 q^{-60} -234 q^{-62} +120 q^{-64} +62 q^{-66} -250 q^{-68} +358 q^{-70} -329 q^{-72} +175 q^{-74} +56 q^{-76} -283 q^{-78} +422 q^{-80} -428 q^{-82} +287 q^{-84} -63 q^{-86} -176 q^{-88} +333 q^{-90} -352 q^{-92} +236 q^{-94} -38 q^{-96} -143 q^{-98} +230 q^{-100} -197 q^{-102} +55 q^{-104} +115 q^{-106} -235 q^{-108} +256 q^{-110} -158 q^{-112} -6 q^{-114} +173 q^{-116} -275 q^{-118} +278 q^{-120} -193 q^{-122} +61 q^{-124} +66 q^{-126} -157 q^{-128} +184 q^{-130} -156 q^{-132} +99 q^{-134} -28 q^{-136} -26 q^{-138} +54 q^{-140} -65 q^{-142} +54 q^{-144} -34 q^{-146} +16 q^{-148} + q^{-150} -8 q^{-152} +10 q^{-154} -10 q^{-156} +6 q^{-158} -3 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 92"];
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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+10 t^2-20 t+25-20 t^{-1} +10 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^6-2 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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{ 89, 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}-4 q^9+8 q^8-12 q^7+14 q^6-15 q^5+14 q^4-10 q^3+7 q^2-3 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} -2 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-2} -z^2 a^{-6} +z^2 a^{-8} + a^{-2} + a^{-4} - a^{-6} } |
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 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8 a^{-8} +3 z^7 a^{-3} +5 z^7 a^{-5} +12 z^7 a^{-7} +10 z^7 a^{-9} +z^6 a^{-2} -8 z^6 a^{-4} -22 z^6 a^{-6} -5 z^6 a^{-8} +8 z^6 a^{-10} -8 z^5 a^{-3} -22 z^5 a^{-5} -32 z^5 a^{-7} -14 z^5 a^{-9} +4 z^5 a^{-11} -3 z^4 a^{-2} +2 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +6 z^3 a^{-3} +18 z^3 a^{-5} +21 z^3 a^{-7} +7 z^3 a^{-9} -2 z^3 a^{-11} +3 z^2 a^{-2} +z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-3} -5 z a^{-5} -5 z a^{-7} -z a^{-9} - a^{-2} + a^{-4} + a^{-6} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a153, K11a224, K11n35, K11n43,}
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 92"];
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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+10 t^2-20 t+25-20 t^{-1} +10 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}-4 q^9+8 q^8-12 q^7+14 q^6-15 q^5+14 q^4-10 q^3+7 q^2-3 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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{K11a153, K11a224, K11n35, K11n43,} |
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, 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 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 92. 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}-4 q^{27}+4 q^{26}+8 q^{25}-27 q^{24}+20 q^{23}+35 q^{22}-83 q^{21}+36 q^{20}+90 q^{19}-147 q^{18}+32 q^{17}+148 q^{16}-178 q^{15}+5 q^{14}+176 q^{13}-159 q^{12}-28 q^{11}+161 q^{10}-103 q^9-47 q^8+109 q^7-41 q^6-41 q^5+48 q^4-6 q^3-18 q^2+11 q+1-3 q^{-1} + q^{-2} } |
| 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^{54}-4 q^{53}+4 q^{52}+4 q^{51}-7 q^{50}-11 q^{49}+19 q^{48}+29 q^{47}-53 q^{46}-55 q^{45}+98 q^{44}+119 q^{43}-163 q^{42}-228 q^{41}+230 q^{40}+390 q^{39}-284 q^{38}-587 q^{37}+289 q^{36}+815 q^{35}-255 q^{34}-1017 q^{33}+162 q^{32}+1187 q^{31}-44 q^{30}-1285 q^{29}-101 q^{28}+1320 q^{27}+247 q^{26}-1290 q^{25}-383 q^{24}+1197 q^{23}+510 q^{22}-1065 q^{21}-598 q^{20}+871 q^{19}+676 q^{18}-680 q^{17}-675 q^{16}+446 q^{15}+651 q^{14}-254 q^{13}-550 q^{12}+78 q^{11}+435 q^{10}+23 q^9-289 q^8-84 q^7+178 q^6+81 q^5-83 q^4-65 q^3+34 q^2+37 q-9-18 q^{-1} +3 q^{-2} +5 q^{-3} + q^{-4} -3 q^{-5} + q^{-6} } |
| 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}-4 q^{87}+4 q^{86}+4 q^{85}-11 q^{84}+9 q^{83}-12 q^{82}+23 q^{81}+8 q^{80}-72 q^{79}+38 q^{78}+2 q^{77}+122 q^{76}+6 q^{75}-342 q^{74}+8 q^{73}+139 q^{72}+583 q^{71}+119 q^{70}-1113 q^{69}-506 q^{68}+286 q^{67}+1884 q^{66}+969 q^{65}-2318 q^{64}-2175 q^{63}-322 q^{62}+3950 q^{61}+3315 q^{60}-2978 q^{59}-4828 q^{58}-2531 q^{57}+5569 q^{56}+6790 q^{55}-2052 q^{54}-7049 q^{53}-5888 q^{52}+5562 q^{51}+9769 q^{50}+180 q^{49}-7594 q^{48}-8868 q^{47}+4062 q^{46}+11021 q^{45}+2543 q^{44}-6525 q^{43}-10458 q^{42}+1939 q^{41}+10549 q^{40}+4362 q^{39}-4508 q^{38}-10679 q^{37}-320 q^{36}+8837 q^{35}+5589 q^{34}-1940 q^{33}-9709 q^{32}-2514 q^{31}+6095 q^{30}+6014 q^{29}+862 q^{28}-7464 q^{27}-4042 q^{26}+2702 q^{25}+5105 q^{24}+3016 q^{23}-4213 q^{22}-4038 q^{21}-240 q^{20}+2925 q^{19}+3490 q^{18}-1173 q^{17}-2522 q^{16}-1515 q^{15}+692 q^{14}+2348 q^{13}+370 q^{12}-773 q^{11}-1170 q^{10}-369 q^9+903 q^8+460 q^7+73 q^6-411 q^5-361 q^4+164 q^3+141 q^2+137 q-53-120 q^{-1} +11 q^{-2} +6 q^{-3} +39 q^{-4} +3 q^{-5} -21 q^{-6} +3 q^{-7} -3 q^{-8} +5 q^{-9} + q^{-10} -3 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. |
|
