8 1

7_7

8_2

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 1 at Knotilus!

[edit 8 1 Quick Notes]

[edit 8 1 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_9,12,10,13 X_3,11,4,10 X_11,3,12,2 X_5,16,6,1 X_7,14,8,15 X_13,8,14,9 X_15,6,16,7
Gauss code	-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5
Dowker-Thistlethwaite code	4 10 16 14 12 2 8 6
Conway Notation	[62]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 10, width is 5,

Braid index is 5

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

[edit Notes on presentations of 8 1]

Knot 8_1.

A graph, knot 8_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.org/wiki/KnotTheory.

In[3]:= K = Knot["8 1"];

In[4]:= PD[K]

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

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

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [62]

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]= [math]\displaystyle{ \textrm{BR}(5,\{-1,-1,-2,1,-2,-3,2,4,-3,4\}) }[/math]

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]= { 5, 10, 5 }

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	1
Bridge index	2
Super bridge index	[math]\displaystyle{ \{4,5\} }[/math]
Nakanishi index	1
Maximal Thurston-Bennequin number	[-7][-3]
Hyperbolic Volume	3.42721
A-Polynomial	See Data:8 1/A-polynomial

[edit Notes for 8 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	[math]\displaystyle{ 1 }[/math]
Topological 4 genus	[math]\displaystyle{ 1 }[/math]
Concordance genus	[math]\displaystyle{ 1 }[/math]
Rasmussen s-Invariant	0

