8 6

8_5

8_7

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 6 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X7,16,8,1 X15,6,16,7 X13,8,14,9
Gauss code	-1, 4, -3, 1, -5, 7, -6, 8, -2, 3, -4, 2, -8, 5, -7, 6
Dowker-Thistlethwaite code	4 10 14 16 12 2 8 6
Conway Notation	[332]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{10, 3}, {4, 2}, {3, 9}, {1, 4}, {8, 10}, {9, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 1}]

[edit Notes on presentations of 8 6]

Knot 8_6.

A graph, knot 8_6.

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 6"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X7,16,8,1 X15,6,16,7 X13,8,14,9

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -5, 7, -6, 8, -2, 3, -4, 2, -8, 5, -7, 6

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 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]=

[332]

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]=

$\textrm{BR}(4,\{-1,-1,-1,-1,-2,1,3,-2,3\})$

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[{10, 3}, {4, 2}, {3, 9}, {1, 4}, {8, 10}, {9, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 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 $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-1] Hyperbolic Volume 7.47524 A-Polynomial See Data:8 6/A-polynomial

[edit Notes for 8 6's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 8 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2 t^2+6 t-7+6 t^{-1} -2 t^{-2}$
Conway polynomial	$-2 z^4-2 z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, -2 }
Jones polynomial	$q-1+3 q^{-1} -4 q^{-2} +4 q^{-3} -4 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+2$
Kauffman polynomial (db, data sources)	$z^4 a^8-2 z^2 a^8+2 z^5 a^7-4 z^3 a^7+z a^7+2 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-z^5 a^5+2 z^3 a^5-z a^5+3 z^6 a^4-6 z^4 a^4+6 z^2 a^4-a^4+z^7 a^3-2 z^5 a^3+5 z^3 a^3-3 z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2$
The A2 invariant	$q^{22}+q^{16}-q^{14}-q^{10}-q^8-q^4+2 q^2+1+ q^{-2} + q^{-4}$
The G2 invariant	$q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-3 q^{102}+6 q^{100}-7 q^{98}+7 q^{96}-4 q^{94}-2 q^{92}+7 q^{90}-9 q^{88}+11 q^{86}-6 q^{84}+q^{82}+5 q^{80}-7 q^{78}+7 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+2 q^{66}+4 q^{64}-9 q^{62}+12 q^{60}-10 q^{58}+5 q^{56}+q^{54}-10 q^{52}+13 q^{50}-13 q^{48}+10 q^{46}-5 q^{44}-3 q^{42}+7 q^{40}-10 q^{38}+6 q^{36}-3 q^{34}-4 q^{32}+5 q^{30}-5 q^{28}-q^{26}+6 q^{24}-8 q^{22}+8 q^{20}-6 q^{18}-q^{16}+6 q^{14}-8 q^{12}+10 q^{10}-5 q^8+3 q^6+2 q^4-2 q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 8_6.

A1 Invariants.

