9 29

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9 28.gif

9_28

9 30.gif

9_30

9 29.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 29's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 29 at Knotilus!

[edit 9 29 Quick Notes]
[edit 9 29 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X16,11,17,12 X10,4,11,3 X2,15,3,16 X14,5,15,6 X18,8,1,7 X4,10,5,9 X12,17,13,18 X8,13,9,14
Gauss code 1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code 6 10 14 18 4 16 8 2 12
Conway Notation [.2.20.2]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif

Length is 9, width is 4,

Braid index is 4

9 29 ML.gif 9 29 AP.gif
[{2, 4}, {1, 3}, {12, 5}, {4, 9}, {10, 6}, {5, 7}, {9, 11}, {6, 8}, {7, 2}, {3, 10}, {8, 12}, {11, 1}]

[edit Notes on presentations of 9 29]


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["9 29"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X16,11,17,12 X10,4,11,3 X2,15,3,16 X14,5,15,6 X18,8,1,7 X4,10,5,9 X12,17,13,18 X8,13,9,14
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6
In[6]:= DTCode[K]
Out[6]= 6 10 14 18 4 16 8 2 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [.2.20.2]
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]=
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, 9, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
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."
9 29 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{2, 4}, {1, 3}, {12, 5}, {4, 9}, {10, 6}, {5, 7}, {9, 11}, {6, 8}, {7, 2}, {3, 10}, {8, 12}, {11, 1}]
In[14]:= Draw[ap]
9 29 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index 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 \{4,7\}}
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][-3]
Hyperbolic Volume 12.2059
A-Polynomial See Data:9 29/A-polynomial

[edit Notes for 9 29'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 2

[edit Notes for 9 29'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 t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-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 z^6+z^4+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 { 51, -2 }
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^3-3 q^2+5 q-7+9 q^{-1} -8 q^{-2} +8 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} }
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 a^2 z^6-a^4 z^4+4 a^2 z^4-2 z^4-2 a^4 z^2+7 a^2 z^2+z^2 a^{-2} -5 z^2-2 a^4+5 a^2+ a^{-2} -3}
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 a^2 z^8+2 z^8+6 a^3 z^7+9 a z^7+3 z^7 a^{-1} +8 a^4 z^6+6 a^2 z^6+z^6 a^{-2} -z^6+6 a^5 z^5-8 a^3 z^5-24 a z^5-10 z^5 a^{-1} +3 a^6 z^4-13 a^4 z^4-24 a^2 z^4-3 z^4 a^{-2} -11 z^4+a^7 z^3-5 a^5 z^3-a^3 z^3+14 a z^3+9 z^3 a^{-1} +8 a^4 z^2+17 a^2 z^2+3 z^2 a^{-2} +12 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} -2 a^4-5 a^2- a^{-2} -3}
The A2 invariant
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^{100}-2 q^{98}+3 q^{96}-4 q^{94}+3 q^{92}-2 q^{90}-q^{88}+8 q^{86}-12 q^{84}+16 q^{82}-19 q^{80}+13 q^{78}-6 q^{76}-9 q^{74}+28 q^{72}-39 q^{70}+44 q^{68}-37 q^{66}+16 q^{64}+13 q^{62}-46 q^{60}+63 q^{58}-61 q^{56}+31 q^{54}+5 q^{52}-41 q^{50}+57 q^{48}-42 q^{46}+9 q^{44}+30 q^{42}-59 q^{40}+56 q^{38}-21 q^{36}-30 q^{34}+78 q^{32}-91 q^{30}+77 q^{28}-24 q^{26}-29 q^{24}+77 q^{22}-96 q^{20}+88 q^{18}-50 q^{16}+47 q^{12}-69 q^{10}+69 q^8-36 q^6-5 q^4+36 q^2-55+41 q^{-2} -7 q^{-4} -39 q^{-6} +68 q^{-8} -70 q^{-10} +39 q^{-12} +10 q^{-14} -56 q^{-16} +77 q^{-18} -70 q^{-20} +40 q^{-22} -3 q^{-24} -32 q^{-26} +47 q^{-28} -40 q^{-30} +27 q^{-32} -6 q^{-34} -7 q^{-36} +11 q^{-38} -10 q^{-40} +6 q^{-42} -2 q^{-44} + q^{-46} }
Further Quantum Invariants
Further quantum knot invariants for 9_29.

