10 26

10_25

10_27

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 26's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 26 at Knotilus!

[edit 10 26 Quick Notes]

[edit 10 26 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_16,8,17,7 X_12,3,13,4 X_2,15,3,16 X_14,5,15,6 X_4,13,5,14 X_20,12,1,11 X_8,20,9,19 X_18,10,19,9 X_10,18,11,17
Gauss code	1, -4, 3, -6, 5, -1, 2, -8, 9, -10, 7, -3, 6, -5, 4, -2, 10, -9, 8, -7
Dowker-Thistlethwaite code	6 12 14 16 18 20 4 2 10 8
Conway Notation	[32113]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{2, 13}, {1, 7}, {12, 4}, {13, 9}, {8, 10}, {9, 11}, {10, 12}, {5, 3}, {4, 6}, {7, 5}, {6, 2}, {3, 8}, {11, 1}]

[edit Notes on presentations of 10 26]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 26"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₆₂₇₁ X_16,8,17,7 X_12,3,13,4 X_2,15,3,16 X_14,5,15,6 X_4,13,5,14 X_20,12,1,11 X_8,20,9,19 X_18,10,19,9 X_10,18,11,17

In[5]:= GaussCode[K]

Out[5]= 1, -4, 3, -6, 5, -1, 2, -8, 9, -10, 7, -3, 6, -5, 4, -2, 10, -9, 8, -7

In[6]:= DTCode[K]

Out[6]= 6 12 14 16 18 20 4 2 10 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [32113]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= 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,-1,2,2,2,3,-2,3\})}

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]= { 4, 11, 4 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{2, 13}, {1, 7}, {12, 4}, {13, 9}, {8, 10}, {9, 11}, {10, 12}, {5, 3}, {4, 6}, {7, 5}, {6, 2}, {3, 8}, {11, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-7]
Hyperbolic Volume	11.352
A-Polynomial	See Data:10 26/A-polynomial

[edit Notes for 10 26's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	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}
Topological 4 genus	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}
Concordance genus	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 3}
Rasmussen s-Invariant	0

[edit Notes for 10 26's four dimensional invariants]

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^2-13 t+17-13 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^4-3 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	{ 61, 0 }
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^6-2 q^5+4 q^4-7 q^3+9 q^2-10 q+10-8 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }
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^{-2} -z^6+a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-6 z^2 a^{-2} +3 z^2 a^{-4} -2 z^2+a^2-3 a^{-2} +2 a^{-4} +1}
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^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8+5 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +2 z^7 a^{-5} +5 a^2 z^6-12 z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -z^6+3 a^3 z^5-7 a z^5-9 z^5 a^{-1} -6 z^5 a^{-3} -7 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+14 z^4 a^{-2} +4 z^4 a^{-4} -4 z^4 a^{-6} -2 z^4-3 a^3 z^3+5 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+4 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} +4 z^2 a^{-6} +z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -a^2+3 a^{-2} +2 a^{-4} +1}
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 q^{12}-q^{10}+q^8+q^6-q^4+3 q^2-1+ q^{-2} - q^{-4} -2 q^{-6} + q^{-8} -2 q^{-10} + q^{-12} + q^{-14} + q^{-18} }
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^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+12 q^{52}-20 q^{50}+28 q^{48}-29 q^{46}+17 q^{44}+q^{42}-27 q^{40}+53 q^{38}-65 q^{36}+62 q^{34}-39 q^{32}+43 q^{28}-73 q^{26}+85 q^{24}-66 q^{22}+28 q^{20}+16 q^{18}-49 q^{16}+60 q^{14}-40 q^{12}+4 q^{10}+37 q^8-57 q^6+47 q^4-6 q^2-47+94 q^{-2} -108 q^{-4} +83 q^{-6} -27 q^{-8} -44 q^{-10} +103 q^{-12} -130 q^{-14} +114 q^{-16} -64 q^{-18} -4 q^{-20} +59 q^{-22} -92 q^{-24} +83 q^{-26} -49 q^{-28} - q^{-30} +38 q^{-32} -57 q^{-34} +41 q^{-36} - q^{-38} -41 q^{-40} +69 q^{-42} -70 q^{-44} +38 q^{-46} +11 q^{-48} -57 q^{-50} +88 q^{-52} -83 q^{-54} +59 q^{-56} -14 q^{-58} -29 q^{-60} +57 q^{-62} -62 q^{-64} +51 q^{-66} -27 q^{-68} +2 q^{-70} +16 q^{-72} -24 q^{-74} +24 q^{-76} -17 q^{-78} +9 q^{-80} - q^{-82} -4 q^{-84} +4 q^{-86} -5 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }

Further Quantum Invariants

Further quantum knot invariants for 10_26.