Weight	Invariant
1	$q^{15}-q^{13}+q^{11}-q^9-q^3+2 q+ q^{-3}$
2	$q^{42}-q^{40}-q^{38}+3 q^{36}-q^{34}-4 q^{32}+3 q^{30}+q^{28}-4 q^{26}+3 q^{24}+2 q^{22}-2 q^{20}+q^{16}+q^{14}-3 q^{12}+q^{10}+4 q^8-4 q^6-q^4+4 q^2-2- q^{-2} +2 q^{-4} + q^{-10}$
3	$q^{81}-q^{79}-q^{77}+q^{75}+2 q^{73}-q^{71}-5 q^{69}+7 q^{65}+3 q^{63}-7 q^{61}-7 q^{59}+6 q^{57}+10 q^{55}-5 q^{53}-10 q^{51}+3 q^{49}+12 q^{47}-q^{45}-10 q^{43}-q^{41}+7 q^{39}+q^{37}-5 q^{35}-3 q^{33}+q^{31}+5 q^{29}+q^{27}-6 q^{25}-4 q^{23}+8 q^{21}+7 q^{19}-7 q^{17}-10 q^{15}+8 q^{13}+10 q^{11}-3 q^9-10 q^7+q^5+9 q^3+q-5 q^{-1} -2 q^{-3} +3 q^{-5} + q^{-7} - q^{-9} - q^{-11} + q^{-13} + q^{-21}$
4	$q^{132}-q^{130}-q^{128}+q^{126}+2 q^{122}-3 q^{120}-3 q^{118}+2 q^{116}+2 q^{114}+9 q^{112}-3 q^{110}-11 q^{108}-6 q^{106}-q^{104}+20 q^{102}+10 q^{100}-7 q^{98}-19 q^{96}-19 q^{94}+20 q^{92}+26 q^{90}+9 q^{88}-21 q^{86}-36 q^{84}+6 q^{82}+28 q^{80}+24 q^{78}-11 q^{76}-39 q^{74}-5 q^{72}+20 q^{70}+25 q^{68}-q^{66}-26 q^{64}-9 q^{62}+8 q^{60}+17 q^{58}+5 q^{56}-10 q^{54}-10 q^{52}-2 q^{50}+9 q^{48}+12 q^{46}+5 q^{44}-16 q^{42}-15 q^{40}+3 q^{38}+19 q^{36}+19 q^{34}-20 q^{32}-28 q^{30}-8 q^{28}+24 q^{26}+36 q^{24}-13 q^{22}-30 q^{20}-19 q^{18}+14 q^{16}+38 q^{14}+4 q^{12}-16 q^{10}-23 q^8-4 q^6+23 q^4+10 q^2-12 q^{-2} -9 q^{-4} +6 q^{-6} +4 q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} + q^{-16} - q^{-18} +2 q^{-20} - q^{-24} + q^{-26} - q^{-28} + q^{-36}$
5	$q^{195}-q^{193}-q^{191}+q^{189}-q^{181}-2 q^{179}+2 q^{177}+5 q^{175}+2 q^{173}-q^{171}-6 q^{169}-9 q^{167}-5 q^{165}+8 q^{163}+17 q^{161}+13 q^{159}-20 q^{155}-28 q^{153}-17 q^{151}+16 q^{149}+42 q^{147}+37 q^{145}+3 q^{143}-45 q^{141}-63 q^{139}-29 q^{137}+37 q^{135}+81 q^{133}+58 q^{131}-18 q^{129}-87 q^{127}-86 q^{125}-11 q^{123}+82 q^{121}+106 q^{119}+33 q^{117}-65 q^{115}-108 q^{113}-54 q^{111}+47 q^{109}+105 q^{107}+63 q^{105}-30 q^{103}-86 q^{101}-64 q^{99}+13 q^{97}+69 q^{95}+56 q^{93}-2 q^{91}-50 q^{89}-47 q^{87}-6 q^{85}+30 q^{83}+38 q^{81}+12 q^{79}-18 q^{77}-30 q^{75}-21 q^{73}+4 q^{71}+28 q^{69}+32 q^{67}+8 q^{65}-27 q^{63}-44 q^{61}-23 q^{59}+29 q^{57}+61 q^{55}+40 q^{53}-24 q^{51}-78 q^{49}-61 q^{47}+20 q^{45}+90 q^{43}+80 q^{41}-93 q^{37}-105 q^{35}-14 q^{33}+81 q^{31}+106 q^{29}+41 q^{27}-59 q^{25}-106 q^{23}-60 q^{21}+32 q^{19}+85 q^{17}+68 q^{15}-58 q^{11}-64 q^9-19 q^7+34 q^5+49 q^3+28 q-7 q^{-1} -31 q^{-3} -26 q^{-5} -4 q^{-7} +14 q^{-9} +18 q^{-11} +10 q^{-13} -5 q^{-15} -10 q^{-17} -7 q^{-19} -2 q^{-21} +5 q^{-23} +6 q^{-25} + q^{-27} - q^{-29} - q^{-31} -3 q^{-33} +2 q^{-37} + q^{-43} - q^{-45} - q^{-47} + q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}+q^{16}-q^{14}-q^{10}-q^8-q^4+2 q^2+1+ q^{-2} + q^{-4}$
1,1	$q^{60}-2 q^{58}+4 q^{56}-8 q^{54}+13 q^{52}-16 q^{50}+22 q^{48}-26 q^{46}+25 q^{44}-24 q^{42}+16 q^{40}-8 q^{38}-5 q^{36}+18 q^{34}-30 q^{32}+40 q^{30}-46 q^{28}+50 q^{26}-44 q^{24}+42 q^{22}-28 q^{20}+18 q^{18}-4 q^{16}-8 q^{14}+15 q^{12}-24 q^{10}+22 q^8-24 q^6+18 q^4-16 q^2+12-6 q^{-2} +8 q^{-4} -2 q^{-6} +4 q^{-8} + q^{-12}$