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^{13}+2 q^{11}-3 q^9+2 q^7+q^3+2 q-2 q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} }
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^{36}-2 q^{34}+q^{32}+4 q^{30}-9 q^{28}+3 q^{26}+11 q^{24}-15 q^{22}-q^{20}+14 q^{18}-9 q^{16}-6 q^{14}+10 q^{12}+4 q^{10}-8 q^8+11 q^4-5 q^2-10+13 q^{-2} + q^{-4} -15 q^{-6} +9 q^{-8} +8 q^{-10} -11 q^{-12} + q^{-14} +7 q^{-16} -3 q^{-18} -2 q^{-20} + q^{-22} }
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^{69}+2 q^{67}-q^{65}-2 q^{63}+3 q^{61}+3 q^{59}-6 q^{57}-8 q^{55}+17 q^{53}+16 q^{51}-30 q^{49}-29 q^{47}+36 q^{45}+50 q^{43}-36 q^{41}-71 q^{39}+25 q^{37}+78 q^{35}-76 q^{31}-23 q^{29}+55 q^{27}+45 q^{25}-32 q^{23}-53 q^{21}+6 q^{19}+58 q^{17}+17 q^{15}-56 q^{13}-32 q^{11}+52 q^9+46 q^7-46 q^5-56 q^3+32 q+70 q^{-1} -19 q^{-3} -74 q^{-5} -4 q^{-7} +74 q^{-9} +28 q^{-11} -61 q^{-13} -45 q^{-15} +40 q^{-17} +54 q^{-19} -14 q^{-21} -50 q^{-23} -7 q^{-25} +34 q^{-27} +17 q^{-29} -16 q^{-31} -18 q^{-33} +4 q^{-35} +10 q^{-37} +2 q^{-39} -3 q^{-41} -2 q^{-43} + q^{-45} }
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^{112}-2 q^{110}+q^{108}+2 q^{106}-5 q^{104}+3 q^{102}+6 q^{98}-3 q^{96}-26 q^{94}+13 q^{92}+27 q^{90}+31 q^{88}-28 q^{86}-108 q^{84}+116 q^{80}+152 q^{78}-38 q^{76}-292 q^{74}-141 q^{72}+179 q^{70}+390 q^{68}+125 q^{66}-400 q^{64}-409 q^{62}+9 q^{60}+492 q^{58}+406 q^{56}-200 q^{54}-488 q^{52}-290 q^{50}+251 q^{48}+476 q^{46}+141 q^{44}-244 q^{42}-395 q^{40}-96 q^{38}+269 q^{36}+317 q^{34}+61 q^{32}-296 q^{30}-286 q^{28}+46 q^{26}+330 q^{24}+214 q^{22}-185 q^{20}-349 q^{18}-75 q^{16}+332 q^{14}+294 q^{12}-117 q^{10}-407 q^8-186 q^6+320 q^4+392 q^2+25-417 q^{-2} -365 q^{-4} +167 q^{-6} +440 q^{-8} +273 q^{-10} -247 q^{-12} -476 q^{-14} -129 q^{-16} +271 q^{-18} +433 q^{-20} +77 q^{-22} -324 q^{-24} -318 q^{-26} -57 q^{-28} +298 q^{-30} +267 q^{-32} -10 q^{-34} -206 q^{-36} -224 q^{-38} +24 q^{-40} +162 q^{-42} +130 q^{-44} +8 q^{-46} -122 q^{-48} -76 q^{-50} +57 q^{-54} +56 q^{-56} -8 q^{-58} -24 q^{-60} -23 q^{-62} -3 q^{-64} +13 q^{-66} +5 q^{-68} +2 q^{-70} -3 q^{-72} -2 q^{-74} + q^{-76} }
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^{165}+2 q^{163}-q^{161}-2 q^{159}+5 q^{157}-q^{155}-6 q^{153}+5 q^{149}+9 q^{147}+6 q^{145}-16 q^{143}-38 q^{141}-10 q^{139}+59 q^{137}+89 q^{135}+14 q^{133}-130 q^{131}-200 q^{129}-67 q^{127}+254 q^{125}+441 q^{123}+189 q^{121}-403 q^{119}-816 q^{117}-496 q^{115}+483 q^{113}+1329 q^{111}+1073 q^{109}-382 q^{107}-1849 q^{105}-1866 q^{103}-78 q^{101}+2147 q^{99}+2768 q^{97}+911 q^{95}-2033 q^{93}-3470 q^{91}-1957 q^{89}+1380 q^{87}+3686 q^{85}+2946 q^{83}-303 q^{81}-3318 q^{79}-3528 q^{77}-872 q^{75}+2349 q^{73}+3541 q^{71}+1880 q^{69}-1126 q^{67}-3004 q^{65}-2413 q^{63}-87 q^{61}+2092 q^{59}+2513 q^{57}+1006 q^{55}-1127 q^{53}-2239 q^{51}-1560 q^{49}+319 q^{47}+1843 q^{45}+1785 q^{43}+217 q^{41}-1502 q^{39}-1837 q^{37}-484 q^{35}+1323 q^{33}+1861 q^{31}+586 q^{29}-1311 q^{27}-1969 q^{25}-675 q^{23}+1392 q^{21}+2211 q^{19}+855 q^{17}-1456 q^{15}-2516 q^{13}-1226 q^{11}+1355 q^9+2841 q^7+1764 q^5-1010 q^3-2984 q-2389 q^{-1} +331 q^{-3} +2866 q^{-5} +2967 q^{-7} +552 q^{-9} -2323 q^{-11} -3273 q^{-13} -1548 q^{-15} +1403 q^{-17} +3166 q^{-19} +2365 q^{-21} -246 q^{-23} -2544 q^{-25} -2779 q^{-27} -907 q^{-29} +1522 q^{-31} +2650 q^{-33} +1737 q^{-35} -353 q^{-37} -1991 q^{-39} -2039 q^{-41} -653 q^{-43} +1031 q^{-45} +1796 q^{-47} +1222 q^{-49} -105 q^{-51} -1161 q^{-53} -1270 q^{-55} -531 q^{-57} +437 q^{-59} +947 q^{-61} +741 q^{-63} +97 q^{-65} -465 q^{-67} -608 q^{-69} -348 q^{-71} +74 q^{-73} +346 q^{-75} +327 q^{-77} +114 q^{-79} -100 q^{-81} -192 q^{-83} -147 q^{-85} -20 q^{-87} +72 q^{-89} +85 q^{-91} +44 q^{-93} -2 q^{-95} -30 q^{-97} -31 q^{-99} -8 q^{-101} +6 q^{-103} +8 q^{-105} +5 q^{-107} +2 q^{-109} -3 q^{-111} -2 q^{-113} + q^{-115} }