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^9-2 q^7+3 q^5-2 q^3+2 q- q^{-3} +2 q^{-5} -3 q^{-7} +2 q^{-9} - q^{-11} + q^{-13} }
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^{26}-2 q^{24}+6 q^{20}-7 q^{18}-3 q^{16}+15 q^{14}-10 q^{12}-9 q^{10}+19 q^8-7 q^6-12 q^4+13 q^2+2-8 q^{-2} - q^{-4} +10 q^{-6} - q^{-8} -13 q^{-10} +12 q^{-12} +8 q^{-14} -18 q^{-16} +6 q^{-18} +12 q^{-20} -14 q^{-22} +9 q^{-26} -6 q^{-28} -2 q^{-30} +4 q^{-32} - q^{-34} - q^{-36} + q^{-38} }
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^{51}-2 q^{49}+3 q^{45}+q^{43}-6 q^{41}-4 q^{39}+13 q^{37}+7 q^{35}-20 q^{33}-13 q^{31}+27 q^{29}+26 q^{27}-37 q^{25}-39 q^{23}+42 q^{21}+55 q^{19}-42 q^{17}-70 q^{15}+33 q^{13}+79 q^{11}-19 q^9-77 q^7+q^5+67 q^3+19 q-48 q^{-1} -35 q^{-3} +26 q^{-5} +51 q^{-7} -4 q^{-9} -55 q^{-11} -19 q^{-13} +62 q^{-15} +37 q^{-17} -60 q^{-19} -55 q^{-21} +49 q^{-23} +66 q^{-25} -36 q^{-27} -75 q^{-29} +19 q^{-31} +76 q^{-33} -67 q^{-37} -17 q^{-39} +57 q^{-41} +26 q^{-43} -38 q^{-45} -30 q^{-47} +22 q^{-49} +26 q^{-51} -9 q^{-53} -19 q^{-55} + q^{-57} +12 q^{-59} + q^{-61} -6 q^{-63} -2 q^{-65} +3 q^{-67} + q^{-69} - q^{-71} - q^{-73} + q^{-75} }
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^{84}-2 q^{82}+3 q^{78}-2 q^{76}+2 q^{74}-7 q^{72}+q^{70}+12 q^{68}-5 q^{66}+4 q^{64}-23 q^{62}+37 q^{58}-2 q^{56}-2 q^{54}-64 q^{52}-4 q^{50}+92 q^{48}+31 q^{46}-15 q^{44}-160 q^{42}-46 q^{40}+180 q^{38}+143 q^{36}+3 q^{34}-302 q^{32}-175 q^{30}+216 q^{28}+307 q^{26}+126 q^{24}-365 q^{22}-348 q^{20}+108 q^{18}+374 q^{16}+291 q^{14}-246 q^{12}-401 q^{10}-89 q^8+250 q^6+356 q^4-14 q^2-276-231 q^{-2} +30 q^{-4} +276 q^{-6} +187 q^{-8} -75 q^{-10} -280 q^{-12} -157 q^{-14} +147 q^{-16} +308 q^{-18} +93 q^{-20} -277 q^{-22} -278 q^{-24} +22 q^{-26} +373 q^{-28} +227 q^{-30} -234 q^{-32} -353 q^{-34} -115 q^{-36} +365 q^{-38} +340 q^{-40} -106 q^{-42} -355 q^{-44} -279 q^{-46} +237 q^{-48} +387 q^{-50} +86 q^{-52} -235 q^{-54} -374 q^{-56} +21 q^{-58} +289 q^{-60} +222 q^{-62} -27 q^{-64} -305 q^{-66} -133 q^{-68} +98 q^{-70} +196 q^{-72} +111 q^{-74} -133 q^{-76} -129 q^{-78} -35 q^{-80} +78 q^{-82} +107 q^{-84} -15 q^{-86} -48 q^{-88} -49 q^{-90} +2 q^{-92} +46 q^{-94} +9 q^{-96} -2 q^{-98} -20 q^{-100} -9 q^{-102} +12 q^{-104} +2 q^{-106} +4 q^{-108} -4 q^{-110} -4 q^{-112} +3 q^{-114} + q^{-118} - q^{-120} - q^{-122} + q^{-124} }
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^{125}-2 q^{123}+3 q^{119}-2 q^{117}-q^{115}+q^{113}-2 q^{111}+7 q^{107}+q^{105}-8 q^{103}-3 q^{101}-q^{99}+5 q^{97}+10 q^{95}+5 q^{93}-16 q^{91}-25 q^{89}+5 q^{87}+36 q^{85}+41 q^{83}-3 q^{81}-74 q^{79}-100 q^{77}-9 q^{75}+156 q^{73}+208 q^{71}+46 q^{69}-247 q^{67}-395 q^{65}-176 q^{63}+334 q^{61}+689 q^{59}+411 q^{57}-375 q^{55}-1006 q^{53}-802 q^{51}+268 q^{49}+1330 q^{47}+1305 q^{45}-1520 q^{41}-1829 q^{39}-461 q^{37}+1502 q^{35}+2271 q^{33}+1028 q^{31}-1234 q^{29}-2494 q^{27}-1578 q^{25}+741 q^{23}+2422 q^{21}+1992 q^{19}-128 q^{17}-2069 q^{15}-2173 q^{13}-467 q^{11}+1491 q^9+2089 q^7+961 q^5-828 q^3-1793 q-1275 q^{-1} +193 q^{-3} +1376 q^{-5} +1409 q^{-7} +361 q^{-9} -938 q^{-11} -1448 q^{-13} -760 q^{-15} +559 q^{-17} +1413 q^{-19} +1068 q^{-21} -247 q^{-23} -1416 q^{-25} -1305 q^{-27} +39 q^{-29} +1416 q^{-31} +1528 q^{-33} +167 q^{-35} -1453 q^{-37} -1766 q^{-39} -382 q^{-41} +1451 q^{-43} +2007 q^{-45} +674 q^{-47} -1363 q^{-49} -2207 q^{-51} -1050 q^{-53} +1121 q^{-55} +2324 q^{-57} +1451 q^{-59} -712 q^{-61} -2258 q^{-63} -1829 q^{-65} +156 q^{-67} +1975 q^{-69} +2084 q^{-71} +451 q^{-73} -1485 q^{-75} -2108 q^{-77} -998 q^{-79} +833 q^{-81} +1889 q^{-83} +1383 q^{-85} -173 q^{-87} -1447 q^{-89} -1484 q^{-91} -401 q^{-93} +875 q^{-95} +1360 q^{-97} +753 q^{-99} -335 q^{-101} -1019 q^{-103} -865 q^{-105} -101 q^{-107} +620 q^{-109} +770 q^{-111} +335 q^{-113} -258 q^{-115} -550 q^{-117} -394 q^{-119} +5 q^{-121} +316 q^{-123} +332 q^{-125} +112 q^{-127} -129 q^{-129} -217 q^{-131} -134 q^{-133} +18 q^{-135} +114 q^{-137} +103 q^{-139} +27 q^{-141} -44 q^{-143} -63 q^{-145} -29 q^{-147} +9 q^{-149} +27 q^{-151} +24 q^{-153} +3 q^{-155} -14 q^{-157} -10 q^{-159} -2 q^{-161} +6 q^{-165} +4 q^{-167} -3 q^{-169} -2 q^{-171} + q^{-173} + q^{-179} - q^{-181} - q^{-183} + q^{-185} }

