8 4

8_3

8_5

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 4 at Knotilus!

[edit 8 4 Quick Notes]

[edit 8 4 Further Notes and Views]

Somewhat symmetric representation

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_14,10,15,9 X_10,3,11,4 X_2,13,3,14 X_12,5,13,6 X_16,8,1,7 X_4,11,5,12 X_8,16,9,15
Gauss code	1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code	6 10 12 16 14 4 2 8
Conway Notation	[413]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 9, width is 4,

Braid index is 4

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

[edit Notes on presentations of 8 4]

Knot 8_4.

A graph, knot 8_4.

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["8 4"];

In[4]:= PD[K]

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

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

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [413]

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

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{-1,-1,-1,2,-1,2,3,-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[{2, 10}, {1, 3}, {4, 2}, {3, 5}, {9, 4}, {10, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	2
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 \{3,5\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[-7][-3]
Hyperbolic Volume	5.50049
A-Polynomial	See Data:8 4/A-polynomial

[edit Notes for 8 4'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 2}
Rasmussen s-Invariant	2

[edit Notes for 8 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^2+5 t-5+5 t^{-1} -2 t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^4-3 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 19, -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-q^2+2 q-3+3 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-z^4-3 z^2-2+z^2 a^{-2} +2 a^{-2} }
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 z^7+z^7 a^{-1} +2 a^2 z^6+z^6 a^{-2} +3 z^6+3 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^4 z^4-3 a^2 z^4-5 z^4 a^{-2} -11 z^4+2 a^5 z^3-5 a^3 z^3-3 a z^3+4 z^3 a^{-1} +a^6 z^2-3 a^4 z^2-a^2 z^2+7 z^2 a^{-2} +10 z^2+2 a^3 z+a z-z a^{-1} +a^4-2 a^{-2} -2}
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^{16}+q^{10}+q^6-q^4-q^2-1- q^{-2} + q^{-4} + q^{-6} + 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^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-2 q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+q^{54}+q^{50}+2 q^{48}-q^{46}+2 q^{44}+2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}+3 q^{28}-3 q^{26}+2 q^{22}-6 q^{20}+5 q^{18}-5 q^{16}+2 q^{12}-4 q^{10}+3 q^8-4 q^6+q^4-q^2-2+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} + q^{-16} +3 q^{-18} -4 q^{-20} +4 q^{-22} -2 q^{-24} + q^{-26} +4 q^{-28} -4 q^{-30} +4 q^{-32} - q^{-34} + q^{-36} + q^{-38} -2 q^{-40} +2 q^{-42} + q^{-46} }

Further Quantum Invariants

Further quantum knot invariants for 8_4.