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^{18}+q^{16}-2 q^{14}-q^{12}+2 q^{10}+4 q^6+q^2-2 q^{-2} + q^{-4} - q^{-6} + q^{-10} }
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^{52}-4 q^{50}+8 q^{48}-12 q^{46}+22 q^{44}-36 q^{42}+48 q^{40}-64 q^{38}+93 q^{36}-118 q^{34}+142 q^{32}-176 q^{30}+194 q^{28}-204 q^{26}+174 q^{24}-128 q^{22}+52 q^{20}+62 q^{18}-170 q^{16}+292 q^{14}-375 q^{12}+438 q^{10}-454 q^8+428 q^6-372 q^4+276 q^2-162+38 q^{-2} +70 q^{-4} -164 q^{-6} +234 q^{-8} -258 q^{-10} +250 q^{-12} -212 q^{-14} +168 q^{-16} -114 q^{-18} +66 q^{-20} -36 q^{-22} +14 q^{-24} -4 q^{-26} + q^{-28} }
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^{46}-q^{44}+q^{42}+2 q^{40}-2 q^{38}-2 q^{36}+3 q^{34}+3 q^{32}-8 q^{30}-8 q^{28}+3 q^{26}+3 q^{24}-7 q^{22}+2 q^{20}+11 q^{18}+4 q^{16}+q^{14}+3 q^{12}+2 q^{10}-4 q^8-q^6+2 q^4-6 q^2-4+5 q^{-2} +2 q^{-4} -6 q^{-6} + q^{-8} +7 q^{-10} +2 q^{-12} -4 q^{-14} - q^{-16} +5 q^{-18} + q^{-20} -3 q^{-22} -2 q^{-24} + q^{-28} }

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^{42}-2 q^{40}-q^{38}+6 q^{36}-3 q^{34}-6 q^{32}+10 q^{30}-3 q^{28}-10 q^{26}+10 q^{24}-3 q^{22}-10 q^{20}+6 q^{18}+q^{16}-2 q^{14}+3 q^{12}+7 q^{10}+7 q^8-6 q^6+3 q^4+7 q^2-11-2 q^{-2} +8 q^{-4} -9 q^{-6} + q^{-8} +7 q^{-10} -6 q^{-12} +2 q^{-14} +2 q^{-16} -2 q^{-18} + q^{-20} }
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^{23}+q^{21}-3 q^{19}-2 q^{15}+2 q^{13}+q^{11}+4 q^9+3 q^7+q^5+q^3-2 q-3 q^{-3} + q^{-5} - q^{-7} + q^{-9} + q^{-13} }