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 q^{12}-q^{10}+q^8+q^6-q^4+3 q^2-1+ q^{-2} - q^{-4} -2 q^{-6} + q^{-8} -2 q^{-10} + q^{-12} + q^{-14} + q^{-18} }
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^{36}-4 q^{34}+10 q^{32}-20 q^{30}+38 q^{28}-64 q^{26}+100 q^{24}-144 q^{22}+197 q^{20}-246 q^{18}+290 q^{16}-322 q^{14}+326 q^{12}-292 q^{10}+226 q^8-124 q^6-12 q^4+172 q^2-332+474 q^{-2} -590 q^{-4} +656 q^{-6} -672 q^{-8} +630 q^{-10} -537 q^{-12} +410 q^{-14} -250 q^{-16} +98 q^{-18} +56 q^{-20} -180 q^{-22} +272 q^{-24} -322 q^{-26} +329 q^{-28} -318 q^{-30} +274 q^{-32} -220 q^{-34} +166 q^{-36} -118 q^{-38} +78 q^{-40} -46 q^{-42} +27 q^{-44} -14 q^{-46} +6 q^{-48} -2 q^{-50} + q^{-52} }
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^{32}-q^{30}-q^{28}+3 q^{26}+q^{24}-3 q^{22}+6 q^{18}-8 q^{14}+3 q^{12}+8 q^{10}-5 q^8-7 q^6+6 q^4+2 q^2-6- q^{-2} +5 q^{-4} -3 q^{-6} -4 q^{-8} +7 q^{-10} +2 q^{-12} -5 q^{-14} +5 q^{-16} +9 q^{-18} -3 q^{-20} -4 q^{-22} +4 q^{-24} +3 q^{-26} -7 q^{-28} -5 q^{-30} +3 q^{-32} + q^{-34} -3 q^{-36} - q^{-38} +2 q^{-40} + q^{-42} + q^{-48} }

