9 17

9_16

9_18

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 17's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 17 at Knotilus!

[edit 9 17 Quick Notes]

[edit 9 17 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_5,12,6,13 X_3,11,4,10 X_11,3,12,2 X_7,14,8,15 X_13,6,14,7 X_15,18,16,1 X_9,17,10,16 X_17,9,18,8
Gauss code	-1, 4, -3, 1, -2, 6, -5, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 12 14 16 2 6 18 8
Conway Notation	[21312]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 9, width is 4,

Braid index is 4

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

[edit Notes on presentations of 9 17]

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 17"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_5,12,6,13 X_3,11,4,10 X_11,3,12,2 X_7,14,8,15 X_13,6,14,7 X_15,18,16,1 X_9,17,10,16 X_17,9,18,8

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [21312]

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	2
Super bridge index	$\{4,7\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[-8][-3]
Hyperbolic Volume	9.47458
A-Polynomial	See Data:9 17/A-polynomial

[edit Notes for 9 17'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 17'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+9 t-9+9 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^4-2 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 39, -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^3-2 q^2+4 q-5+6 q^{-1} -7 q^{-2} +6 q^{-3} -4 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+5 a^2 z^2+z^2 a^{-2} -6 z^2+2 a^2+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 a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +4 a^4 z^6+4 a^2 z^6+z^6 a^{-2} +z^6+4 a^5 z^5-2 a^3 z^5-13 a z^5-7 z^5 a^{-1} +3 a^6 z^4-3 a^4 z^4-14 a^2 z^4-4 z^4 a^{-2} -12 z^4+a^7 z^3-3 a^5 z^3-4 a^3 z^3+6 a z^3+6 z^3 a^{-1} -2 a^6 z^2-a^4 z^2+9 a^2 z^2+5 z^2 a^{-2} +13 z^2+a^5 z+3 a^3 z+a z-z a^{-1} -2 a^2-2 a^{-2} -3}
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^{18}+q^{16}+q^{12}+2 q^{10}-q^8+q^6-2 q^4- q^{-2} + q^{-4} + q^{-8} + q^{-10} }
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}+q^{92}-3 q^{88}+8 q^{86}-9 q^{84}+12 q^{82}-9 q^{80}+3 q^{78}+4 q^{76}-9 q^{74}+14 q^{72}-20 q^{70}+17 q^{68}-12 q^{66}+q^{64}+10 q^{62}-18 q^{60}+22 q^{58}-16 q^{56}+8 q^{54}+q^{52}-14 q^{50}+16 q^{48}-7 q^{46}-2 q^{44}+15 q^{42}-18 q^{40}+15 q^{38}+3 q^{36}-17 q^{34}+30 q^{32}-35 q^{30}+26 q^{28}-6 q^{26}-14 q^{24}+33 q^{22}-38 q^{20}+32 q^{18}-16 q^{16}-4 q^{14}+15 q^{12}-26 q^{10}+22 q^8-12 q^6-4 q^4+14 q^2-17+10 q^{-2} +3 q^{-4} -17 q^{-6} +23 q^{-8} -24 q^{-10} +11 q^{-12} +5 q^{-14} -21 q^{-16} +32 q^{-18} -27 q^{-20} +17 q^{-22} - q^{-24} -11 q^{-26} +19 q^{-28} -18 q^{-30} +14 q^{-32} -4 q^{-34} -2 q^{-36} +5 q^{-38} -5 q^{-40} +4 q^{-42} - q^{-44} + q^{-46} }

Further Quantum Invariants

Further quantum knot invariants for 9_17.