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

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^{42}+2 q^{40}-3 q^{38}+6 q^{36}-9 q^{34}+10 q^{32}-14 q^{30}+13 q^{28}-12 q^{26}+8 q^{24}-3 q^{22}-2 q^{20}+10 q^{18}-15 q^{16}+22 q^{14}-23 q^{12}+27 q^{10}-23 q^8+20 q^6-13 q^4+7 q^2-1-6 q^{-2} +10 q^{-4} -13 q^{-6} +13 q^{-8} -13 q^{-10} +10 q^{-12} -8 q^{-14} +6 q^{-16} -2 q^{-18} + q^{-20} }
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^{68}-2 q^{64}-2 q^{62}+q^{60}+6 q^{58}+3 q^{56}-6 q^{54}-8 q^{52}+12 q^{48}+5 q^{46}-10 q^{44}-11 q^{42}+4 q^{40}+13 q^{38}-13 q^{34}-6 q^{32}+10 q^{30}+7 q^{28}-7 q^{26}-9 q^{24}+5 q^{22}+11 q^{20}+q^{18}-8 q^{16}+2 q^{14}+10 q^{12}+3 q^{10}-9 q^8-4 q^6+9 q^4+9 q^2-7-15 q^{-2} +14 q^{-6} +6 q^{-8} -11 q^{-10} -11 q^{-12} +6 q^{-14} +12 q^{-16} -8 q^{-20} -4 q^{-22} +5 q^{-24} +4 q^{-26} -2 q^{-28} -2 q^{-30} + q^{-34} }

D4 Invariants.

Weight Invariant
1,0,0,0

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^{100}-2 q^{98}+3 q^{96}-4 q^{94}+3 q^{92}-2 q^{90}-q^{88}+8 q^{86}-12 q^{84}+16 q^{82}-19 q^{80}+13 q^{78}-6 q^{76}-9 q^{74}+28 q^{72}-39 q^{70}+44 q^{68}-37 q^{66}+16 q^{64}+13 q^{62}-46 q^{60}+63 q^{58}-61 q^{56}+31 q^{54}+5 q^{52}-41 q^{50}+57 q^{48}-42 q^{46}+9 q^{44}+30 q^{42}-59 q^{40}+56 q^{38}-21 q^{36}-30 q^{34}+78 q^{32}-91 q^{30}+77 q^{28}-24 q^{26}-29 q^{24}+77 q^{22}-96 q^{20}+88 q^{18}-50 q^{16}+47 q^{12}-69 q^{10}+69 q^8-36 q^6-5 q^4+36 q^2-55+41 q^{-2} -7 q^{-4} -39 q^{-6} +68 q^{-8} -70 q^{-10} +39 q^{-12} +10 q^{-14} -56 q^{-16} +77 q^{-18} -70 q^{-20} +40 q^{-22} -3 q^{-24} -32 q^{-26} +47 q^{-28} -40 q^{-30} +27 q^{-32} -6 q^{-34} -7 q^{-36} +11 q^{-38} -10 q^{-40} +6 q^{-42} -2 q^{-44} + q^{-46} }

.

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 29"];
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 t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-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 z^6+z^4+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]= { 51, -2 }
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^3-3 q^2+5 q-7+9 q^{-1} -8 q^{-2} +8 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} }
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 a^2 z^6-a^4 z^4+4 a^2 z^4-2 z^4-2 a^4 z^2+7 a^2 z^2+z^2 a^{-2} -5 z^2-2 a^4+5 a^2+ a^{-2} -3}
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 2 a^2 z^8+2 z^8+6 a^3 z^7+9 a z^7+3 z^7 a^{-1} +8 a^4 z^6+6 a^2 z^6+z^6 a^{-2} -z^6+6 a^5 z^5-8 a^3 z^5-24 a z^5-10 z^5 a^{-1} +3 a^6 z^4-13 a^4 z^4-24 a^2 z^4-3 z^4 a^{-2} -11 z^4+a^7 z^3-5 a^5 z^3-a^3 z^3+14 a z^3+9 z^3 a^{-1} +8 a^4 z^2+17 a^2 z^2+3 z^2 a^{-2} +12 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} -2 a^4-5 a^2- a^{-2} -3}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_28, 10_163, K11n87,}