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 q^{28}-2 q^{26}+5 q^{22}-6 q^{20}-q^{18}+12 q^{16}-10 q^{14}-4 q^{12}+16 q^{10}-8 q^8-6 q^6+14 q^4-2 q^2-5+4 q^{-2} +3 q^{-4} -4 q^{-6} -9 q^{-8} +6 q^{-10} +2 q^{-12} -15 q^{-14} +8 q^{-16} +9 q^{-18} -12 q^{-20} +8 q^{-22} +8 q^{-24} -8 q^{-26} +3 q^{-28} +2 q^{-30} -5 q^{-32} + q^{-34} + q^{-36} - q^{-38} + q^{-40} }

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^{15}-q^{13}+2 q^{11}-q^9+2 q^7-q^5+3 q^3+ q^{-1} -2 q^{-5} - q^{-7} -3 q^{-9} + q^{-11} -2 q^{-13} +2 q^{-15} +2 q^{-19} + q^{-23} }

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^{34}-q^{32}-q^{30}+3 q^{28}-4 q^{24}+2 q^{22}+7 q^{20}-2 q^{18}-6 q^{16}+6 q^{14}+8 q^{12}-7 q^{10}-5 q^8+12 q^6+q^4-8 q^2+5+7 q^{-2} -8 q^{-4} -5 q^{-6} +6 q^{-8} -4 q^{-10} -12 q^{-12} +3 q^{-14} +8 q^{-16} -9 q^{-18} -3 q^{-20} +13 q^{-22} +4 q^{-24} -6 q^{-26} +3 q^{-28} +7 q^{-30} -2 q^{-32} -5 q^{-34} + q^{-36} + q^{-38} -3 q^{-40} +2 q^{-44} + q^{-50} }

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^{18}-q^{16}+2 q^{14}+2 q^8-q^6+3 q^4+2-2 q^{-6} -2 q^{-8} -2 q^{-10} -3 q^{-12} + q^{-14} -2 q^{-16} +2 q^{-18} + q^{-20} + q^{-22} +2 q^{-24} + q^{-28} }

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 q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+10 q^{20}-13 q^{18}+16 q^{16}-16 q^{14}+16 q^{12}-12 q^{10}+8 q^8-6 q^4+16 q^2-23+28 q^{-2} -31 q^{-4} +30 q^{-6} -29 q^{-8} +22 q^{-10} -16 q^{-12} +7 q^{-14} -7 q^{-18} +12 q^{-20} -14 q^{-22} +16 q^{-24} -14 q^{-26} +13 q^{-28} -10 q^{-30} +7 q^{-32} -5 q^{-34} +3 q^{-36} - q^{-38} + q^{-40} }