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}-q^9+2 q^7-q^5-q^3+q- q^{-1} +2 q^{-3} - 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}-4 q^{28}+q^{26}+7 q^{24}-7 q^{22}-q^{20}+7 q^{18}-5 q^{16}-4 q^{14}+4 q^{12}+2 q^{10}-3 q^8-q^6+6 q^4-q^2-5+7 q^{-2} + q^{-4} -8 q^{-6} +4 q^{-8} +4 q^{-10} -6 q^{-12} + q^{-14} +4 q^{-16} -2 q^{-18} - 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}-2 q^{61}+q^{59}+2 q^{57}-5 q^{55}+3 q^{53}+7 q^{51}-3 q^{49}-12 q^{47}+7 q^{45}+17 q^{43}-8 q^{41}-22 q^{39}+4 q^{37}+24 q^{35}-22 q^{31}-8 q^{29}+18 q^{27}+12 q^{25}-6 q^{23}-17 q^{21}+18 q^{17}+7 q^{15}-16 q^{13}-13 q^{11}+15 q^9+16 q^7-11 q^5-21 q^3+9 q+22 q^{-1} -2 q^{-3} -23 q^{-5} -4 q^{-7} +23 q^{-9} +11 q^{-11} -19 q^{-13} -15 q^{-15} +12 q^{-17} +18 q^{-19} -4 q^{-21} -18 q^{-23} - q^{-25} +13 q^{-27} +6 q^{-29} -8 q^{-31} -7 q^{-33} +3 q^{-35} +5 q^{-37} -2 q^{-41} - 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^{102}-4 q^{100}-q^{98}-9 q^{94}+9 q^{92}+9 q^{88}+q^{86}-28 q^{84}+q^{82}+12 q^{80}+36 q^{78}+9 q^{76}-60 q^{74}-30 q^{72}+17 q^{70}+76 q^{68}+39 q^{66}-80 q^{64}-74 q^{62}-6 q^{60}+96 q^{58}+80 q^{56}-50 q^{54}-86 q^{52}-55 q^{50}+56 q^{48}+93 q^{46}+14 q^{44}-44 q^{42}-75 q^{40}-14 q^{38}+51 q^{36}+57 q^{34}+19 q^{32}-56 q^{30}-64 q^{28}+q^{26}+70 q^{24}+53 q^{22}-30 q^{20}-80 q^{18}-29 q^{16}+71 q^{14}+66 q^{12}-12 q^{10}-85 q^8-51 q^6+64 q^4+77 q^2+18-78 q^{-2} -78 q^{-4} +34 q^{-6} +76 q^{-8} +58 q^{-10} -45 q^{-12} -91 q^{-14} -18 q^{-16} +40 q^{-18} +85 q^{-20} +13 q^{-22} -64 q^{-24} -52 q^{-26} -16 q^{-28} +64 q^{-30} +50 q^{-32} -7 q^{-34} -38 q^{-36} -52 q^{-38} +12 q^{-40} +38 q^{-42} +27 q^{-44} + q^{-46} -37 q^{-48} -16 q^{-50} +4 q^{-52} +18 q^{-54} +17 q^{-56} -8 q^{-58} -9 q^{-60} -7 q^{-62} + q^{-64} +7 q^{-66} + q^{-68} -2 q^{-72} - 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}-3 q^{155}-2 q^{153}+3 q^{151}+6 q^{149}+3 q^{147}+2 q^{145}-6 q^{143}-13 q^{141}-9 q^{139}+8 q^{137}+27 q^{135}+16 q^{133}-9 q^{131}-40 q^{129}-46 q^{127}+10 q^{125}+78 q^{123}+71 q^{121}-7 q^{119}-103 q^{117}-128 q^{115}-13 q^{113}+159 q^{111}+194 q^{109}+37 q^{107}-194 q^{105}-279 q^{103}-97 q^{101}+218 q^{99}+372 q^{97}+176 q^{95}-211 q^{93}-442 q^{91}-274 q^{89}+152 q^{87}+468 q^{85}+377 q^{83}-56 q^{81}-433 q^{79}-435 q^{77}-72 q^{75}+338 q^{73}+439 q^{71}+185 q^{69}-189 q^{67}-386 q^{65}-271 q^{63}+43 q^{61}+283 q^{59}+291 q^{57}+103 q^{55}-157 q^{53}-285 q^{51}-193 q^{49}+43 q^{47}+239 q^{45}+251 q^{43}+50 q^{41}-205 q^{39}-281 q^{37}-101 q^{35}+174 q^{33}+292 q^{31}+133 q^{29}-160 q^{27}-311 q^{25}-151 q^{23}+168 q^{21}+325 q^{19}+176 q^{17}-153 q^{15}-356 q^{13}-214 q^{11}+143 q^9+372 q^7+260 q^5-95 q^3-375 q-322 q^{-1} +27 q^{-3} +348 q^{-5} +369 q^{-7} +69 q^{-9} -286 q^{-11} -391 q^{-13} -169 q^{-15} +186 q^{-17} +376 q^{-19} +257 q^{-21} -64 q^{-23} -313 q^{-25} -300 q^{-27} -65 q^{-29} +206 q^{-31} +300 q^{-33} +166 q^{-35} -81 