2,0	$q^{56}+q^{48}-3 q^{44}-q^{42}+q^{40}-2 q^{36}+2 q^{32}-q^{28}+2 q^{26}+2 q^{24}+2 q^{20}+2 q^{18}-q^{16}+q^{12}-q^{10}-4 q^8-2 q^6+q^4-2 q^2-1+2 q^{-2} +3 q^{-4} + q^{-6} + q^{-8} + q^{-10} + q^{-12}$
3,0	$q^{102}+q^{92}-2 q^{90}-2 q^{88}-3 q^{86}+q^{84}+5 q^{82}+2 q^{80}-2 q^{78}-7 q^{76}-q^{74}+6 q^{72}+5 q^{70}-2 q^{68}-8 q^{66}+10 q^{62}+8 q^{60}-q^{58}-7 q^{56}+6 q^{52}+q^{50}-7 q^{48}-7 q^{46}+q^{42}-3 q^{40}-4 q^{38}+q^{36}+3 q^{34}-2 q^{32}-2 q^{30}+2 q^{28}+7 q^{26}+3 q^{24}-6 q^{22}-q^{20}+7 q^{18}+12 q^{16}+q^{14}-8 q^{12}-2 q^{10}+6 q^8+7 q^6-5 q^4-10 q^2-5+2 q^{-2} +3 q^{-4} - q^{-6} -3 q^{-8} +2 q^{-12} +3 q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} + q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-q^{46}+q^{42}-3 q^{40}+q^{38}+3 q^{36}-3 q^{34}+q^{32}+3 q^{30}-2 q^{28}+2 q^{24}+q^{22}+q^{16}-2 q^{14}-5 q^{12}+q^{10}-2 q^8-4 q^6+4 q^4+2 q^2+1+3 q^{-2} +2 q^{-4} + q^{-8}$
1,0,0	$q^{29}+q^{25}+q^{21}-q^{19}-q^{15}-q^{13}-q^{11}-q^9-q^5+2 q^3+q+2 q^{-1} + q^{-3} + q^{-5}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{58}+q^{56}+q^{54}-2 q^{52}-q^{50}+2 q^{48}+q^{46}-3 q^{44}-q^{42}+2 q^{40}-q^{38}-2 q^{36}+3 q^{34}+3 q^{32}+4 q^{28}+4 q^{26}+q^{24}-q^{22}+q^{20}-2 q^{18}-7 q^{16}-5 q^{14}-2 q^{12}-4 q^{10}-4 q^8+2 q^6+3 q^4+3 q^2+3+4 q^{-2} +3 q^{-4} +2 q^{-6} + q^{-8} + q^{-10}$
1,0,0,0	$q^{36}+q^{32}+q^{30}+q^{26}-q^{24}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}-q^6+2 q^4+q^2+2+2 q^{-2} + q^{-4} + q^{-6}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-q^{46}+2 q^{44}-3 q^{42}+3 q^{40}-3 q^{38}+3 q^{36}-q^{34}+q^{32}+q^{30}-2 q^{28}+4 q^{26}-6 q^{24}+5 q^{22}-6 q^{20}+4 q^{18}-5 q^{16}+2 q^{14}-q^{12}-q^{10}+2 q^8-2 q^6+4 q^4-2 q^2+3- q^{-2} +2 q^{-4} + q^{-8}$
1,0	$q^{78}-q^{74}-q^{72}+q^{70}+2 q^{68}-q^{66}-3 q^{64}-q^{62}+3 q^{60}+3 q^{58}-2 q^{56}-3 q^{54}+3 q^{50}+q^{48}-2 q^{46}-q^{44}+2 q^{42}+2 q^{40}-q^{38}-q^{36}+q^{34}+3 q^{32}-2 q^{28}-q^{26}+2 q^{24}-3 q^{20}-3 q^{18}+q^{16}+2 q^{14}-2 q^{12}-4 q^{10}-q^8+4 q^6+2 q^4-1+ q^{-2} +2 q^{-4} +2 q^{-6} + q^{-14}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-q^{64}+q^{62}-2 q^{60}+2 q^{58}-3 q^{56}+2 q^{54}-2 q^{52}+3 q^{50}-q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}-3 q^{38}+3 q^{36}-3 q^{34}+5 q^{32}-4 q^{30}+4 q^{28}-4 q^{26}+4 q^{24}-3 q^{22}-4 q^{18}-2 q^{16}-q^{14}-3 q^{12}-3 q^8+4 q^6+4 q^2+1+4 q^{-2} + q^{-4} +2 q^{-6} + q^{-10}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-3 q^{102}+6 q^{100}-7 q^{98}+7 q^{96}-4 q^{94}-2 q^{92}+7 q^{90}-9 q^{88}+11 q^{86}-6 q^{84}+q^{82}+5 q^{80}-7 q^{78}+7 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+2 q^{66}+4 q^{64}-9 q^{62}+12 q^{60}-10 q^{58}+5 q^{56}+q^{54}-10 q^{52}+13 q^{50}-13 q^{48}+10 q^{46}-5 q^{44}-3 q^{42}+7 q^{40}-10 q^{38}+6 q^{36}-3 q^{34}-4 q^{32}+5 q^{30}-5 q^{28}-q^{26}+6 q^{24}-8 q^{22}+8 q^{20}-6 q^{18}-q^{16}+6 q^{14}-8 q^{12}+10 q^{10}-5 q^8+3 q^6+2 q^4-2 q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18}$