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 29"];
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 t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-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^3-3 q^2+5 q-7+9 q^{-1} -8 q^{-2} +8 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} } }
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]= {9_28, 10_163, K11n87,}
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: (1, -2)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
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 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 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 \frac{62}{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{14}{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 -64} 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{352}{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 -80} 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{32}{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 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 \frac{248}{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{56}{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{634}{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 \frac{11822}{45}} 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{209}{18}} 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{751}{30}}

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=} -2 is the signature of 9 29. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
7         11
5        2 -2
3       31 2
1      42  -2
-1     53   2
-3    45    1
-5   44     0
-7  24      2
-9 14       -3
-11 2        2
-131         -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=-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 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=-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}}
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}^{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=-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}^{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=-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^{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=-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}^{5}\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}^{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=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}\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=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}\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 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}_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^{10}-3 q^9-q^8+11 q^7-9 q^6-13 q^5+30 q^4-8 q^3-37 q^2+46 q+4-60 q^{-1} +51 q^{-2} +20 q^{-3} -71 q^{-4} +43 q^{-5} +32 q^{-6} -65 q^{-7} +27 q^{-8} +29 q^{-9} -42 q^{-10} +12 q^{-11} +15 q^{-12} -16 q^{-13} +4 q^{-14} +3 q^{-15} -3 q^{-16} + q^{-17} }