q^{-37} -244 q^{-39} -212 q^{-41} -42 q^{-43} +146 q^{-45} +211 q^{-47} +122 q^{-49} -43 q^{-51} -154 q^{-53} -147 q^{-55} -50 q^{-57} +76 q^{-59} +132 q^{-61} +90 q^{-63} -3 q^{-65} -79 q^{-67} -92 q^{-69} -44 q^{-71} +27 q^{-73} +66 q^{-75} +55 q^{-77} +9 q^{-79} -31 q^{-81} -41 q^{-83} -26 q^{-85} +5 q^{-87} +24 q^{-89} +21 q^{-91} +5 q^{-93} -6 q^{-95} -11 q^{-97} -9 q^{-99} + q^{-101} +5 q^{-103} +3 q^{-105} + q^{-107} -2 q^{-111} - 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}+q^{12}+2 q^{10}-q^8+q^6-2 q^4- q^{-2} + q^{-4} + q^{-8} + 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}+18 q^{44}-28 q^{42}+36 q^{40}-40 q^{38}+45 q^{36}-56 q^{34}+60 q^{32}-56 q^{30}+60 q^{28}-50 q^{26}+40 q^{24}-16 q^{22}-17 q^{20}+44 q^{18}-88 q^{16}+114 q^{14}-141 q^{12}+160 q^{10}-156 q^8+148 q^6-117 q^4+92 q^2-50+6 q^{-2} +31 q^{-4} -62 q^{-6} +80 q^{-8} -94 q^{-10} +92 q^{-12} -78 q^{-14} +64 q^{-16} -46 q^{-18} +29 q^{-20} -16 q^{-22} +8 q^{-24} -2 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}-q^{38}+2 q^{34}+3 q^{32}-2 q^{30}+4 q^{26}+2 q^{24}-6 q^{22}-q^{20}+3 q^{18}-3 q^{16}-3 q^{14}+2 q^{10}-2 q^8+2 q^6+4 q^4+q^2+1+4 q^{-2} -5 q^{-6} +2 q^{-10} - q^{-12} -3 q^{-14} + q^{-16} +3 q^{-18} -2 q^{-22} + q^{-26} + 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}+4 q^{36}-3 q^{34}-2 q^{32}+7 q^{30}-q^{28}-5 q^{26}+6 q^{24}-2 q^{22}-6 q^{20}+3 q^{18}+q^{16}+q^{12}+5 q^{10}+3 q^8-5 q^6+q^4+3 q^2-7- q^{-2} +4 q^{-4} -5 q^{-6} + q^{-8} +4 q^{-10} -2 q^{-12} +2 q^{-14} +2 q^{-16} - 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}-q^{19}+2 q^{17}+2 q^{13}+q^9-q^5-2 q-2 q^{-3} + q^{-5} +2 q^{-9} + q^{-11} + 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}+2 q^{46}+q^{44}-4 q^{42}-q^{40}+5 q^{38}+2 q^{36}-2 q^{34}+3 q^{32}+6 q^{30}-3 q^{28}-5 q^{26}+q^{24}-3 q^{22}-7 q^{20}+2 q^{18}+4 q^{16}-q^{14}+3 q^{12}+8 q^{10}+3 q^8-3 q^6+2 q^4+4 q^2-4-5 q^{-2} - q^{-6} -4 q^{-8} - q^{-10} + q^{-12} + q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} + q^{-22} + q^{-24} + 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}-q^{24}+q^{22}+q^{20}+2 q^{16}+2 q^{12}+q^8-q^6-2 q^2-2- q^{-2} -2 q^{-4} + q^{-6} +2 q^{-10} +2 q^{-12} + q^{-14} + 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}+4 q^{36}-5 q^{34}+6 q^{32}-7 q^{30}+7 q^{28}-5 q^{26}+4 q^{24}-2 q^{20}+7 q^{18}-9 q^{16}+12 q^{14}-13 q^{12}+13 q^{10}-13 q^8+9 q^6-7 q^4+3 q^2-1-3 q^{-2} +6 q^{-4} -7 q^{-6} +7 q^{-8} -6 q^{-10} +6 q^{-12} -4 q^{-14} +4 q^{-16} - 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}+4 q^{58}+q^{56}-4 q^{54}-4 q^{52}+2 q^{50}+7 q^{48}+3 q^{46}-5 q^{44}-5 q^{42}+2 q^{40}+7 q^{38}-q^{36}-7 q^{34}-3 q^{32}+5 q^{30}+3 q^{28}-4 q^{26}-4 q^{24}+3 q^{22}+6 q^{20}-3 q^{16}+q^{14}+5 q^{12}-5 q^8-2 q^6+5 q^4+4 q^2-4-8 q^{-2} +8 q^{-6} +3 q^{-8} -6 q^{-10} -6 q^{-12} +3 q^{-14} +6 q^{-16} + q^{-18} -4 q^{-20} -2 q^{-22} +3 q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-34} }