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^{11}-q^9+q^7- q^{-1} + q^{-3} + 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^{30}-q^{28}+2 q^{24}-2 q^{22}+q^{18}-2 q^{16}+q^{12}-q^8+q^6+2 q^4+2 q^{-2} -2 q^{-6} + q^{-8} -2 q^{-12} + q^{-14} + q^{-16} - q^{-18} + 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^{57}-q^{55}+q^{51}-q^{47}-q^{45}+q^{43}-q^{41}-q^{39}+2 q^{37}+q^{35}-2 q^{33}-q^{31}+3 q^{29}+2 q^{27}-q^{25}-2 q^{23}-q^{21}+2 q^{19}+2 q^{17}-3 q^{13}+3 q^9-3 q^5-q^3+2 q+ q^{-1} - q^{-3} - q^{-5} +2 q^{-7} +2 q^{-9} -2 q^{-13} + q^{-15} +3 q^{-17} + q^{-19} -3 q^{-21} -2 q^{-23} +2 q^{-25} +2 q^{-27} - q^{-29} -3 q^{-31} +2 q^{-35} + q^{-37} - q^{-39} - q^{-41} + 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^{92}-q^{90}+q^{86}-q^{84}+q^{82}-2 q^{80}-2 q^{74}+4 q^{72}-2 q^{66}-4 q^{64}+5 q^{62}+4 q^{60}+2 q^{58}-5 q^{56}-8 q^{54}+3 q^{52}+7 q^{50}+6 q^{48}-2 q^{46}-9 q^{44}-3 q^{42}+2 q^{40}+5 q^{38}+3 q^{36}-2 q^{34}-3 q^{32}-5 q^{30}+5 q^{26}+4 q^{24}-q^{22}-6 q^{20}-3 q^{18}+3 q^{16}+6 q^{14}+q^{12}-4 q^{10}-2 q^8+4 q^6+6 q^4-q^2-3-2 q^{-2} +3 q^{-4} +5 q^{-6} -2 q^{-8} -4 q^{-10} -5 q^{-12} +6 q^{-16} + q^{-18} -5 q^{-22} -4 q^{-24} +4 q^{-26} +4 q^{-28} +5 q^{-30} -2 q^{-32} -6 q^{-34} -2 q^{-36} + q^{-38} +7 q^{-40} +3 q^{-42} -3 q^{-44} -4 q^{-46} -4 q^{-48} +3 q^{-50} +4 q^{-52} +2 q^{-54} - q^{-56} -5 q^{-58} - q^{-60} + q^{-62} +2 q^{-64} +2 q^{-66} - q^{-68} - q^{-70} - q^{-72} + 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^{135}-q^{133}+q^{129}-q^{127}-q^{121}-q^{119}+q^{115}+2 q^{113}+2 q^{111}-q^{109}-5 q^{107}-4 q^{105}+4 q^{103}+8 q^{101}+5 q^{99}-3 q^{97}-12 q^{95}-8 q^{93}+5 q^{91}+16 q^{89}+11 q^{87}-4 q^{85}-17 q^{83}-17 q^{81}-q^{79}+18 q^{77}+20 q^{75}+4 q^{73}-14 q^{71}-21 q^{69}-11 q^{67}+7 q^{65}+19 q^{63}+14 q^{61}-q^{59}-11 q^{57}-13 q^{55}-7 q^{53}+4 q^{51}+11 q^{49}+12 q^{47}+4 q^{45}-3 q^{43}-11 q^{41}-11 q^{39}+10 q^{35}+11 q^{33}+3 q^{31}-9 q^{29}-11 q^{27}-3 q^{25}+7 q^{23}+9 q^{21}+2 q^{19}-7 q^{17}-7 q^{15}+8 q^{11}+8 q^9-2 q^7-10 q^5-10 q^3+q+12 q^{-1} +12 q^{-3} -11 q^{-7} -13 q^{-9} -2 q^{-11} +11 q^{-13} +16 q^{-15} +7 q^{-17} -5 q^{-19} -13 q^{-21} -11 q^{-23} +2 q^{-25} +12 q^{-27} +11 q^{-29} +4 q^{-31} -7 q^{-33} -13 q^{-35} -10 q^{-37} + q^{-39} +9 q^{-41} +10 q^{-43} +6 q^{-45} -4 q^{-47} -11 q^{-49} -10 q^{-51} - q^{-53} +7 q^{-55} +11 q^{-57} +8 q^{-59} - q^{-61} -9 q^{-63} -10 q^{-65} -4 q^{-67} +4 q^{-69} +9 q^{-71} +8 q^{-73} + q^{-75} -5 q^{-77} -8 q^{-79} -5 q^{-81} + q^{-83} +5 q^{-85} +6 q^{-87} +3 q^{-89} -2 q^{-91} -5 q^{-93} -3 q^{-95} - q^{-97} + q^{-99} +3 q^{-101} +2 q^{-103} - q^{-107} - q^{-109} - q^{-111} + q^{-115} }
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^{186}-q^{184}+q^{180}-q^{178}-q^{174}+q^{172}-2 q^{170}-q^{168}+3 q^{166}+2 q^{162}+q^{160}-q^{158}-7 q^{156}-3 q^{154}+6 q^{152}+5 q^{150}+7 q^{148}+q^{146}-7 q^{144}-16 q^{142}-5 q^{140}+13 q^{138}+15 q^{136}+13 q^{134}-4 q^{132}-21 q^{130}-28 q^{128}-6 q^{126}+24 q^{124}+30 q^{122}+22 q^{120}-6 q^{118}-35 q^{116}-42 q^{114}-15 q^{112}+26 q^{110}+43 q^{108}+38 q^{106}+8 q^{104}-32 q^{102}-52 q^{100}-35 q^{98}+5 q^{96}+37 q^{94}+49 q^{92}+31 q^{90}-5 q^{88}-38 q^{86}-44 q^{84}-23 q^{82}+3 q^{80}+28 q^{78}+35 q^{76}+24 q^{74}+q^{72}-19 q^{70}-25 q^{68}-24 q^{66}-10 q^{64}+9 q^{62}+26 q^{60}+25 q^{58}+12 q^{56}-8 q^{54}-25 q^{52}-25 q^{50}-11 q^{48}+13 q^{46}+22 q^{44}+18 q^{42}-16 q^{38}-17 q^{36}-7 q^{34}+12 q^{32}+16 q^{30}+7 q^{28}-9 q^{26}-16 q^{24}-9 q^{22}+5 q^{20}+22 q^{18}+20 q^{16}+2 q^{14}-19 q^{12}-24 q^{10}-15 q^8+6 q^6+28 q^4+30 q^2+10-18 q^{-2} -30 q^{-4} -28 q^{-6} -6 q^{-8} +24 q^{-10} +38 q^{-12} +24 q^{-14} -6 q^{-16} -27 q^{-18} -37 q^{-20} -23 q^{-22} +7 q^{-24} +34 q^{-26} +35 q^{-28} +13 q^{-30} -9 q^{-32} -31 q^{-34} -33 q^{-36} -15 q^{-38} +15 q^{-40} +32 q^{-42} +28 q^{-44} +17 q^{-46} -9 q^{-48} -27 q^{-50} -30 q^{-52} -12 q^{-54} +7 q^{-56} +19 q^{-58} +28 q^{-60} +16 q^{-62} - q^{-64} -20 q^{-66} -23 q^{-68} -18 q^{-70} -7 q^{-72} +14 q^{-74} +21 q^{-76} +21 q^{-78} +7 q^{-80} -5 q^{-82} -18 q^{-84} -23 q^{-86} -11 q^{-88} + q^{-90} +15 q^{-92} +17 q^{-94} +17 q^{-96} +4 q^{-98} -11 q^{-100} -15 q^{-102} -15 q^{-104} -6 q^{-106} +2 q^{-108} +14 q^{-110} +14 q^{-112} +7 q^{-114} -7 q^{-118} -10 q^{-120} -11 q^{-122} - q^{-124} +4 q^{-126} +7 q^{-128} +7 q^{-130} +4 q^{-132} -6 q^{-136} -4 q^{-138} -3 q^{-140} - q^{-142} + q^{-144} +3 q^{-146} +3 q^{-148} - q^{-154} - q^{-156} - q^{-158} + q^{-162} }