[edit Notes for 8 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t+7-3 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-3 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 13, 0 }
Jones polynomial	[math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-z^2 a^4-a^4-z^2 a^2-z^2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-4 a^3 z^5+a z^5-5 a^6 z^4-8 a^4 z^4-2 a^2 z^4+z^4+7 a^5 z^3+5 a^3 z^3-a z^3+z^3 a^{-1} +6 a^6 z^2+7 a^4 z^2+z^2 a^{-2} -3 a^5 z-3 a^3 z-a^6-a^4- a^{-2} }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}-q^{12}-q^{10}+ q^{-2} + q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}+q^{90}-q^{88}+2 q^{80}-2 q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2 q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^8+ q^{-2} - q^{-6} + q^{-10} + q^{-14} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-q^7+ q^{-1} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{38}-q^{34}-q^{28}+q^{24}+q^{12}+q^{10}-1+ q^{-8} + q^{-14} }[/math]
3	[math]\displaystyle{ q^{75}-q^{71}-q^{69}+q^{65}-q^{61}+q^{57}+q^{55}-q^{51}+q^{37}+q^{35}-q^{31}-q^{25}-q^{23}-q^{17}+q^{13}+2 q^{11}+q^9+q^7-q^5+q+ q^{-1} - q^{-5} + q^{-9} - q^{-13} - q^{-15} + q^{-19} + q^{-27} }[/math]
4	[math]\displaystyle{ q^{124}-q^{120}-q^{118}-q^{116}+q^{114}+q^{112}+q^{110}-2 q^{106}+q^{102}+q^{100}+q^{98}-q^{96}-q^{94}-q^{92}+q^{88}+q^{74}+q^{72}-q^{68}-2 q^{66}+q^{62}-q^{58}-2 q^{56}+2 q^{52}+q^{50}-q^{46}+2 q^{42}+q^{40}+q^{32}-q^{28}-q^{26}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}+q^{10}+2 q^8+q^6-q^4-2 q^2+2 q^{-2} +4 q^{-4} + q^{-6} -2 q^{-8} - q^{-10} + q^{-12} +3 q^{-14} -2 q^{-18} - q^{-20} +2 q^{-24} - q^{-28} - q^{-30} - q^{-32} + q^{-34} + q^{-44} }[/math]
5	[math]\displaystyle{ q^{185}-q^{181}-q^{179}-q^{177}+q^{173}+2 q^{171}+q^{169}-q^{165}-2 q^{163}-q^{161}+q^{159}+2 q^{157}+q^{155}-q^{151}-2 q^{149}-q^{147}+q^{143}+q^{141}+q^{139}-q^{135}+q^{123}+q^{121}-q^{117}-2 q^{115}-2 q^{113}+2 q^{109}+2 q^{107}-2 q^{103}-2 q^{101}-q^{99}+2 q^{97}+4 q^{95}+2 q^{93}-q^{91}-2 q^{89}-2 q^{87}+2 q^{83}+2 q^{81}-2 q^{77}-2 q^{75}+q^{71}+2 q^{69}+q^{67}-q^{65}-2 q^{63}-q^{61}+q^{57}+q^{55}-q^{53}-2 q^{51}-q^{49}+q^{45}-q^{41}+q^{37}+3 q^{35}+2 q^{33}-q^{25}+q^{23}+q^{21}+q^{19}+q^{17}-3 q^{13}-3 q^{11}-q^9+q^7+3 q^5+2 q^3-3 q^{-1} -3 q^{-3} +3 q^{-7} +3 q^{-9} -2 q^{-13} -2 q^{-15} +3 q^{-19} +2 q^{-21} -2 q^{-25} +2 q^{-29} +2 q^{-31} + q^{-33} - q^{-35} -2 q^{-37} - q^{-39} + q^{-41} + q^{-43} - q^{-49} - q^{-51} + q^{-65} }[/math]