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

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}+q^{92}-3 q^{88}+8 q^{86}-9 q^{84}+12 q^{82}-9 q^{80}+3 q^{78}+4 q^{76}-9 q^{74}+14 q^{72}-20 q^{70}+17 q^{68}-12 q^{66}+q^{64}+10 q^{62}-18 q^{60}+22 q^{58}-16 q^{56}+8 q^{54}+q^{52}-14 q^{50}+16 q^{48}-7 q^{46}-2 q^{44}+15 q^{42}-18 q^{40}+15 q^{38}+3 q^{36}-17 q^{34}+30 q^{32}-35 q^{30}+26 q^{28}-6 q^{26}-14 q^{24}+33 q^{22}-38 q^{20}+32 q^{18}-16 q^{16}-4 q^{14}+15 q^{12}-26 q^{10}+22 q^8-12 q^6-4 q^4+14 q^2-17+10 q^{-2} +3 q^{-4} -17 q^{-6} +23 q^{-8} -24 q^{-10} +11 q^{-12} +5 q^{-14} -21 q^{-16} +32 q^{-18} -27 q^{-20} +17 q^{-22} - q^{-24} -11 q^{-26} +19 q^{-28} -18 q^{-30} +14 q^{-32} -4 q^{-34} -2 q^{-36} +5 q^{-38} -5 q^{-40} +4 q^{-42} - 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.

In[3]:= K = Knot["9 17"];

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+9 t-9+9 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^4-2 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]= { 39, -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^3-2 q^2+4 q-5+6 q^{-1} -7 q^{-2} +6 q^{-3} -4 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+5 a^2 z^2+z^2 a^{-2} -6 z^2+2 a^2+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 a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +4 a^4 z^6+4 a^2 z^6+z^6 a^{-2} +z^6+4 a^5 z^5-2 a^3 z^5-13 a z^5-7 z^5 a^{-1} +3 a^6 z^4-3 a^4 z^4-14 a^2 z^4-4 z^4 a^{-2} -12 z^4+a^7 z^3-3 a^5 z^3-4 a^3 z^3+6 a z^3+6 z^3 a^{-1} -2 a^6 z^2-a^4 z^2+9 a^2 z^2+5 z^2 a^{-2} +13 z^2+a^5 z+3 a^3 z+a z-z a^{-1} -2 a^2-2 a^{-2} -3}

"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["9 17"];

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+9 t-9+9 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^3-2 q^2+4 q-5+6 q^{-1} -7 q^{-2} +6 q^{-3} -4 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]= {}

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:

(-2, 0)

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 -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 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 32}