. </div></div>

Computer Talk

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

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

In[1]:=

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

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2 t^2+6 t-7+6 t^{-1} -2 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-2 z^4-2 z^2+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 23, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-1+3 q^{-1} -4 q^{-2} +4 q^{-3} -4 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^4 a^8-2 z^2 a^8+2 z^5 a^7-4 z^3 a^7+z a^7+2 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-z^5 a^5+2 z^3 a^5-z a^5+3 z^6 a^4-6 z^4 a^4+6 z^2 a^4-a^4+z^7 a^3-2 z^5 a^3+5 z^3 a^3-3 z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n20, K11n151, K11n152,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

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

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

In[1]:=

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

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 6"];

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]=

{ $-2 t^2+6 t-7+6 t^{-1} -2 t^{-2}$, $q-1+3 q^{-1} -4 q^{-2} +4 q^{-3} -4 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7}$ }

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]=

{K11n20, K11n151, K11n152,}

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, 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-8$ $24$ $32$ $\frac{68}{3}$ $\frac{100}{3}$ $-192$ $-336$ $-96$ $-104$ $-\frac{256}{3}$ $288$ $-\frac{544}{3}$ $-\frac{800}{3}$ $\frac{10529}{15}$ $\frac{2804}{15}$ $-\frac{964}{45}$ $\frac{1231}{9}$ $-\frac{1471}{15}$

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 $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of 8 6. 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 3                 1 1 1                   0 -1             3 1   2 -3           2 1     -1 -5         2 2       0 -7       2 2         0 -9     1 2           -1 -11   1 2             1 -13   1               -1 -15 1                 1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