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^{46}-2 q^{42}-2 q^{40}+2 q^{38}+6 q^{36}+q^{34}-8 q^{32}-7 q^{30}+6 q^{28}+14 q^{26}+q^{24}-15 q^{22}-10 q^{20}+11 q^{18}+16 q^{16}-2 q^{14}-16 q^{12}-4 q^{10}+13 q^8+10 q^6-9 q^4-10 q^2+6+11 q^{-2} -2 q^{-4} -11 q^{-6} - q^{-8} +9 q^{-10} +2 q^{-12} -11 q^{-14} -5 q^{-16} +10 q^{-18} +8 q^{-20} -10 q^{-22} -14 q^{-24} +5 q^{-26} +18 q^{-28} +4 q^{-30} -15 q^{-32} -11 q^{-34} +10 q^{-36} +16 q^{-38} -12 q^{-42} -6 q^{-44} +7 q^{-46} +7 q^{-48} -2 q^{-50} -6 q^{-52} -2 q^{-54} +3 q^{-56} +2 q^{-58} - q^{-60} - q^{-62} + q^{-66} }

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^{38}-2 q^{36}+2 q^{34}-3 q^{32}+6 q^{30}-8 q^{28}+8 q^{26}-9 q^{24}+14 q^{22}-13 q^{20}+11 q^{18}-11 q^{16}+13 q^{14}-6 q^{12}+3 q^{10}-q^8-q^6+12 q^4-12 q^2+17-20 q^{-2} +24 q^{-4} -23 q^{-6} +21 q^{-8} -27 q^{-10} +18 q^{-12} -19 q^{-14} +11 q^{-16} -13 q^{-18} +3 q^{-20} + q^{-22} -2 q^{-24} +8 q^{-26} -7 q^{-28} +15 q^{-30} -9 q^{-32} +13 q^{-34} -11 q^{-36} +11 q^{-38} -9 q^{-40} +6 q^{-42} -7 q^{-44} +4 q^{-46} -3 q^{-48} +2 q^{-50} - q^{-52} + q^{-54} }

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^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+12 q^{52}-20 q^{50}+28 q^{48}-29 q^{46}+17 q^{44}+q^{42}-27 q^{40}+53 q^{38}-65 q^{36}+62 q^{34}-39 q^{32}+43 q^{28}-73 q^{26}+85 q^{24}-66 q^{22}+28 q^{20}+16 q^{18}-49 q^{16}+60 q^{14}-40 q^{12}+4 q^{10}+37 q^8-57 q^6+47 q^4-6 q^2-47+94 q^{-2} -108 q^{-4} +83 q^{-6} -27 q^{-8} -44 q^{-10} +103 q^{-12} -130 q^{-14} +114 q^{-16} -64 q^{-18} -4 q^{-20} +59 q^{-22} -92 q^{-24} +83 q^{-26} -49 q^{-28} - q^{-30} +38 q^{-32} -57 q^{-34} +41 q^{-36} - q^{-38} -41 q^{-40} +69 q^{-42} -70 q^{-44} +38 q^{-46} +11 q^{-48} -57 q^{-50} +88 q^{-52} -83 q^{-54} +59 q^{-56} -14 q^{-58} -29 q^{-60} +57 q^{-62} -62 q^{-64} +51 q^{-66} -27 q^{-68} +2 q^{-70} +16 q^{-72} -24 q^{-74} +24 q^{-76} -17 q^{-78} +9 q^{-80} - q^{-82} -4 q^{-84} +4 q^{-86} -5 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 26"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[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 -2 t^3+7 t^2-13 t+17-13 t^{-1} +7 t^{-2} -2 t^{-3} }

In[5]:= Conway[K][z]

Out[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 -2 z^6-5 z^4-3 z^2+1}

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= 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]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 61, 0 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= 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^5+4 q^4-7 q^3+9 q^2-10 q+10-8 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= 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^{-2} -z^6+a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-6 z^2 a^{-2} +3 z^2 a^{-4} -2 z^2+a^2-3 a^{-2} +2 a^{-4} +1}

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= 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^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8+5 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +2 z^7 a^{-5} +5 a^2 z^6-12 z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -z^6+3 a^3 z^5-7 a z^5-9 z^5 a^{-1} -6 z^5 a^{-3} -7 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+14 z^4 a^{-2} +4 z^4 a^{-4} -4 z^4 a^{-6} -2 z^4-3 a^3 z^3+5 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+4 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} +4 z^2 a^{-6} +z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -a^2+3 a^{-2} +2 a^{-4} +1}