Failed to parse (SVG (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{116}{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{28}{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 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 32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 -\frac{256}{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 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 -\frac{928}{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 -\frac{4951}{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{1796}{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{2764}{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{7}{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 \frac{89}{15}}

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=} -2 is the signature of 9 17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

1

-1

3

1

2

1

2

1

-1

4

3

1

-3

4

3

-1

-5

2

3

-1

-7

2

4

2

-9

1

2

-1

-11

2

-13

1

-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}^{2}\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^{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}^{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=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}^{3}\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}^{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}^{3}\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}\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}	Failed to parse (SVG (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}	Failed to parse (SVG (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}-2 q^9-q^8+7 q^7-5 q^6-8 q^5+17 q^4-5 q^3-20 q^2+26 q+1-32 q^{-1} +30 q^{-2} +8 q^{-3} -39 q^{-4} +28 q^{-5} +13 q^{-6} -37 q^{-7} +20 q^{-8} +12 q^{-9} -25 q^{-10} +12 q^{-11} +6 q^{-12} -11 q^{-13} +6 q^{-14} + 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}-2 q^{20}-q^{19}+2 q^{18}+6 q^{17}-4 q^{16}-11 q^{15}+q^{14}+20 q^{13}+3 q^{12}-25 q^{11}-16 q^{10}+34 q^9+25 q^8-31 q^7-43 q^6+30 q^5+55 q^4-19 q^3-70 q^2+11 q+76+5 q^{-1} -83 q^{-2} -19 q^{-3} +86 q^{-4} +32 q^{-5} -84 q^{-6} -47 q^{-7} +83 q^{-8} +55 q^{-9} -73 q^{-10} -65 q^{-11} +66 q^{-12} +66 q^{-13} -55 q^{-14} -59 q^{-15} +40 q^{-16} +52 q^{-17} -33 q^{-18} -35 q^{-19} +20 q^{-20} +26 q^{-21} -19 q^{-22} -10 q^{-23} +10 q^{-24} +7 q^{-25} -10 q^{-26} +6 q^{-28} - q^{-29} -3 q^{-30} - 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}-2 q^{35}-q^{34}+2 q^{33}+q^{32}+7 q^{31}-8 q^{30}-9 q^{29}+2 q^{27}+32 q^{26}-7 q^{25}-23 q^{24}-20 q^{23}-19 q^{22}+70 q^{21}+19 q^{20}-12 q^{19}-46 q^{18}-83 q^{17}+84 