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^{21}-3 q^{20}-q^{19}+5 q^{18}+9 q^{17}-9 q^{16}-23 q^{15}+7 q^{14}+42 q^{13}+8 q^{12}-64 q^{11}-36 q^{10}+78 q^9+76 q^8-78 q^7-121 q^6+62 q^5+165 q^4-32 q^3-199 q^2-8 q+220+57 q^{-1} -237 q^{-2} -96 q^{-3} +230 q^{-4} +149 q^{-5} -231 q^{-6} -180 q^{-7} +206 q^{-8} +222 q^{-9} -190 q^{-10} -232 q^{-11} +147 q^{-12} +243 q^{-13} -113 q^{-14} -222 q^{-15} +69 q^{-16} +190 q^{-17} -37 q^{-18} -144 q^{-19} +16 q^{-20} +94 q^{-21} -2 q^{-22} -58 q^{-23} +2 q^{-24} +29 q^{-25} -3 q^{-26} -12 q^{-27} +3 q^{-28} +4 q^{-29} - q^{-30} -3 q^{-31} +3 q^{-32} - q^{-33} }
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^{36}-3 q^{35}-q^{34}+5 q^{33}+3 q^{32}+9 q^{31}-19 q^{30}-21 q^{29}+4 q^{28}+19 q^{27}+73 q^{26}-18 q^{25}-78 q^{24}-72 q^{23}-27 q^{22}+203 q^{21}+104 q^{20}-46 q^{19}-210 q^{18}-275 q^{17}+221 q^{16}+300 q^{15}+231 q^{14}-179 q^{13}-630 q^{12}-40 q^{11}+294 q^{10}+632 q^9+177 q^8-792 q^7-440 q^6-53 q^5+861 q^4+697 q^3-625 q^2-713 q-585+809 q^{-1} +1139 q^{-2} -258 q^{-3} -785 q^{-4} -1091 q^{-5} +588 q^{-6} +1429 q^{-7} +153 q^{-8} -747 q^{-9} -1498 q^{-10} +314 q^{-11} +1593 q^{-12} +552 q^{-13} -631 q^{-14} -1782 q^{-15} -18 q^{-16} +1583 q^{-17} +909 q^{-18} -375 q^{-19} -1830 q^{-20} -383 q^{-21} +1284 q^{-22} +1060 q^{-23} +10 q^{-24} -1495 q^{-25} -608 q^{-26} +743 q^{-27} +862 q^{-28} +298 q^{-29} -889 q^{-30} -522 q^{-31} +260 q^{-32} +444 q^{-33} +307 q^{-34} -364 q^{-35} -257 q^{-36} +49 q^{-37} +124 q^{-38} +156 q^{-39} -110 q^{-40} -67 q^{-41} +13 q^{-42} +8 q^{-43} +48 q^{-44} -30 q^{-45} -8 q^{-46} +9 q^{-47} -6 q^{-48} +9 q^{-49} -7 q^{-50} + q^{-51} +3 q^{-52} -3 q^{-53} + q^{-54} }
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^{55}-3 q^{54}-q^{53}+5 q^{52}+3 q^{51}+3 q^{50}-q^{49}-17 q^{48}-24 q^{47}+6 q^{46}+31 q^{45}+49 q^{44}+40 q^{43}-30 q^{42}-116 q^{41}-121 q^{40}-14 q^{39}+141 q^{38}+254 q^{37}+183 q^{36}-97 q^{35}-393 q^{34}-436 q^{33}-119 q^{32}+397 q^{31}+745 q^{30}+547 q^{29}-187 q^{28}-946 q^{27}-1087 q^{26}-342 q^{25}+854 q^{24}+1603 q^{23}+1140 q^{22}-372 q^{21}-1852 q^{20}-2026 q^{19}-532 q^{18}+1651 q^{17}+2778 q^{16}+1718 q^{15}-939 q^{14}-3154 q^{13}-2961 q^{12}-221 q^{11}+3013 q^{10}+4016 q^9+1672 q^8-2353 q^7-4724 q^6-3172 q^5+1288 q^4+4966 q^3+4547 q^2+62 q-4825-5707 q^{-1} -1432 q^{-2} +4371 q^{-3} +6521 q^{-4} +2836 q^{-5} -3748 q^{-6} -7193 q^{-7} -4013 q^{-8} +3081 q^{-9} +7581 q^{-10} +5147 q^{-11} -2392 q^{-12} -8012 q^{-13} -6080 q^{-14} +1787 q^{-15} +8239 q^{-16} +7044 q^{-17} -1117 q^{-18} -8550 q^{-19} -7887 q^{-20} +434 q^{-21} +8574 q^{-22} +8763 q^{-23} +451 q^{-24} -8492 q^{-25} -9411 q^{-26} -1445 q^{-27} +7895 q^{-28} +9875 q^{-29} +2584 q^{-30} -6985 q^{-31} -9832 q^{-32} -3624 q^{-33} +5569 q^{-34} +9284 q^{-35} +4462 q^{-36} -3979 q^{-37} -8171 q^{-38} -4816 q^{-39} +2348 q^{-40} +6628 q^{-41} +4672 q^{-42} -964 q^{-43} -4922 q^{-44} -4076 q^{-45} +42 q^{-46} +3291 q^{-47} +3159 q^{-48} +473 q^{-49} -1978 q^{-50} -2219 q^{-51} -579 q^{-52} +1066 q^{-53} +1371 q^{-54} +490 q^{-55} -511 q^{-56} -764 q^{-57} -323 q^{-58} +220 q^{-59} +392 q^{-60} +170 q^{-61} -98 q^{-62} -172 q^{-63} -71 q^{-64} +33 q^{-65} +71 q^{-66} +37 q^{-67} -28 q^{-68} -28 q^{-69} +4 q^{-70} +3 q^{-71} +2 q^{-72} +9 q^{-73} -6 q^{-74} -6 q^{-75} +7 q^{-76} - q^{-77} -3 q^{-78} +3 q^{-79} - q^{-80} }