"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]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["10 26"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { 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^2-13 t+17-13 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^6-2 q^5+4 q^4-7 q^3+9 q^2-10 q+10-8 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} } }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(-3, -2)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 -12}

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 -16}

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 72}

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 130}

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 78}

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 192}

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 \frac{1184}{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 \frac{224}{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 144}

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 -288}

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 128}

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 -1560}

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 -936}

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 -\frac{10351}{10}}

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 \frac{10826}{15}}

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 -\frac{28462}{15}}

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 \frac{2159}{6}}

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 -\frac{5231}{10}}

V_2,1 through V_6,9 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=} 0 is the signature of 10 26. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

1

-1

9

3

1

2

7

4

1

-3

5

3

2

3

5

4

-1

1

5

0

-1

4

6

2

-3

2

4

-2

-5

1

4

3

-7

2

-2

-9

1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=-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 i=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 r=-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 {\mathbb Z}}
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=-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 {\mathbb Z}^{2}\oplus{\mathbb Z}_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 {\mathbb Z}}
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=-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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_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 {\mathbb Z}}
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=6}	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 {\mathbb Z}_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 {\mathbb Z}}

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^{18}-2 q^{17}+6 q^{15}-8 q^{14}-4 q^{13}+21 q^{12}-17 q^{11}-18 q^{10}+47 q^9-23 q^8-42 q^7+73 q^6-19 q^5-67 q^4+85 q^3-8 q^2-78 q+78+2 q^{-1} -67 q^{-2} +53 q^{-3} +7 q^{-4} -41 q^{-5} +25 q^{-6} +6 q^{-7} -16 q^{-8} +7 q^{-9} +2 q^{-10} -3 q^{-11} + q^{-12} }