q^{16}+50 q^{15}+45 q^{14}-32 q^{13}-163 q^{12}+48 q^{11}+38 q^{10}+122 q^9+40 q^8-208 q^7-10 q^6-35 q^5+168 q^4+143 q^3-190 q^2-52 q-147+168 q^{-1} +239 q^{-2} -131 q^{-3} -65 q^{-4} -262 q^{-5} +134 q^{-6} +312 q^{-7} -53 q^{-8} -60 q^{-9} -362 q^{-10} +83 q^{-11} +362 q^{-12} +30 q^{-13} -43 q^{-14} -431 q^{-15} +18 q^{-16} +370 q^{-17} +105 q^{-18} -5 q^{-19} -437 q^{-20} -47 q^{-21} +309 q^{-22} +136 q^{-23} +53 q^{-24} -358 q^{-25} -84 q^{-26} +198 q^{-27} +105 q^{-28} +89 q^{-29} -228 q^{-30} -68 q^{-31} +96 q^{-32} +37 q^{-33} +83 q^{-34} -109 q^{-35} -31 q^{-36} +37 q^{-37} -10 q^{-38} +53 q^{-39} -40 q^{-40} -4 q^{-41} +13 q^{-42} -21 q^{-43} +24 q^{-44} -11 q^{-45} +4 q^{-46} +4 q^{-47} -12 q^{-48} +6 q^{-49} -2 q^{-50} +3 q^{-51} + 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}-2 q^{54}-q^{53}+2 q^{52}+q^{51}+2 q^{50}+3 q^{49}-6 q^{48}-11 q^{47}+6 q^{45}+13 q^{44}+19 q^{43}-3 q^{42}-30 q^{41}-31 q^{40}-9 q^{39}+23 q^{38}+59 q^{37}+43 q^{36}-19 q^{35}-70 q^{34}-80 q^{33}-25 q^{32}+72 q^{31}+119 q^{30}+74 q^{29}-28 q^{28}-136 q^{27}-151 q^{26}-25 q^{25}+112 q^{24}+185 q^{23}+137 q^{22}-47 q^{21}-216 q^{20}-213 q^{19}-58 q^{18}+153 q^{17}+300 q^{16}+200 q^{15}-82 q^{14}-307 q^{13}-329 q^{12}-82 q^{11}+287 q^{10}+449 q^9+239 q^8-188 q^7-519 q^6-437 q^5+65 q^4+554 q^3+594 q^2+112 q-540-758 q^{-1} -284 q^{-2} +501 q^{-3} +874 q^{-4} +467 q^{-5} -428 q^{-6} -987 q^{-7} -641 q^{-8} +359 q^{-9} +1077 q^{-10} +796 q^{-11} -279 q^{-12} -1144 q^{-13} -960 q^{-14} +199 q^{-15} +1228 q^{-16} +1089 q^{-17} -120 q^{-18} -1262 q^{-19} -1235 q^{-20} +19 q^{-21} +1304 q^{-22} +1344 q^{-23} +81 q^{-24} -1274 q^{-25} -1431 q^{-26} -217 q^{-27} +1212 q^{-28} +1472 q^{-29} +341 q^{-30} -1086 q^{-31} -1439 q^{-32} -457 q^{-33} +898 q^{-34} +1357 q^{-35} +538 q^{-36} -712 q^{-37} -1185 q^{-38} -558 q^{-39} +488 q^{-40} +994 q^{-41} +540 q^{-42} -335 q^{-43} -752 q^{-44} -467 q^{-45} +172 q^{-46} +568 q^{-47} +372 q^{-48} -104 q^{-49} -365 q^{-50} -271 q^{-51} +18 q^{-52} +253 q^{-53} +190 q^{-54} -19 q^{-55} -134 q^{-56} -114 q^{-57} -17 q^{-58} +81 q^{-59} +75 q^{-60} +6 q^{-61} -38 q^{-62} -36 q^{-63} -10 q^{-64} +13 q^{-65} +19 q^{-66} +12 q^{-67} -7 q^{-68} -11 q^{-69} + q^{-70} -6 q^{-71} +2 q^{-72} +8 q^{-73} -3 q^{-75} +2 q^{-76} -3 q^{-77} - 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}-2 q^{77}-q^{76}+2 q^{75}+q^{74}+2 q^{73}-2 q^{72}+5 q^{71}-8 q^{70}-11 q^{69}+3 q^{68}+5 q^{67}+14 q^{66}+3 q^{65}+24 q^{64}-17 q^{63}-39 q^{62}-22 q^{61}-11 q^{60}+25 q^{59}+19 q^{58}+100 q^{57}+17 q^{56}-49 q^{55}-73 q^{54}-90 q^{53}-39 q^{52}-31 q^{51}+197 q^{50}+133 q^{49}+68 q^{48}-29 q^{47}-136 