<p>

$n$	$J_n$
2	$q^4-q^3+3 q-4- q^{-1} +9 q^{-2} -9 q^{-3} -4 q^{-4} +17 q^{-5} -12 q^{-6} -8 q^{-7} +21 q^{-8} -12 q^{-9} -9 q^{-10} +19 q^{-11} -8 q^{-12} -8 q^{-13} +12 q^{-14} -3 q^{-15} -6 q^{-16} +5 q^{-17} -2 q^{-19} + q^{-20}$
3	$q^9-q^8+2 q^5-3 q^4+2 q^2+4 q-8-3 q^{-1} +8 q^{-2} +12 q^{-3} -16 q^{-4} -14 q^{-5} +15 q^{-6} +25 q^{-7} -18 q^{-8} -32 q^{-9} +18 q^{-10} +39 q^{-11} -17 q^{-12} -44 q^{-13} +16 q^{-14} +46 q^{-15} -13 q^{-16} -48 q^{-17} +12 q^{-18} +44 q^{-19} -7 q^{-20} -42 q^{-21} +4 q^{-22} +35 q^{-23} +2 q^{-24} -29 q^{-25} -5 q^{-26} +22 q^{-27} +7 q^{-28} -14 q^{-29} -9 q^{-30} +9 q^{-31} +7 q^{-32} -4 q^{-33} -5 q^{-34} +2 q^{-35} +2 q^{-36} -2 q^{-38} + q^{-39}$
4	$q^{16}-q^{15}-q^{12}+3 q^{11}-3 q^{10}+q^9+2 q^8-4 q^7+5 q^6-8 q^5+3 q^4+9 q^3-5 q^2+7 q-23+21 q^{-2} +5 q^{-3} +20 q^{-4} -50 q^{-5} -19 q^{-6} +28 q^{-7} +25 q^{-8} +54 q^{-9} -74 q^{-10} -52 q^{-11} +17 q^{-12} +42 q^{-13} +103 q^{-14} -86 q^{-15} -84 q^{-16} -3 q^{-17} +50 q^{-18} +142 q^{-19} -86 q^{-20} -100 q^{-21} -21 q^{-22} +49 q^{-23} +163 q^{-24} -79 q^{-25} -103 q^{-26} -32 q^{-27} +41 q^{-28} +163 q^{-29} -64 q^{-30} -91 q^{-31} -41 q^{-32} +24 q^{-33} +146 q^{-34} -39 q^{-35} -65 q^{-36} -46 q^{-37} - q^{-38} +112 q^{-39} -11 q^{-40} -30 q^{-41} -42 q^{-42} -23 q^{-43} +70 q^{-44} +4 q^{-45} -25 q^{-47} -29 q^{-48} +31 q^{-49} +4 q^{-50} +12 q^{-51} -8 q^{-52} -19 q^{-53} +10 q^{-54} - q^{-55} +7 q^{-56} -7 q^{-58} +3 q^{-59} - q^{-60} +2 q^{-61} -2 q^{-63} + q^{-64}$
5	$q^{25}-q^{24}-q^{21}+3 q^{19}-2 q^{18}+2 q^{16}-3 q^{15}-3 q^{14}+5 q^{13}-2 q^{12}+2 q^{11}+7 q^{10}-4 q^9-10 q^8-5 q^6+7 q^5+22 q^4+4 q^3-14 q^2-18 q-27+2 q^{-1} +46 q^{-2} +39 q^{-3} +7 q^{-4} -33 q^{-5} -80 q^{-6} -43 q^{-7} +52 q^{-8} +97 q^{-9} +75 q^{-10} -16 q^{-11} -133 q^{-12} -135 q^{-13} +6 q^{-14} +144 q^{-15} +175 q^{-16} +49 q^{-17} -158 q^{-18} -230 q^{-19} -85 q^{-20} +156 q^{-21} +268 q^{-22} +129 q^{-23} -148 q^{-24} -300 q^{-25} -166 q^{-26} +139 q^{-27} +322 q^{-28} +193 q^{-29} -127 q^{-30} -332 q^{-31} -218 q^{-32} +118 q^{-33} +339 q^{-34} +228 q^{-35} -103 q^{-36} -336 q^{-37} -242 q^{-38} +93 q^{-39} +330 q^{-40} +240 q^{-41} -73 q^{-42} -310 q^{-43} -250 q^{-44} +57 q^{-45} +289 q^{-46} +237 q^{-47} -25 q^{-48} -252 q^{-49} -237 q^{-50} + q^{-51} +212 q^{-52} +215 q^{-53} +31 q^{-54} -159 q^{-55} -195 q^{-56} -57 q^{-57} +111 q^{-58} +161 q^{-59} +74 q^{-60} -61 q^{-61} -122 q^{-62} -81 q^{-63} +18 q^{-64} +86 q^{-65} +73 q^{-66} +8 q^{-67} -46 q^{-68} -58 q^{-69} -26 q^{-70} +20 q^{-71} +39 q^{-72} +26 q^{-73} +2 q^{-74} -24 q^{-75} -21 q^{-76} -6 q^{-77} +6 q^{-78} +15 q^{-79} +10 q^{-80} -4 q^{-81} -8 q^{-82} -2 q^{-83} -3 q^{-84} +2 q^{-85} +6 q^{-86} - q^{-87} -3 q^{-88} + q^{-89} - q^{-91} +2 q^{-92} -2 q^{-94} + q^{-95}$