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^{16}+q^{10}+q^6-q^4-q^2-1- q^{-2} + q^{-4} + q^{-6} + 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^{44}-2 q^{42}+2 q^{40}-2 q^{38}+5 q^{36}-4 q^{34}+4 q^{32}-4 q^{30}+5 q^{28}-4 q^{26}+2 q^{24}-4 q^{22}+2 q^{20}-2 q^{18}-2 q^{16}+2 q^{14}-6 q^{12}+10 q^{10}-10 q^8+16 q^6-10 q^4+16 q^2-8+8 q^{-2} -7 q^{-4} -6 q^{-10} +5 q^{-12} -8 q^{-14} +10 q^{-16} -8 q^{-18} +6 q^{-20} -4 q^{-22} +4 q^{-24} + 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^{40}+q^{34}+q^{32}-q^{28}-q^{24}-3 q^{22}+q^{18}-q^{14}+2 q^6+3 q^4+3 q^2+2+3 q^{-2} -2 q^{-6} -2 q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} + q^{-18} + q^{-20} + q^{-24} + q^{-26} + q^{-28} }
3,0	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^{72}+q^{66}+q^{64}+q^{62}-2 q^{60}-q^{58}-q^{54}-2 q^{52}-3 q^{50}+q^{48}+3 q^{46}-4 q^{42}-4 q^{40}+3 q^{38}+6 q^{36}+4 q^{34}-q^{32}+3 q^{28}+3 q^{26}+q^{24}-2 q^{22}+q^{18}-q^{14}-2 q^8-5 q^6-3 q^4-q^2-1-3 q^{-2} -3 q^{-4} +2 q^{-6} +5 q^{-8} +6 q^{-10} +2 q^{-12} +3 q^{-14} +5 q^{-16} +6 q^{-18} +3 q^{-20} -2 q^{-22} -4 q^{-24} -3 q^{-26} - q^{-28} -2 q^{-32} -3 q^{-34} -2 q^{-36} +2 q^{-40} + q^{-42} - q^{-46} + q^{-50} + q^{-52} + q^{-54} }