q^{46}-190 q^{45}-267 q^{44}+131 q^{43}+158 q^{42}+264 q^{41}+216 q^{40}+93 q^{39}-162 q^{38}-544 q^{37}-172 q^{36}-175 q^{35}+177 q^{34}+397 q^{33}+591 q^{32}+315 q^{31}-406 q^{30}-331 q^{29}-750 q^{28}-436 q^{27}+5 q^{26}+852 q^{25}+1007 q^{24}+335 q^{23}+181 q^{22}-959 q^{21}-1230 q^{20}-1058 q^{19}+349 q^{18}+1248 q^{17}+1241 q^{16}+1369 q^{15}-319 q^{14}-1523 q^{13}-2269 q^{12}-848 q^{11}+615 q^{10}+1642 q^9+2680 q^8+1030 q^7-965 q^6-2990 q^5-2193 q^4-702 q^3+1264 q^2+3551 q+2555+213 q^{-1} -3010 q^{-2} -3198 q^{-3} -2203 q^{-4} +349 q^{-5} +3838 q^{-6} +3830 q^{-7} +1550 q^{-8} -2571 q^{-9} -3766 q^{-10} -3511 q^{-11} -685 q^{-12} +3770 q^{-13} +4771 q^{-14} +2724 q^{-15} -2030 q^{-16} -4101 q^{-17} -4553 q^{-18} -1573 q^{-19} +3631 q^{-20} +5526 q^{-21} +3697 q^{-22} -1559 q^{-23} -4399 q^{-24} -5450 q^{-25} -2336 q^{-26} +3485 q^{-27} +6189 q^{-28} +4596 q^{-29} -1024 q^{-30} -4589 q^{-31} -6235 q^{-32} -3156 q^{-33} +3069 q^{-34} +6557 q^{-35} +5428 q^{-36} -169 q^{-37} -4292 q^{-38} -6629 q^{-39} -4019 q^{-40} +2108 q^{-41} +6172 q^{-42} +5837 q^{-43} +941 q^{-44} -3233 q^{-45} -6149 q^{-46} -4501 q^{-47} +771 q^{-48} +4842 q^{-49} +5341 q^{-50} +1762 q^{-51} -1720 q^{-52} -4702 q^{-53} -4120 q^{-54} -303 q^{-55} +3028 q^{-56} +3959 q^{-57} +1809 q^{-58} -469 q^{-59} -2869 q^{-60} -2983 q^{-61} -661 q^{-62} +1502 q^{-63} +2333 q^{-64} +1232 q^{-65} +114 q^{-66} -1398 q^{-67} -1719 q^{-68} -483 q^{-69} +630 q^{-70} +1112 q^{-71} +573 q^{-72} +203 q^{-73} -565 q^{-74} -825 q^{-75} -202 q^{-76} +246 q^{-77} +452 q^{-78} +175 q^{-79} +132 q^{-80} -194 q^{-81} -356 q^{-82} -42 q^{-83} +91 q^{-84} +162 q^{-85} +25 q^{-86} +70 q^{-87} -55 q^{-88} -141 q^{-89} +4 q^{-90} +24 q^{-91} +52 q^{-92} -9 q^{-93} +37 q^{-94} -11 q^{-95} -49 q^{-96} +7 q^{-97} +15 q^{-99} -8 q^{-100} +16 q^{-101} -14 q^{-103} +4 q^{-104} -3 q^{-105} +3 q^{-106} -2 q^{-107} +3 q^{-108} + 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}-2 q^{104}-q^{103}+2 q^{102}+q^{101}+2 q^{100}-2 q^{99}+3 q^{97}-8 q^{96}-8 q^{95}+2 q^{94}+5 q^{93}+16 q^{92}+5 q^{91}+q^{90}+13 q^{89}-23 q^{88}-34 q^{87}-25 q^{86}-13 q^{85}+38 q^{84}+38 q^{83}+37 q^{82}+70 q^{81}-2 q^{80}-59 q^{79}-93 q^{78}-132 q^{77}-23 q^{76}+24 q^{75}+75 q^{74}+209 q^{73}+150 q^{72}+74 q^{71}-50 q^{70}-267 q^{69}-229 q^{68}-208 q^{67}-133 q^{66}+198 q^{65}+294 q^{64}+399 q^{63}+367 q^{62}-25 q^{61}-184 q^{60}-453 q^{59}-643 q^{58}-324 q^{57}-123 q^{56}+322 q^{55}+799 q^{54}+676 q^{53}+635 q^{52}+114 q^{51}-670 q^{50}-874 q^{49}-1205 q^{48}-849 q^{47}+115 q^{46}+738 q^{45}+1635 q^{44}+1699 q^{43}+788 q^{42}-41 q^{41}-1577 q^{40}-2446 q^{39}-2017 q^{38}-1187 