6	$q^{36}-q^{35}-q^{32}+4 q^{29}-3 q^{28}+2 q^{26}-3 q^{25}-2 q^{24}-2 q^{23}+10 q^{22}-4 q^{21}+7 q^{19}-6 q^{18}-9 q^{17}-9 q^{16}+18 q^{15}-3 q^{14}+5 q^{13}+21 q^{12}-7 q^{11}-23 q^{10}-32 q^9+18 q^8-6 q^7+19 q^6+61 q^5+14 q^4-30 q^3-75 q^2-16 q-53+19 q^{-1} +131 q^{-2} +94 q^{-3} +25 q^{-4} -100 q^{-5} -79 q^{-6} -193 q^{-7} -69 q^{-8} +173 q^{-9} +224 q^{-10} +193 q^{-11} -21 q^{-12} -97 q^{-13} -399 q^{-14} -290 q^{-15} +92 q^{-16} +318 q^{-17} +430 q^{-18} +197 q^{-19} +18 q^{-20} -576 q^{-21} -581 q^{-22} -126 q^{-23} +300 q^{-24} +632 q^{-25} +472 q^{-26} +251 q^{-27} -651 q^{-28} -832 q^{-29} -385 q^{-30} +192 q^{-31} +738 q^{-32} +698 q^{-33} +492 q^{-34} -643 q^{-35} -977 q^{-36} -587 q^{-37} +75 q^{-38} +763 q^{-39} +826 q^{-40} +664 q^{-41} -608 q^{-42} -1031 q^{-43} -701 q^{-44} -8 q^{-45} +749 q^{-46} +876 q^{-47} +757 q^{-48} -565 q^{-49} -1030 q^{-50} -753 q^{-51} -64 q^{-52} +706 q^{-53} +879 q^{-54} +802 q^{-55} -498 q^{-56} -978 q^{-57} -770 q^{-58} -126 q^{-59} +613 q^{-60} +837 q^{-61} +823 q^{-62} -377 q^{-63} -852 q^{-64} -750 q^{-65} -213 q^{-66} +440 q^{-67} +723 q^{-68} +814 q^{-69} -192 q^{-70} -630 q^{-71} -664 q^{-72} -303 q^{-73} +198 q^{-74} +519 q^{-75} +735 q^{-76} +7 q^{-77} -339 q^{-78} -486 q^{-79} -330 q^{-80} -41 q^{-81} +256 q^{-82} +556 q^{-83} +127 q^{-84} -70 q^{-85} -251 q^{-86} -247 q^{-87} -172 q^{-88} +27 q^{-89} +317 q^{-90} +116 q^{-91} +74 q^{-92} -52 q^{-93} -100 q^{-94} -158 q^{-95} -79 q^{-96} +118 q^{-97} +35 q^{-98} +77 q^{-99} +33 q^{-100} +11 q^{-101} -77 q^{-102} -68 q^{-103} +26 q^{-104} -22 q^{-105} +28 q^{-106} +26 q^{-107} +39 q^{-108} -19 q^{-109} -28 q^{-110} +11 q^{-111} -24 q^{-112} +4 q^{-114} +22 q^{-115} -3 q^{-116} -8 q^{-117} +9 q^{-118} -9 q^{-119} -2 q^{-120} -2 q^{-121} +8 q^{-122} -2 q^{-123} -4 q^{-124} +5 q^{-125} -2 q^{-126} - q^{-128} +2 q^{-129} -2 q^{-131} + q^{-132}$