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^{36}-q^{34}-q^{32}+2 q^{30}-q^{26}+2 q^{24}-q^{20}+q^{18}+q^{16}+q^{12}+q^{10}-2 q^6-q^4+q^2-2+ q^{-4} - q^{-6} + q^{-10} + q^{-14} +2 q^{-16} + 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^{21}+q^{17}+q^{13}+q^9-q^5-q^3-2 q- q^{-1} - q^{-3} + q^{-5} + q^{-7} +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^{46}-q^{42}+q^{38}+q^{36}+q^{32}+2 q^{30}-q^{26}-2 q^{20}+q^{10}+2 q^8+2 q^4+3 q^2+1-2 q^{-2} - q^{-4} -2 q^{-6} -4 q^{-8} -3 q^{-10} - q^{-12} + q^{-14} + q^{-16} +3 q^{-18} +3 q^{-20} +2 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^{26}+q^{22}+q^{20}+q^{16}+q^{12}-q^6-q^4-2 q^2-2- q^{-2} - q^{-4} + q^{-6} + q^{-8} +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^{36}-q^{34}+q^{32}-2 q^{30}+2 q^{28}-q^{26}+2 q^{24}+q^{20}+q^{18}-q^{16}+2 q^{14}-3 q^{12}+3 q^{10}-4 q^8+2 q^6-3 q^4+q^2-2+ q^{-4} - q^{-6} +2 q^{-8} - q^{-10} +2 q^{-12} - q^{-14} +2 q^{-16} + 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^{58}-q^{54}-q^{52}+2 q^{48}+q^{46}-q^{44}-q^{42}+2 q^{38}-q^{34}-q^{32}+q^{30}+q^{28}-q^{24}+2 q^{20}+q^{18}-q^{16}-q^{14}+q^{12}-q^8-q^6+q^4+q^2-2 q^{-2} +2 q^{-6} + q^{-8} - q^{-10} -2 q^{-12} + q^{-16} + q^{-18} - q^{-20} - q^{-22} + q^{-24} +2 q^{-26} + 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^{50}-q^{48}-q^{44}+2 q^{42}-q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30}+q^{24}+2 q^{20}-q^{18}+3 q^{16}-2 q^{14}+3 q^{12}-3 q^{10}+q^8-3 q^6-3 q^2- q^{-2} - q^{-4} - q^{-8} +2 q^{-10} +2 q^{-14} +3 q^{-18} +2 q^{-22} + 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^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-2 q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+q^{54}+q^{50}+2 q^{48}-q^{46}+2 q^{44}+2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}+3 q^{28}-3 q^{26}+2 q^{22}-6 q^{20}+5 q^{18}-5 q^{16}+2 q^{12}-4 q^{10}+3 q^8-4 q^6+q^4-q^2-2+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} + q^{-16} +3 q^{-18} -4 q^{-20} +4 q^{-22} -2 q^{-24} + q^{-26} +4 q^{-28} -4 q^{-30} +4 q^{-32} - q^{-34} + q^{-36} + q^{-38} -2 q^{-40} +2 q^{-42} + 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["8 4"];

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

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

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^2+5 t-5+5 t^{-1} -2 t^{-2} }

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

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^4-3 z^2+1}

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

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

Out[6]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 19, -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-q^2+2 q-3+3 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5} }

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

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-z^4-3 z^2-2+z^2 a^{-2} +2 a^{-2} }

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 z^7+z^7 a^{-1} +2 a^2 z^6+z^6 a^{-2} +3 z^6+3 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^4 z^4-3 a^2 z^4-5 z^4 a^{-2} -11 z^4+2 a^5 z^3-5 a^3 z^3-3 a z^3+4 z^3 a^{-1} +a^6 z^2-3 a^4 z^2-a^2 z^2+7 z^2 a^{-2} +10 z^2+2 a^3 z+a z-z a^{-1} +a^4-2 a^{-2} -2}

"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["8 4"];

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

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

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

Out[4]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^2+5 t-5+5 t^{-1} -2 t^{-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^3-q^2+2 q-3+3 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5} } }

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

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

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

Out[5]= {}

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

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

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(-3, 1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -12}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 72}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 114}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 54}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -96}

Failed to parse (SVG (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{496}{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{160}{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 -24}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -288}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 -1368}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -648}

Failed to parse (SVG (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{11871}{10}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2182}{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{20902}{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{1055}{6}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{3391}{10}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -2 is the signature of 8 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