6	[math]\displaystyle{ q^{258}-q^{254}-q^{252}-q^{250}+2 q^{244}+2 q^{242}+q^{240}-q^{236}-2 q^{234}-3 q^{232}+q^{228}+2 q^{226}+2 q^{224}+q^{222}-3 q^{218}-2 q^{216}-q^{214}+q^{210}+2 q^{208}+2 q^{206}-q^{200}-q^{198}-q^{196}+q^{192}+q^{184}+q^{182}-q^{178}-2 q^{176}-2 q^{174}-2 q^{172}+q^{170}+3 q^{168}+3 q^{166}+2 q^{164}-q^{162}-3 q^{160}-4 q^{158}-q^{156}+2 q^{154}+4 q^{152}+5 q^{150}+2 q^{148}-2 q^{146}-5 q^{144}-4 q^{142}-2 q^{140}+q^{138}+4 q^{136}+3 q^{134}+q^{132}-2 q^{130}-3 q^{128}-3 q^{126}-q^{124}+3 q^{122}+3 q^{120}+3 q^{118}+q^{116}-q^{114}-3 q^{112}-4 q^{110}-q^{108}+q^{106}+2 q^{104}+2 q^{102}+q^{100}-2 q^{98}-4 q^{96}-2 q^{94}+q^{92}+3 q^{90}+3 q^{88}+2 q^{86}-q^{84}-4 q^{82}-q^{80}+2 q^{78}+3 q^{76}+3 q^{74}+2 q^{72}-q^{70}-3 q^{68}-q^{66}+q^{64}+q^{62}-q^{58}-2 q^{56}-2 q^{54}+q^{50}-q^{46}-2 q^{44}-2 q^{42}-q^{40}+q^{38}+2 q^{36}+3 q^{34}+2 q^{32}+q^{30}-q^{26}-3 q^{24}-3 q^{22}+3 q^{18}+4 q^{16}+4 q^{14}+3 q^{12}-2 q^{10}-4 q^8-4 q^6-q^4+2 q^2+5+6 q^{-2} + q^{-4} -2 q^{-6} -5 q^{-8} -4 q^{-10} -2 q^{-12} +2 q^{-14} +4 q^{-16} + q^{-18} -2 q^{-22} - q^{-24} - q^{-26} + q^{-30} - q^{-32} + q^{-34} + q^{-36} +2 q^{-38} - q^{-42} - q^{-44} -2 q^{-46} + q^{-48} +2 q^{-50} +3 q^{-52} + q^{-54} - q^{-58} -2 q^{-60} + q^{-66} - q^{-74} - q^{-78} + q^{-90} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}-q^{12}-q^{10}+ q^{-2} + q^{-6} + q^{-8} }[/math]
1,1	[math]\displaystyle{ q^{52}+2 q^{48}-2 q^{46}+2 q^{44}-4 q^{42}+2 q^{40}-2 q^{38}-2 q^{32}+2 q^{30}-3 q^{28}+4 q^{26}-2 q^{24}+4 q^{22}-2 q^{20}+4 q^{18}+2 q^{14}+2 q^{12}-2 q^6-2 q^4-2 q^2-2+ q^{-4} +2 q^{-8} +2 q^{-12} +2 q^{-16} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{52}+q^{50}+q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{18}+2 q^{16}+q^{14}+q^{12}+q^{10}-q^4-2 q^2-2- q^{-2} + q^{-4} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}+q^{36}-q^{32}-q^{30}-q^{28}+q^{24}+2 q^{22}+q^{20}+2 q^{18}-q^{14}-q^{12}-q^{10}-q^8-q^6+ q^{-4} + q^{-8} +2 q^{-10} + q^{-12} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ q^{27}+q^{25}+q^{23}-q^{17}-q^{15}-q^{13}+ q^{-3} + q^{-7} + q^{-9} + q^{-11} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{40}+q^{36}+q^{32}-q^{30}+q^{28}-q^{24}-q^{20}-2 q^{16}+q^{14}-q^{12}+q^{10}-q^8+q^6+ q^{-4} + q^{-8} + q^{-12} + q^{-16} }[/math]
1,0	[math]\displaystyle{ q^{66}+q^{58}-q^{54}-q^{52}-q^{46}-q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+q^{30}+q^{28}-q^{18}-q^{16}-q^{10}-q^8+ q^{-6} + q^{-14} + q^{-16} + q^{-18} + q^{-26} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{94}+q^{90}-q^{88}+2 q^{80}-2 q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2 q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^8+ q^{-2} - q^{-6} + q^{-10} + q^{-14} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} }[/math]