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^{36}-2 q^{35}+2 q^{33}+3 q^{32}-7 q^{31}-4 q^{30}+9 q^{29}+14 q^{28}-18 q^{27}-24 q^{26}+19 q^{25}+49 q^{24}-22 q^{23}-76 q^{22}+11 q^{21}+113 q^{20}+9 q^{19}-150 q^{18}-39 q^{17}+180 q^{16}+85 q^{15}-207 q^{14}-133 q^{13}+219 q^{12}+187 q^{11}-224 q^{10}-237 q^9+214 q^8+284 q^7-199 q^6-318 q^5+178 q^4+335 q^3-144 q^2-343 q+117+322 q^{-1} -77 q^{-2} -295 q^{-3} +51 q^{-4} +244 q^{-5} -19 q^{-6} -197 q^{-7} +5 q^{-8} +141 q^{-9} +9 q^{-10} -100 q^{-11} -8 q^{-12} +60 q^{-13} +11 q^{-14} -37 q^{-15} -7 q^{-16} +20 q^{-17} +4 q^{-18} -10 q^{-19} - q^{-20} +3 q^{-21} +2 q^{-22} -3 q^{-23} + q^{-24} }
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^{60}-2 q^{59}+2 q^{57}-q^{56}+4 q^{55}-9 q^{54}+10 q^{52}-3 q^{51}+14 q^{50}-30 q^{49}-11 q^{48}+28 q^{47}+8 q^{46}+51 q^{45}-74 q^{44}-62 q^{43}+29 q^{42}+41 q^{41}+173 q^{40}-103 q^{39}-175 q^{38}-65 q^{37}+37 q^{36}+417 q^{35}-18 q^{34}-273 q^{33}-296 q^{32}-135 q^{31}+695 q^{30}+231 q^{29}-206 q^{28}-564 q^{27}-530 q^{26}+834 q^{25}+552 q^{24}+95 q^{23}-714 q^{22}-1046 q^{21}+758 q^{20}+801 q^{19}+541 q^{18}-689 q^{17}-1526 q^{16}+520 q^{15}+920 q^{14}+1002 q^{13}-543 q^{12}-1877 q^{11}+220 q^{10}+921 q^9+1372 q^8-328 q^7-2038 q^6-84 q^5+798 q^4+1577 q^3-66 q^2-1949 q-330+537 q^{-1} +1532 q^{-2} +196 q^{-3} -1579 q^{-4} -436 q^{-5} +198 q^{-6} +1220 q^{-7} +351 q^{-8} -1042 q^{-9} -353 q^{-10} -68 q^{-11} +764 q^{-12} +334 q^{-13} -551 q^{-14} -172 q^{-15} -159 q^{-16} +373 q^{-17} +207 q^{-18} -246 q^{-19} -32 q^{-20} -122 q^{-21} +147 q^{-22} +93 q^{-23} -101 q^{-24} +14 q^{-25} -61 q^{-26} +51 q^{-27} +33 q^{-28} -39 q^{-29} +14 q^{-30} -22 q^{-31} +14 q^{-32} +10 q^{-33} -12 q^{-34} +5 q^{-35} -5 q^{-36} +3 q^{-37} +2 q^{-38} -3 q^{-39} + q^{-40} }
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^{90}-2 q^{89}+2 q^{87}-q^{86}+2 q^{84}-5 q^{83}-q^{82}+9 q^{81}+q^{80}-6 q^{79}-13 q^{77}-5 q^{76}+26 q^{75}+22 q^{74}-3 q^{73}-18 q^{72}-51 q^{71}-39 q^{70}+45 q^{69}+93 q^{68}+73 q^{67}-7 q^{66}-147 q^{65}-191 q^{64}-38 q^{63}+181 q^{62}+314 q^{61}+213 q^{60}-163 q^{59}-502 q^{58}-437 q^{57}+25 q^{56}+606 q^{55}+806 q^{54}+272 q^{53}-652 q^{52}-1158 q^{51}-739 q^{50}+452 q^{49}+1490 q^{48}+1360 q^{47}-45 q^{46}-1643 q^{45}-2015 q^{44}-631 q^{43}+1527 q^{42}+2634 q^{41}+1511 q^{40}-1137 q^{39}-3071 q^{38}-2462 q^{37}+417 q^{36}+3257 q^{35}+3447 q^{34}+496 q^{33}-3180 q^{32}-4281 q^{31}-1568 q^{30}+2828 q^{29}+4993 q^{28}+2659 q^{27}-2307 q^{26}-5484 q^{25}-3739 q^{24}+1671 q^{23}+5837 q^{22}+4696 q^{21}-974 q^{20}-6040 q^{19}-5572 q^{18}+287 q^{17}+6150 q^{16}+6316 q^{15}+387 q^{14}-6152 q^{13}-6949 q^{12}-1057 q^{11}+6039 q^{10}+7485 q^9+1702 q^8-5807 q^7-7803 q^6-2376 q^5+5351 q^4+7995 q^3+3001 q^2-4759 q-7836-3559 q^{-1} +3883 q^{-2} +7477 q^{-3} +3966 q^{-4} -2970 q^{-5} -6708 q^{-6} -4157 q^{-7} +1925 q^{-8} +5771 q^{-9} +4070 q^{-10} -1029 q^{-11} -4588 q^{-12} -3727 q^{-13} +244 q^{-14} +3452 q^{-15} +3154 q^{-16} +231 q^{-17} -2326 q^{-18} -2484 q^{-19} -525 q^{-20} +1489 q^{-21} +1786 q^{-22} +540 q^{-23} -806 q^{-24} -1179 q^{-25} -500 q^{-26} +427 q^{-27} +716 q^{-28} +336 q^{-29} -175 q^{-30} -393 q^{-31} -222 q^{-32} +72 q^{-33} +206 q^{-34} +117 q^{-35} -27 q^{-36} -100 q^{-37} -60 q^{-38} +20 q^{-39} +41 q^{-40} +26 q^{-41} - q^{-42} -29 q^{-43} -16 q^{-44} +15 q^{-45} +10 q^{-46} -4 q^{-47} +8 q^{-48} -8 q^{-49} -11 q^{-50} +10 q^{-51} +4 q^{-52} -6 q^{-53} +3 q^{-54} + q^{-55} -5 q^{-56} +3 q^{-57} +2 q^{-58} -3 q^{-59} + q^{-60} }