0

3

2

1

-1

2

0

-3

2

0

-5

1

0

-7

1

2

1

-9

1

-1

-11

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=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{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=-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}\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=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}^{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}^{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}^{2}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_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=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}}
Failed to parse (SVG (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}-q^9-q^8+3 q^7-q^6-4 q^5+5 q^4-7 q^2+7 q+2-9 q^{-1} +7 q^{-2} +4 q^{-3} -10 q^{-4} +5 q^{-5} +5 q^{-6} -9 q^{-7} +4 q^{-8} +3 q^{-9} -6 q^{-10} +3 q^{-11} + q^{-12} -2 q^{-13} + q^{-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 q^{21}-q^{20}-q^{19}+3 q^{17}-3 q^{15}-3 q^{14}+5 q^{13}+3 q^{12}-3 q^{11}-7 q^{10}+4 q^9+7 q^8-q^7-9 q^6+q^5+9 q^4+q^3-9 q^2-2 q+9+3 q^{-1} -8 q^{-2} -5 q^{-3} +7 q^{-4} +6 q^{-5} -5 q^{-6} -8 q^{-7} +4 q^{-8} +9 q^{-9} -3 q^{-10} -8 q^{-11} + q^{-12} +8 q^{-13} -2 q^{-14} -5 q^{-15} +2 q^{-16} +4 q^{-17} -3 q^{-18} -2 q^{-19} +3 q^{-20} + q^{-21} -3 q^{-22} + q^{-24} + q^{-25} -2 q^{-26} + q^{-27} }
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}-q^{35}-q^{34}+4 q^{31}-q^{30}-2 q^{29}-2 q^{28}-4 q^{27}+8 q^{26}+2 q^{25}-3 q^{23}-11 q^{22}+8 q^{21}+3 q^{20}+6 q^{19}+q^{18}-17 q^{17}+5 q^{16}-q^{15}+10 q^{14}+8 q^{13}-18 q^{12}+5 q^{11}-9 q^{10}+9 q^9+13 q^8-17 q^7+10 q^6-15 q^5+4 q^4+14 q^3-15 q^2+17 q-17- q^{-1} +13 q^{-2} -13 q^{-3} +24 q^{-4} -19 q^{-5} -7 q^{-6} +11 q^{-7} -8 q^{-8} +29 q^{-9} -22 q^{-10} -13 q^{-11} +8 q^{-12} -3 q^{-13} +34 q^{-14} -21 q^{-15} -18 q^{-16} +3 q^{-17} - q^{-18} +35 q^{-19} -16 q^{-20} -16 q^{-21} -6 q^{-23} +29 q^{-24} -9 q^{-25} -8 q^{-26} + q^{-27} -10 q^{-28} +18 q^{-29} -6 q^{-30} - q^{-31} +3 q^{-32} -9 q^{-33} +9 q^{-34} -4 q^{-35} + q^{-36} +3 q^{-37} -5 q^{-38} +3 q^{-39} -2 q^{-40} + q^{-41} + q^{-42} -2 q^{-43} + q^{-44} }
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}-q^{54}-q^{53}+q^{50}+3 q^{49}-3 q^{47}-2 q^{46}-2 q^{45}-q^{44}+6 q^{43}+5 q^{42}-3 q^{40}-6 q^{39}-7 q^{38}+3 q^{37}+8 q^{36}+6 q^{35}+4 q^{34}-5 q^{33}-12 q^{32}-5 q^{31}+2 q^{30}+7 q^{29}+12 q^{28}+4 q^{27}-9 q^{26}-9 q^{25}-6 q^{24}-2 q^{23}+11 q^{22}+11 q^{21}+q^{20}-5 q^{19}-7 q^{18}-10 q^{17}+8 q^{15}+7 q^{14}+6 q^{13}-9 q^{11}-10 q^{10}-5 q^9+5 