7	$q^{49}-q^{48}-q^{45}+q^{42}+3 q^{41}-3 q^{40}+2 q^{38}-3 q^{37}-q^{36}-2 q^{35}+2 q^{34}+9 q^{33}-6 q^{32}-q^{31}+5 q^{30}-5 q^{29}-3 q^{28}-9 q^{27}+3 q^{26}+21 q^{25}-5 q^{24}-q^{23}+7 q^{22}-11 q^{21}-8 q^{20}-26 q^{19}-2 q^{18}+41 q^{17}+9 q^{16}+17 q^{15}+18 q^{14}-20 q^{13}-27 q^{12}-70 q^{11}-37 q^{10}+45 q^9+32 q^8+78 q^7+88 q^6+11 q^5-33 q^4-150 q^3-152 q^2-45 q-14+148 q^{-1} +253 q^{-2} +185 q^{-3} +98 q^{-4} -168 q^{-5} -329 q^{-6} -300 q^{-7} -286 q^{-8} +52 q^{-9} +399 q^{-10} +512 q^{-11} +511 q^{-12} +84 q^{-13} -366 q^{-14} -625 q^{-15} -824 q^{-16} -409 q^{-17} +264 q^{-18} +776 q^{-19} +1136 q^{-20} +720 q^{-21} -33 q^{-22} -742 q^{-23} -1441 q^{-24} -1179 q^{-25} -293 q^{-26} +712 q^{-27} +1682 q^{-28} +1562 q^{-29} +675 q^{-30} -504 q^{-31} -1844 q^{-32} -1981 q^{-33} -1089 q^{-34} +285 q^{-35} +1939 q^{-36} +2297 q^{-37} +1475 q^{-38} -13 q^{-39} -1953 q^{-40} -2551 q^{-41} -1826 q^{-42} -250 q^{-43} +1927 q^{-44} +2743 q^{-45} +2094 q^{-46} +476 q^{-47} -1871 q^{-48} -2851 q^{-49} -2301 q^{-50} -679 q^{-51} +1813 q^{-52} +2931 q^{-53} +2444 q^{-54} +813 q^{-55} -1752 q^{-56} -2956 q^{-57} -2540 q^{-58} -929 q^{-59} +1698 q^{-60} +2977 q^{-61} +2596 q^{-62} +1002 q^{-63} -1642 q^{-64} -2961 q^{-65} -2633 q^{-66} -1077 q^{-67} +1584 q^{-68} +2941 q^{-69} +2651 q^{-70} +1128 q^{-71} -1499 q^{-72} -2878 q^{-73} -2659 q^{-74} -1210 q^{-75} +1391 q^{-76} +2805 q^{-77} +2640 q^{-78} +1273 q^{-79} -1227 q^{-80} -2653 q^{-81} -2606 q^{-82} -1387 q^{-83} +1031 q^{-84} +2475 q^{-85} +2521 q^{-86} +1464 q^{-87} -767 q^{-88} -2190 q^{-89} -2400 q^{-90} -1567 q^{-91} +474 q^{-92} +1873 q^{-93} +2206 q^{-94} +1607 q^{-95} -159 q^{-96} -1474 q^{-97} -1937 q^{-98} -1616 q^{-99} -146 q^{-100} +1059 q^{-101} +1615 q^{-102} +1539 q^{-103} +383 q^{-104} -637 q^{-105} -1234 q^{-106} -1378 q^{-107} -559 q^{-108} +265 q^{-109} +846 q^{-110} +1153 q^{-111} +623 q^{-112} +17 q^{-113} -481 q^{-114} -873 q^{-115} -583 q^{-116} -209 q^{-117} +176 q^{-118} +596 q^{-119} +480 q^{-120} +280 q^{-121} +25 q^{-122} -338 q^{-123} -327 q^{-124} -269 q^{-125} -145 q^{-126} +152 q^{-127} +181 q^{-128} +201 q^{-129} +176 q^{-130} -34 q^{-131} -65 q^{-132} -122 q^{-133} -144 q^{-134} -17 q^{-135} -16 q^{-136} +43 q^{-137} +111 q^{-138} +35 q^{-139} +31 q^{-140} -6 q^{-141} -52 q^{-142} -10 q^{-143} -50 q^{-144} -26 q^{-145} +34 q^{-146} +10 q^{-147} +27 q^{-148} +12 q^{-149} -7 q^{-150} +14 q^{-151} -19 q^{-152} -24 q^{-153} +5 q^{-154} -2 q^{-155} +10 q^{-156} +3 q^{-157} -5 q^{-158} +13 q^{-159} -2 q^{-160} -8 q^{-161} -2 q^{-163} +4 q^{-164} -5 q^{-166} +4 q^{-167} +2 q^{-168} -2 q^{-169} - q^{-171} +2 q^{-172} -2 q^{-174} + q^{-175}$

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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