6	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^{126}-2 q^{125}+2 q^{123}-q^{122}-2 q^{120}+6 q^{119}-6 q^{118}-2 q^{117}+11 q^{116}-3 q^{115}-4 q^{114}-14 q^{113}+17 q^{112}-11 q^{111}-5 q^{110}+39 q^{109}+5 q^{108}-10 q^{107}-56 q^{106}+19 q^{105}-38 q^{104}-19 q^{103}+111 q^{102}+74 q^{101}+32 q^{100}-125 q^{99}-21 q^{98}-180 q^{97}-146 q^{96}+182 q^{95}+276 q^{94}+297 q^{93}-40 q^{92}+8 q^{91}-535 q^{90}-663 q^{89}-105 q^{88}+400 q^{87}+886 q^{86}+617 q^{85}+707 q^{84}-701 q^{83}-1620 q^{82}-1354 q^{81}-445 q^{80}+1073 q^{79}+1762 q^{78}+2839 q^{77}+603 q^{76}-1846 q^{75}-3253 q^{74}-3091 q^{73}-875 q^{72}+1671 q^{71}+5699 q^{70}+4221 q^{69}+819 q^{68}-3447 q^{67}-6282 q^{66}-5716 q^{65}-2183 q^{64}+6247 q^{63}+8317 q^{62}+6902 q^{61}+849 q^{60}-6364 q^{59}-11004 q^{58}-9989 q^{57}+1541 q^{56}+8808 q^{55}+13402 q^{54}+9446 q^{53}-478 q^{52}-12397 q^{51}-18276 q^{50}-7858 q^{49}+3090 q^{48}+15938 q^{47}+18619 q^{46}+10345 q^{45}-7659 q^{44}-22779 q^{43}-18115 q^{42}-7413 q^{41}+12715 q^{40}+24440 q^{39}+22182 q^{38}+1482 q^{37}-22100 q^{36}-25658 q^{35}-18883 q^{34}+5635 q^{33}+25911 q^{32}+31729 q^{31}+11443 q^{30}-18097 q^{29}-29698 q^{28}-28415 q^{27}-2122 q^{26}+24733 q^{25}+38251 q^{24}+19818 q^{23}-13356 q^{22}-31571 q^{21}-35451 q^{20}-8775 q^{19}+22785 q^{18}+42667 q^{17}+26403 q^{16}-8864 q^{15}-32380 q^{14}-40746 q^{13}-14621 q^{12}+20206 q^{11}+45409 q^{10}+32020 q^9-3705 q^8-31467 q^7-44351 q^6-20643 q^5+15511 q^4+45193 q^3+36470 q^2+3228 q-26904-44552 q^{-1} -26330 q^{-2} +7636 q^{-3} +39858 q^{-4} +37603 q^{-5} +10926 q^{-6} -17871 q^{-7} -39064 q^{-8} -28980 q^{-9} -1762 q^{-10} +29034 q^{-11} +32992 q^{-12} +16016 q^{-13} -6794 q^{-14} -27995 q^{-15} -25997 q^{-16} -8665 q^{-17} +16005 q^{-18} +23124 q^{-19} +15693 q^{-20} +1745 q^{-21} -15209 q^{-22} -18130 q^{-23} -10219 q^{-24} +5736 q^{-25} +12094 q^{-26} +10841 q^{-27} +4982 q^{-28} -5564 q^{-29} -9403 q^{-30} -7460 q^{-31} +724 q^{-32} +4225 q^{-33} +5165 q^{-34} +4105 q^{-35} -931 q^{-36} -3431 q^{-37} -3749 q^{-38} -347 q^{-39} +669 q^{-40} +1516 q^{-41} +2098 q^{-42} +232 q^{-43} -809 q^{-44} -1349 q^{-45} -61 q^{-46} -183 q^{-47} +130 q^{-48} +762 q^{-49} +175 q^{-50} -108 q^{-51} -391 q^{-52} +160 q^{-53} -152 q^{-54} -117 q^{-55} +237 q^{-56} +35 q^{-57} -15 q^{-58} -126 q^{-59} +138 q^{-60} -48 q^{-61} -77 q^{-62} +79 q^{-63} +2 q^{-64} -9 q^{-65} -53 q^{-66} +64 q^{-67} -11 q^{-68} -30 q^{-69} +28 q^{-70} - q^{-71} - q^{-72} -22 q^{-73} +21 q^{-74} -12 q^{-76} +9 q^{-77} - q^{-78} + q^{-79} -5 q^{-80} +3 q^{-81} +2 q^{-82} -3 q^{-83} + q^{-84} }

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).

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10 26

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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