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^{78}-3 q^{77}-q^{76}+5 q^{75}+3 q^{74}+3 q^{73}-7 q^{72}+q^{71}-20 q^{70}-22 q^{69}+18 q^{68}+32 q^{67}+54 q^{66}+18 q^{65}+24 q^{64}-90 q^{63}-164 q^{62}-101 q^{61}-6 q^{60}+180 q^{59}+241 q^{58}+393 q^{57}+74 q^{56}-324 q^{55}-581 q^{54}-647 q^{53}-271 q^{52}+207 q^{51}+1217 q^{50}+1242 q^{49}+702 q^{48}-376 q^{47}-1558 q^{46}-2121 q^{45}-1870 q^{44}+492 q^{43}+2230 q^{42}+3431 q^{41}+2741 q^{40}+412 q^{39}-2883 q^{38}-5620 q^{37}-4131 q^{36}-1195 q^{35}+3704 q^{34}+6942 q^{33}+7131 q^{32}+2474 q^{31}-5071 q^{30}-8939 q^{29}-9751 q^{28}-3787 q^{27}+4702 q^{26}+12713 q^{25}+12976 q^{24}+4715 q^{23}-5335 q^{22}-15491 q^{21}-16099 q^{20}-7864 q^{19}+8110 q^{18}+19072 q^{17}+18793 q^{16}+9107 q^{15}-9630 q^{14}-22746 q^{13}-24071 q^{12}-7497 q^{11}+12872 q^{10}+26434 q^9+26530 q^8+7049 q^7-17140 q^6-33580 q^5-25683 q^4-3472 q^3+22509 q^2+37520 q+26018-2422 q^{-1} -32653 q^{-2} -38225 q^{-3} -21762 q^{-4} +10767 q^{-5} +39573 q^{-6} +40200 q^{-7} +13842 q^{-8} -25216 q^{-9} -43540 q^{-10} -36056 q^{-11} -2035 q^{-12} +36528 q^{-13} +48571 q^{-14} +26735 q^{-15} -17057 q^{-16} -45092 q^{-17} -45769 q^{-18} -12014 q^{-19} +33152 q^{-20} +54119 q^{-21} +36146 q^{-22} -10914 q^{-23} -46555 q^{-24} -53611 q^{-25} -19825 q^{-26} +30914 q^{-27} +59547 q^{-28} +44895 q^{-29} -5020 q^{-30} -48015 q^{-31} -61616 q^{-32} -28849 q^{-33} +26676 q^{-34} +63623 q^{-35} +54687 q^{-36} +4746 q^{-37} -44936 q^{-38} -67352 q^{-39} -40453 q^{-40} +15693 q^{-41} +60548 q^{-42} +61882 q^{-43} +19047 q^{-44} -32354 q^{-45} -63985 q^{-46} -49510 q^{-47} -1443 q^{-48} +45690 q^{-49} +58777 q^{-50} +30939 q^{-51} -12630 q^{-52} -47747 q^{-53} -47621 q^{-54} -15894 q^{-55} +23543 q^{-56} +42812 q^{-57} +31684 q^{-58} +3796 q^{-59} -25265 q^{-60} -33676 q^{-61} -19131 q^{-62} +5635 q^{-63} +22233 q^{-64} +21648 q^{-65} +9222 q^{-66} -8177 q^{-67} -16791 q^{-68} -12912 q^{-69} -1642 q^{-70} +7705 q^{-71} +9869 q^{-72} +6473 q^{-73} -987 q^{-74} -5801 q^{-75} -5525 q^{-76} -1818 q^{-77} +1708 q^{-78} +2954 q^{-79} +2617 q^{-80} +272 q^{-81} -1472 q^{-82} -1551 q^{-83} -611 q^{-84} +282 q^{-85} +558 q^{-86} +695 q^{-87} +117 q^{-88} -337 q^{-89} -294 q^{-90} -77 q^{-91} +68 q^{-92} +43 q^{-93} +137 q^{-94} +12 q^{-95} -81 q^{-96} -32 q^{-97} +8 q^{-98} +24 q^{-99} -18 q^{-100} +23 q^{-101} +3 q^{-102} -20 q^{-103} +3 q^{-104} +3 q^{-105} +6 q^{-106} -7 q^{-107} + q^{-108} +3 q^{-109} -3 q^{-110} + q^{-111} }
7 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^{105}-3 q^{104}-q^{103}+5 q^{102}+3 q^{101}+3 q^{100}-7 q^{99}-5 q^{98}-2 q^{97}-18 q^{96}-10 q^{95}+19 q^{94}+37 q^{93}+57 q^{92}+19 q^{91}-20 q^{90}-35 q^{89}-133 q^{88}-152 q^{87}-89 q^{86}+42 q^{85}+268 q^{84}+342 q^{83}+314 q^{82}+226 q^{81}-187 q^{80}-602 q^{79}-875 q^{78}-884 q^{77}-221 q^{76}+540 q^{75}+1308 q^{74}+1944 q^{73}+1583 q^{72}+493 q^{71}-1225 q^{70}-3091 q^{69}-3567 q^{68}-2798 q^{67}-617 q^{66}+2878 