q^8+14 q^7+12 q^6-15 q^4-18 q^3-6 q^2+16 q+23+12 q^{-1} -15 q^{-2} -29 q^{-3} -17 q^{-4} +16 q^{-5} +31 q^{-6} +22 q^{-7} -15 q^{-8} -37 q^{-9} -24 q^{-10} +16 q^{-11} +40 q^{-12} +29 q^{-13} -17 q^{-14} -47 q^{-15} -32 q^{-16} +18 q^{-17} +52 q^{-18} +37 q^{-19} -18 q^{-20} -57 q^{-21} -43 q^{-22} +18 q^{-23} +60 q^{-24} +44 q^{-25} -10 q^{-26} -58 q^{-27} -50 q^{-28} +7 q^{-29} +54 q^{-30} +46 q^{-31} -43 q^{-33} -45 q^{-34} -5 q^{-35} +36 q^{-36} +36 q^{-37} +7 q^{-38} -25 q^{-39} -29 q^{-40} -7 q^{-41} +17 q^{-42} +20 q^{-43} +7 q^{-44} -12 q^{-45} -14 q^{-46} -2 q^{-47} +6 q^{-48} +7 q^{-49} +3 q^{-50} -3 q^{-51} -6 q^{-52} + q^{-53} +2 q^{-54} - q^{-55} +2 q^{-56} + q^{-57} -3 q^{-58} + q^{-59} + q^{-60} -2 q^{-61} + q^{-62} + q^{-63} -2 q^{-64} + q^{-65} }
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}-q^{77}-q^{76}+q^{73}+4 q^{71}-q^{70}-3 q^{69}-2 q^{68}-2 q^{67}-2 q^{65}+10 q^{64}+3 q^{63}-2 q^{61}-5 q^{60}-5 q^{59}-12 q^{58}+11 q^{57}+6 q^{56}+7 q^{55}+5 q^{54}+2 q^{53}-5 q^{52}-24 q^{51}+3 q^{50}-3 q^{49}+7 q^{48}+9 q^{47}+17 q^{46}+8 q^{45}-24 q^{44}+q^{43}-17 q^{42}-5 q^{41}-3 q^{40}+22 q^{39}+21 q^{38}-12 q^{37}+15 q^{36}-17 q^{35}-12 q^{34}-24 q^{33}+11 q^{32}+16 q^{31}-9 q^{30}+34 q^{29}-33 q^{26}-q^{25}-3 q^{24}-27 q^{23}+37 q^{22}+18 q^{21}+26 q^{20}-22 q^{19}+3 q^{18}-20 q^{17}-57 q^{16}+19 q^{15}+20 q^{14}+48 q^{13}+25 q^{11}-21 q^{10}-84 q^9-11 q^8+6 q^7+58 q^6+21 q^5+55 q^4-7 q^3-98 q^2-41 q-16+56 q^{-1} +33 q^{-2} +83 q^{-3} +13 q^{-4} -100 q^{-5} -63 q^{-6} -37 q^{-7} +47 q^{-8} +38 q^{-9} +104 q^{-10} +31 q^{-11} -98 q^{-12} -80 q^{-13} -51 q^{-14} +40 q^{-15} +45 q^{-16} +120 q^{-17} +40 q^{-18} -102 q^{-19} -99 q^{-20} -61 q^{-21} +41 q^{-22} +61 q^{-23} +138 q^{-24} +44 q^{-25} -111 q^{-26} -123 q^{-27} -75 q^{-28} +41 q^{-29} +78 q^{-30} +158 q^{-31} +57 q^{-32} -110 q^{-33} -140 q^{-34} -94 q^{-35} +27 q^{-36} +77 q^{-37} +164 q^{-38} +77 q^{-39} -87 q^{-40} -129 q^{-41} -101 q^{-42} +2 q^{-43} +51 q^{-44} +143 q^{-45} +83 q^{-46} -54 q^{-47} -93 q^{-48} -83 q^{-49} -10 q^{-50} +19 q^{-51} +103 q^{-52} +66 q^{-53} -34 q^{-54} -53 q^{-55} -53 q^{-56} -5 q^{-57} +2 q^{-58} +62 q^{-59} +39 q^{-60} -27 q^{-61} -24 q^{-62} -25 q^{-63} +3 q^{-64} -4 q^{-65} +32 q^{-66} +17 q^{-67} -20 q^{-68} -7 q^{-69} -8 q^{-70} +5 q^{-71} -6 q^{-72} +14 q^{-73} +6 q^{-74} -11 q^{-75} + q^{-76} -2 q^{-77} +3 q^{-78} -5 q^{-79} +5 q^{-80} +2 q^{-81} -5 q^{-82} +3 q^{-83} - q^{-84} + q^{-85} -2 q^{-86} + q^{-87} + q^{-88} -2 q^{-89} + q^{-90} }