.

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 1"];

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

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

Out[4]= [math]\displaystyle{ -3 t+7-3 t^{-1} }[/math]

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

Out[5]= [math]\displaystyle{ 1-3 z^2 }[/math]

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]= [math]\displaystyle{ \{1\} }[/math]

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

Out[7]= { 13, 0 }

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

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

Out[8]= [math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math]

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

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

Out[9]= [math]\displaystyle{ a^6-z^2 a^4-a^4-z^2 a^2-z^2+ a^{-2} }[/math]

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

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

Out[10]= [math]\displaystyle{ a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-4 a^3 z^5+a z^5-5 a^6 z^4-8 a^4 z^4-2 a^2 z^4+z^4+7 a^5 z^3+5 a^3 z^3-a z^3+z^3 a^{-1} +6 a^6 z^2+7 a^4 z^2+z^2 a^{-2} -3 a^5 z-3 a^3 z-a^6-a^4- a^{-2} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n70,}

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 1"];

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]= { [math]\displaystyle{ -3 t+7-3 t^{-1} }[/math], [math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math] }

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]= {K11n70,}

Vassiliev invariants

V₂ and V₃:

(-3, 3)

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

[math]\displaystyle{ -12 }[/math]

[math]\displaystyle{ 24 }[/math]

[math]\displaystyle{ 72 }[/math]

[math]\displaystyle{ 66 }[/math]

[math]\displaystyle{ 38 }[/math]

[math]\displaystyle{ -288 }[/math]

[math]\displaystyle{ -432 }[/math]

[math]\displaystyle{ -96 }[/math]

[math]\displaystyle{ -104 }[/math]

[math]\displaystyle{ -288 }[/math]

[math]\displaystyle{ 288 }[/math]

[math]\displaystyle{ -792 }[/math]

[math]\displaystyle{ -456 }[/math]

[math]\displaystyle{ \frac{1009}{10} }[/math]

[math]\displaystyle{ \frac{2626}{15} }[/math]

[math]\displaystyle{ -\frac{6182}{15} }[/math]

[math]\displaystyle{ \frac{943}{6} }[/math]

[math]\displaystyle{ -\frac{1551}{10} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 8 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