q^{65}+5492 q^{64}+6556 q^{63}+4805 q^{62}-155 q^{61}-5479 q^{60}-9858 q^{59}-10883 q^{58}-6485 q^{57}+1127 q^{56}+10330 q^{55}+16875 q^{54}+15988 q^{53}+8580 q^{52}-4633 q^{51}-18514 q^{50}-25401 q^{49}-22994 q^{48}-8982 q^{47}+12056 q^{46}+29687 q^{45}+37723 q^{44}+29101 q^{43}+5120 q^{42}-23262 q^{41}-46602 q^{40}-51133 q^{39}-31706 q^{38}+3323 q^{37}+43079 q^{36}+67283 q^{35}+62149 q^{34}+29443 q^{33}-22796 q^{32}-69988 q^{31}-88369 q^{30}-69424 q^{29}-13652 q^{28}+53920 q^{27}+101325 q^{26}+107869 q^{25}+61475 q^{24}-18257 q^{23}-94844 q^{22}-135477 q^{21}-112112 q^{20}-32813 q^{19}+66891 q^{18}+144984 q^{17}+155949 q^{16}+91657 q^{15}-20006 q^{14}-132996 q^{13}-185483 q^{12}-149479 q^{11}-39161 q^{10}+100893 q^9+196107 q^8+198060 q^7+102902 q^6-53039 q^5-187532 q^4-232727 q^3-163734 q^2-3501 q+162847+251327 q^{-1} +215796 q^{-2} +62457 q^{-3} -126646 q^{-4} -255411 q^{-5} -256846 q^{-6} -118052 q^{-7} +85277 q^{-8} +248069 q^{-9} +285755 q^{-10} +166696 q^{-11} -42882 q^{-12} -233495 q^{-13} -304972 q^{-14} -206976 q^{-15} +4237 q^{-16} +216215 q^{-17} +316237 q^{-18} +238621 q^{-19} +29174 q^{-20} -199228 q^{-21} -323576 q^{-22} -263598 q^{-23} -55747 q^{-24} +185744 q^{-25} +329272 q^{-26} +283380 q^{-27} +76834 q^{-28} -176163 q^{-29} -336443 q^{-30} -301601 q^{-31} -93702 q^{-32} +171077 q^{-33} +345916 q^{-34} +320128 q^{-35} +109928 q^{-36} -167583 q^{-37} -358376 q^{-38} -341928 q^{-39} -128398 q^{-40} +163029 q^{-41} +371166 q^{-42} +366896 q^{-43} +153100 q^{-44} -152125 q^{-45} -381130 q^{-46} -394208 q^{-47} -184955 q^{-48} +131056 q^{-49} +381977 q^{-50} +419089 q^{-51} +223868 q^{-52} -96262 q^{-53} -368829 q^{-54} -435963 q^{-55} -264871 q^{-56} +48726 q^{-57} +336921 q^{-58} +437136 q^{-59} +301531 q^{-60} +8157 q^{-61} -286250 q^{-62} -418086 q^{-63} -325264 q^{-64} -65768 q^{-65} +220557 q^{-66} +376410 q^{-67} +329393 q^{-68} +115480 q^{-69} -147897 q^{-70} -316152 q^{-71} -311118 q^{-72} -148664 q^{-73} +78719 q^{-74} +244798 q^{-75} +272303 q^{-76} +161037 q^{-77} -21770 q^{-78} -172351 q^{-79} -220234 q^{-80} -153536 q^{-81} -16964 q^{-82} +108723 q^{-83} +163644 q^{-84} +130712 q^{-85} +36969 q^{-86} -59298 q^{-87} -111331 q^{-88} -100865 q^{-89} -41494 q^{-90} +26537 q^{-91} +69152 q^{-92} +70437 q^{-93} +36131 q^{-94} -7869 q^{-95} -39001 q^{-96} -44848 q^{-97} -26780 q^{-98} -477 q^{-99} +20036 q^{-100} +26152 q^{-101} +17297 q^{-102} +2775 q^{-103} -9348 q^{-104} -13916 q^{-105} -9910 q^{-106} -2628 q^{-107} +4013 q^{-108} +6942 q^{-109} +5121 q^{-110} +1609 q^{-111} -1701 q^{-112} -3180 q^{-113} -2289 q^{-114} -809 q^{-115} +627 q^{-116} +1386 q^{-117} +983 q^{-118} +359 q^{-119} -325 q^{-120} -623 q^{-121} -283 q^{-122} -74 q^{-123} +97 q^{-124} +201 q^{-125} +102 q^{-126} +64 q^{-127} -71 q^{-128} -128 q^{-129} +9 q^{-130} +27 q^{-131} +12 q^{-132} +11 q^{-133} -10 q^{-134} +22 q^{-135} -9 q^{-136} -29 q^{-137} +15 q^{-138} +8 q^{-139} -3 q^{-141} -6 q^{-142} +7 q^{-143} - q^{-144} -3 q^{-145} +3 q^{-146} - q^{-147} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

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