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}-q^{104}-q^{103}+q^{100}+q^{98}+3 q^{97}-q^{96}-3 q^{95}-2 q^{94}-3 q^{93}+q^{92}+9 q^{89}+4 q^{88}-2 q^{86}-8 q^{85}-3 q^{84}-6 q^{83}-8 q^{82}+9 q^{81}+9 q^{80}+9 q^{79}+10 q^{78}-5 q^{77}-9 q^{75}-22 q^{74}-4 q^{73}-2 q^{72}+7 q^{71}+20 q^{70}+7 q^{69}+16 q^{68}+6 q^{67}-21 q^{66}-10 q^{65}-21 q^{64}-16 q^{63}+10 q^{62}+q^{61}+25 q^{60}+27 q^{59}+10 q^{57}-16 q^{56}-30 q^{55}-9 q^{54}-26 q^{53}+5 q^{52}+24 q^{51}+6 q^{50}+37 q^{49}+14 q^{48}-9 q^{47}-41 q^{45}-23 q^{44}-5 q^{43}-22 q^{42}+28 q^{41}+31 q^{40}+24 q^{39}+43 q^{38}-17 q^{37}-21 q^{36}-21 q^{35}-67 q^{34}-15 q^{33}+3 q^{32}+26 q^{31}+80 q^{30}+32 q^{29}+21 q^{28}-85 q^{26}-59 q^{25}-52 q^{24}-20 q^{23}+81 q^{22}+67 q^{21}+76 q^{20}+55 q^{19}-63 q^{18}-71 q^{17}-102 q^{16}-85 q^{15}+40 q^{14}+67 q^{13}+114 q^{12}+116 q^{11}-13 q^{10}-51 q^9-122 q^8-143 q^7-17 q^6+34 q^5+123 q^4+163 q^3+44 q^2-9 q-115-180 q^{-1} -74 q^{-2} -14 q^{-3} +108 q^{-4} +190 q^{-5} +95 q^{-6} +41 q^{-7} -92 q^{-8} -200 q^{-9} -120 q^{-10} -61 q^{-11} +82 q^{-12} +201 q^{-13} +136 q^{-14} +84 q^{-15} -68 q^{-16} -210 q^{-17} -153 q^{-18} -94 q^{-19} +62 q^{-20} +210 q^{-21} +166 q^{-22} +108 q^{-23} -60 q^{-24} -223 q^{-25} -178 q^{-26} -110 q^{-27} +63 q^{-28} +232 q^{-29} +194 q^{-30} +118 q^{-31} -72 q^{-32} -250 q^{-33} -214 q^{-34} -122 q^{-35} +80 q^{-36} +267 q^{-37} +233 q^{-38} +134 q^{-39} -77 q^{-40} -281 q^{-41} -258 q^{-42} -151 q^{-43} +74 q^{-44} +284 q^{-45} +268 q^{-46} +171 q^{-47} -50 q^{-48} -270 q^{-49} -278 q^{-50} -191 q^{-51} +25 q^{-52} +248 q^{-53} +265 q^{-54} +196 q^{-55} +4 q^{-56} -205 q^{-57} -239 q^{-58} -200 q^{-59} -29 q^{-60} +171 q^{-61} +205 q^{-62} +179 q^{-63} +37 q^{-64} -128 q^{-65} -161 q^{-66} -154 q^{-67} -43 q^{-68} +101 q^{-69} +123 q^{-70} +122 q^{-71} +32 q^{-72} -79 q^{-73} -87 q^{-74} -88 q^{-75} -19 q^{-76} +61 q^{-77} +56 q^{-78} +62 q^{-79} +12 q^{-80} -53 q^{-81} -39 q^{-82} -36 q^{-83} +3 q^{-84} +40 q^{-85} +16 q^{-86} +23 q^{-87} -2 q^{-88} -32 q^{-89} -10 q^{-90} -14 q^{-91} +9 q^{-92} +23 q^{-93} + q^{-94} +5 q^{-95} -4 q^{-96} -14 q^{-97} + q^{-98} -6 q^{-99} +4 q^{-100} +12 q^{-101} -4 q^{-102} + q^{-103} - q^{-104} -4 q^{-105} +3 q^{-106} -4 q^{-107} + q^{-108} +5 q^{-109} -4 q^{-110} + q^{-111} + q^{-112} - q^{-113} + q^{-114} -2 q^{-115} + q^{-116} + q^{-117} -2 q^{-118} + q^{-119} }

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages

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

See/edit the Rolfsen_Splice_Base (expert).

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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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