5

1

3

0

1

2

1

-1

1

0

-3

1

0

-5

1

0

-7

1

-1

-9

1

0

-11

0

-13

1

Integral Khovanov Homology

(db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]	[math]\displaystyle{ i=-1 }[/math]	[math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-6 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^6-q^5+2 q^3-2 q^2+2-3 q^{-1} + q^{-2} +2 q^{-3} -3 q^{-4} + q^{-5} +3 q^{-6} -3 q^{-7} +3 q^{-9} -3 q^{-10} +3 q^{-12} -2 q^{-13} - q^{-14} +2 q^{-15} - q^{-16} - q^{-17} + q^{-18} }[/math]
3	[math]\displaystyle{ q^{12}-q^{11}+2 q^8-2 q^7-q^6+3 q^4-q^3-2 q^2-q+4-2 q^{-2} -2 q^{-3} +3 q^{-4} +2 q^{-5} -2 q^{-6} - q^{-7} +2 q^{-8} + q^{-9} -3 q^{-10} +2 q^{-12} -3 q^{-14} + q^{-15} +2 q^{-16} - q^{-17} -2 q^{-18} +2 q^{-19} +2 q^{-20} -2 q^{-21} -2 q^{-22} +2 q^{-23} +2 q^{-24} -2 q^{-25} -2 q^{-26} + q^{-27} +3 q^{-28} - q^{-29} -2 q^{-30} +2 q^{-32} - q^{-34} - q^{-35} + q^{-36} }[/math]
4	[math]\displaystyle{ q^{20}-q^{19}+2 q^{15}-3 q^{14}+q^{11}+4 q^{10}-5 q^9-q^8-q^7+3 q^6+7 q^5-7 q^4-3 q^3-2 q^2+6 q+10-9 q^{-1} -5 q^{-2} -4 q^{-3} +7 q^{-4} +12 q^{-5} -8 q^{-6} -6 q^{-7} -6 q^{-8} +7 q^{-9} +12 q^{-10} -8 q^{-11} -5 q^{-12} -5 q^{-13} +6 q^{-14} +11 q^{-15} -8 q^{-16} -4 q^{-17} -4 q^{-18} +5 q^{-19} +11 q^{-20} -8 q^{-21} -3 q^{-22} -3 q^{-23} +3 q^{-24} +10 q^{-25} -7 q^{-26} -2 q^{-27} -2 q^{-28} + q^{-29} +8 q^{-30} -6 q^{-31} - q^{-32} - q^{-33} +6 q^{-35} -5 q^{-36} +5 q^{-40} -5 q^{-41} +5 q^{-45} -4 q^{-46} - q^{-47} - q^{-48} +5 q^{-50} -2 q^{-51} - q^{-52} - q^{-53} - q^{-54} +3 q^{-55} - q^{-58} - q^{-59} + q^{-60} }[/math]
5	[math]\displaystyle{ q^{30}-q^{29}+q^{24}-2 q^{23}+q^{21}+q^{19}+q^{18}-4 q^{17}-q^{16}+2 q^{15}+2 q^{14}+2 q^{13}+q^{12}-6 q^{11}-3 q^{10}+4 q^9+4 q^8+3 q^7-2 q^6-8 q^5-3 q^4+6 q^3+7 q^2+3 q-5-11 q^{-1} -3 q^{-2} +9 q^{-3} +9 q^{-4} +4 q^{-5} -7 q^{-6} -13 q^{-7} -5 q^{-8} +9 q^{-9} +12 q^{-10} +5 q^{-11} -7 q^{-12} -13 q^{-13} -5 q^{-14} +7 q^{-15} +13 q^{-16} +5 q^{-17} -7 q^{-18} -11 q^{-19} -4 q^{-20} +5 q^{-21} +12 q^{-22} +4 q^{-23} -6 q^{-24} -10 q^{-25} -5 q^{-26} +4 q^{-27} +11 q^{-28} +5 q^{-29} -4 q^{-30} -10 q^{-31} -6 q^{-32} +3 q^{-33} +10 q^{-34} +6 q^{-35} -2 q^{-36} -9 q^{-37} -7 q^{-38} +2 q^{-39} +8 q^{-40} +6 q^{-41} -7 q^{-43} -7 q^{-44} +6 q^{-46} +6 q^{-47} + q^{-48} -4 q^{-49} -5 q^{-50} -2 q^{-51} +3 q^{-52} +5 q^{-53} + q^{-54} -2 q^{-55} -3 q^{-56} -2 q^{-57} + q^{-58} +3 q^{-59} + q^{-60} - q^{-61} -2 q^{-62} - q^{-63} + q^{-64} +2 q^{-65} + q^{-66} - q^{-67} -2 q^{-68} - q^{-69} +3 q^{-71} +2 q^{-72} - q^{-73} -2 q^{-74} -2 q^{-75} - q^{-76} +2 q^{-77} +3 q^{-78} - q^{-80} - q^{-81} -2 q^{-82} +2 q^{-84} + q^{-85} - q^{-88} - q^{-89} + q^{-90} }[/math]
6	[math]\displaystyle{ q^{42}-q^{41}-q^{36}+2 q^{35}-2 q^{34}+q^{33}+q^{30}-2 q^{29}+2 q^{28}-4 q^{27}+2 q^{26}+q^{25}+q^{24}+3 q^{23}-3 q^{22}+q^{21}-7 q^{20}+3 q^{19}+q^{18}+2 q^{17}+5 q^{16}-4 q^{15}+q^{14}-9 q^{13}+5 q^{12}+q^{10}+5 q^9-5 q^8+3 q^7-8 q^6+8 q^5-2 q^4-3 q^3+3 q^2-6 q+6-5 q^{-1} +13 q^{-2} -3 q^{-3} -6 q^{-4} -9 q^{-6} +6 q^{-7} -3 q^{-8} +18 q^{-9} -2 q^{-10} -6 q^{-11} - q^{-12} -12 q^{-13} +3 q^{-14} -3 q^{-15} +20 q^{-16} - q^{-17} -5 q^{-18} -11 q^{-20} +2 q^{-21} -4 q^{-22} +18 q^{-23} -2 q^{-24} -5 q^{-25} + q^{-26} -10 q^{-27} +3 q^{-28} -5 q^{-29} +16 q^{-30} -2 q^{-31} -4 q^{-32} +2 q^{-33} -9 q^{-34} +3 q^{-35} -7 q^{-36} +14 q^{-37} - q^{-39} +3 q^{-40} -9 q^{-41} +2 q^{-42} -10 q^{-43} +11 q^{-44} +3 q^{-45} +2 q^{-46} +4 q^{-47} -9 q^{-48} -13 q^{-50} +9 q^{-51} +5 q^{-52} +5 q^{-53} +5 q^{-54} -9 q^{-55} - q^{-56} -15 q^{-57} +6 q^{-58} +6 q^{-59} +7 q^{-60} +7 q^{-61} -7 q^{-62} - q^{-63} -15 q^{-64} +2 q^{-65} +4 q^{-66} +7 q^{-67} +8 q^{-68} -4 q^{-69} + q^{-70} -14 q^{-71} - q^{-72} + q^{-73} +5 q^{-74} +7 q^{-75} - q^{-76} +5 q^{-77} -11 q^{-78} -2 q^{-79} - q^{-80} +2 q^{-81} +4 q^{-82} +7 q^{-84} -8 q^{-85} - q^{-86} - q^{-87} + q^{-89} +7 q^{-91} -7 q^{-92} +7 q^{-98} -6 q^{-99} - q^{-100} - q^{-101} + q^{-104} +7 q^{-105} -4 q^{-106} - q^{-107} -2 q^{-108} - q^{-109} - q^{-110} +6 q^{-112} - q^{-113} - q^{-115} - q^{-116} -2 q^{-117} - q^{-118} +3 q^{-119} + q^{-121} - q^{-124} - q^{-125} + q^{-126} }[/math]

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

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Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

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