q^{37}+992 q^{36}+2731 q^{35}+3090 q^{34}+2807 q^{33}+427 q^{32}-2300 q^{31}-3897 q^{30}-4558 q^{29}-2308 q^{28}+1094 q^{27}+3862 q^{26}+5999 q^{25}+4607 q^{24}+949 q^{23}-3046 q^{22}-6888 q^{21}-6771 q^{20}-3467 q^{19}+1263 q^{18}+6854 q^{17}+8590 q^{16}+6257 q^{15}+1220 q^{14}-5947 q^{13}-9686 q^{12}-8868 q^{11}-4215 q^{10}+4155 q^9+9981 q^8+11106 q^7+7335 q^6-1728 q^5-9431 q^4-12709 q^3-10385 q^2-1134 q+8236+13694 q^{-1} +13062 q^{-2} +4086 q^{-3} -6474 q^{-4} -14025 q^{-5} -15370 q^{-6} -7019 q^{-7} +4495 q^{-8} +13940 q^{-9} +17159 q^{-10} +9672 q^{-11} -2401 q^{-12} -13477 q^{-13} -18579 q^{-14} -12072 q^{-15} +411 q^{-16} +12923 q^{-17} +19706 q^{-18} +14102 q^{-19} +1349 q^{-20} -12339 q^{-21} -20591 q^{-22} -15887 q^{-23} -2930 q^{-24} +11890 q^{-25} +21484 q^{-26} +17463 q^{-27} +4192 q^{-28} -11593 q^{-29} -22293 q^{-30} -18922 q^{-31} -5408 q^{-32} +11376 q^{-33} +23229 q^{-34} +20426 q^{-35} +6522 q^{-36} -11219 q^{-37} -24079 q^{-38} -21880 q^{-39} -7831 q^{-40} +10781 q^{-41} +24815 q^{-42} +23452 q^{-43} +9340 q^{-44} -10065 q^{-45} -25156 q^{-46} -24782 q^{-47} -11094 q^{-48} +8682 q^{-49} +24854 q^{-50} +25852 q^{-51} +13009 q^{-52} -6760 q^{-53} -23769 q^{-54} -26211 q^{-55} -14777 q^{-56} +4299 q^{-57} +21694 q^{-58} +25700 q^{-59} +16178 q^{-60} -1574 q^{-61} -18846 q^{-62} -24203 q^{-63} -16777 q^{-64} -1005 q^{-65} +15380 q^{-66} +21658 q^{-67} +16509 q^{-68} +3183 q^{-69} -11778 q^{-70} -18449 q^{-71} -15254 q^{-72} -4551 q^{-73} +8329 q^{-74} +14823 q^{-75} +13296 q^{-76} +5144 q^{-77} -5491 q^{-78} -11273 q^{-79} -10798 q^{-80} -4997 q^{-81} +3279 q^{-82} +8091 q^{-83} +8324 q^{-84} +4310 q^{-85} -1887 q^{-86} -5491 q^{-87} -5933 q^{-88} -3371 q^{-89} +951 q^{-90} +3542 q^{-91} +4074 q^{-92} +2435 q^{-93} -581 q^{-94} -2220 q^{-95} -2544 q^{-96} -1575 q^{-97} +327 q^{-98} +1291 q^{-99} +1583 q^{-100} +1000 q^{-101} -301 q^{-102} -792 q^{-103} -889 q^{-104} -524 q^{-105} +235 q^{-106} +425 q^{-107} +497 q^{-108} +306 q^{-109} -207 q^{-110} -261 q^{-111} -272 q^{-112} -143 q^{-113} +180 q^{-114} +139 q^{-115} +130 q^{-116} +69 q^{-117} -116 q^{-118} -66 q^{-119} -85 q^{-120} -51 q^{-121} +99 q^{-122} +49 q^{-123} +28 q^{-124} +13 q^{-125} -46 q^{-126} -8 q^{-127} -26 q^{-128} -27 q^{-129} +41 q^{-130} +13 q^{-131} +5 q^{-132} +5 q^{-133} -14 q^{-134} +4 q^{-135} -9 q^{-136} -10 q^{-137} +12 q^{-138} +2 q^{-139} - q^{-140} +3 q^{-141} -3 q^{-142} +2 q^{-143} -3 q^{-144} - 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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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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