8 2

8_1

8_3

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 2 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9
Gauss code	-1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6
Dowker-Thistlethwaite code	4 10 12 14 16 2 6 8
Conway Notation	[512]

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

[edit Notes on presentations of 8 2]

Knot 8_2.

A graph, knot 8_2.

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 2"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 16 2 6 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[512]

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]=

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

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]=

{ 3, 8, 3 }

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

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 $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-11][1] Hyperbolic Volume 4.93524 A-Polynomial See Data:8 2/A-polynomial

[edit Notes for 8 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 8 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, -4 }
Jones polynomial	$1-q^{-1}+2q^{-2}-2q^{-3}+3q^{-4}-3q^{-5}+2q^{-6}-2q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{6}+3z^{2}a^{6}+a^{6}-z^{6}a^{4}-5z^{4}a^{4}-7z^{2}a^{4}-3a^{4}+z^{4}a^{2}+4z^{2}a^{2}+3a^{2}$
Kauffman polynomial (db, data sources)	$z^{2}a^{10}+2z^{3}a^{9}-za^{9}+2z^{4}a^{8}-z^{2}a^{8}+2z^{5}a^{7}-2z^{3}a^{7}-za^{7}+2z^{6}a^{6}-5z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{7}a^{5}-2z^{5}a^{5}-z^{3}a^{5}+za^{5}+3z^{6}a^{4}-12z^{4}a^{4}+12z^{2}a^{4}-3a^{4}+z^{7}a^{3}-4z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}-3a^{2}$
The A2 invariant	$q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^{6}+q^{4}+q^{2}+1$
The G2 invariant	$q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2q^{120}-2q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2q^{92}-q^{84}+q^{82}-2q^{78}+q^{76}-q^{74}+q^{70}-4q^{68}+2q^{66}-3q^{64}-3q^{58}+3q^{56}-2q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2q^{30}-2q^{28}+3q^{26}-q^{24}+q^{22}+3q^{20}-3q^{18}+4q^{16}+q^{12}+q^{10}-q^{8}+2q^{6}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 8_2.

A1 Invariants.

Weight	Invariant
1	$q^{17}-q^{15}-q^{11}+q^{7}+q^{3}+q^{-1}$
2	$q^{46}-q^{44}-q^{42}+q^{40}-q^{38}+q^{36}+q^{34}-q^{32}+q^{28}-q^{24}-q^{18}-q^{16}+q^{14}+2q^{8}+q^{6}-q^{4}+q^{2}+1-q^{-2}+q^{-6}$
3	$q^{87}-q^{85}-q^{83}+q^{79}+q^{77}-q^{75}-q^{71}+q^{69}+q^{67}-2q^{63}+2q^{59}-q^{55}-q^{53}+q^{49}+2q^{47}-q^{45}-q^{43}+2q^{39}-q^{37}-2q^{35}+q^{31}-q^{29}-q^{27}+q^{23}+q^{21}-q^{19}-q^{17}+q^{15}+2q^{13}+q^{11}-q^{9}-q^{7}+2q^{5}+2q^{3}-2q^{-1}+2q^{-5}+q^{-7}-q^{-9}-q^{-11}+q^{-15}$
4	$q^{140}-q^{138}-q^{136}+3q^{130}-q^{128}-q^{126}-2q^{124}-q^{122}+5q^{120}+q^{118}-q^{116}-4q^{114}-2q^{112}+5q^{110}+4q^{108}-q^{106}-6q^{104}-4q^{102}+4q^{100}+5q^{98}+2q^{96}-4q^{94}-6q^{92}+3q^{88}+4q^{86}-2q^{82}-q^{80}-2q^{78}+q^{76}+4q^{74}+2q^{72}-q^{70}-3q^{68}-q^{66}+3q^{64}+2q^{62}-2q^{60}-3q^{58}-q^{56}+3q^{54}+2q^{52}-3q^{50}-3q^{48}+4q^{44}+3q^{42}-3q^{40}-4q^{38}-2q^{36}+2q^{34}+3q^{32}-q^{30}-q^{28}-2q^{26}+3q^{22}+2q^{20}+2q^{18}-q^{16}-2q^{14}+4q^{8}+2q^{6}-q^{4}-2q^{2}-3+2q^{-2}+3q^{-4}+2q^{-6}-4q^{-10}-q^{-12}+q^{-14}+2q^{-16}+2q^{-18}-q^{-20}-q^{-22}-q^{-24}+q^{-28}$
5	$q^{205}-q^{203}-q^{201}+2q^{195}+q^{193}-q^{191}-3q^{189}-q^{187}+q^{185}+4q^{183}+3q^{181}-q^{179}-5q^{177}-5q^{175}+3q^{173}+6q^{171}+4q^{169}-2q^{167}-9q^{165}-7q^{163}+5q^{161}+11q^{159}+5q^{157}-5q^{155}-11q^{153}-9q^{151}+5q^{149}+14q^{147}+10q^{145}-2q^{143}-12q^{141}-10q^{139}-q^{137}+9q^{135}+12q^{133}+5q^{131}-5q^{129}-9q^{127}-8q^{125}-2q^{123}+4q^{121}+5q^{119}+4q^{117}-5q^{113}-5q^{111}-3q^{109}+2q^{107}+6q^{105}+6q^{103}-q^{101}-5q^{99}-3q^{97}+q^{95}+5q^{93}+4q^{91}-3q^{89}-5q^{87}-q^{85}+3q^{83}+6q^{81}+3q^{79}-4q^{77}-8q^{75}-3q^{73}+4q^{71}+8q^{69}+5q^{67}-3q^{65}-10q^{63}-7q^{61}+q^{59}+8q^{57}+7q^{55}-8q^{51}-9q^{49}-3q^{47}+7q^{45}+9q^{43}+4q^{41}-3q^{39}-8q^{37}-5q^{35}+5q^{31}+6q^{29}+2q^{27}-q^{25}-3q^{23}-3q^{21}-q^{19}+2q^{17}+4q^{15}+3q^{13}+3q^{11}-3q^{7}-4q^{5}-2q^{3}+q+4q^{-1}+5q^{-3}+2q^{-5}-2q^{-7}-5q^{-9}-4q^{-11}+3q^{-15}+5q^{-17}+3q^{-19}-q^{-21}-4q^{-23}-3q^{-25}-q^{-27}+q^{-29}+3q^{-31}+2q^{-33}-q^{-37}-q^{-39}-q^{-41}+q^{-45}$
6	$q^{282}-q^{280}-q^{278}+2q^{272}+q^{268}-3q^{266}-2q^{264}+q^{262}+q^{260}+4q^{258}+2q^{256}+q^{254}-7q^{252}-4q^{250}+2q^{246}+6q^{244}+3q^{242}-q^{240}-9q^{238}-3q^{236}+2q^{234}+4q^{232}+6q^{230}-6q^{226}-10q^{224}+2q^{222}+11q^{220}+10q^{218}+6q^{216}-8q^{214}-17q^{212}-14q^{210}+4q^{208}+19q^{206}+19q^{204}+8q^{202}-10q^{200}-24q^{198}-24q^{196}-5q^{194}+14q^{192}+24q^{190}+20q^{188}+4q^{186}-14q^{184}-24q^{182}-15q^{180}-q^{178}+11q^{176}+18q^{174}+15q^{172}+5q^{170}-7q^{168}-11q^{166}-11q^{164}-7q^{162}+q^{160}+8q^{158}+12q^{156}+8q^{154}-8q^{150}-12q^{148}-8q^{146}+9q^{142}+9q^{140}+q^{138}-6q^{136}-8q^{134}-4q^{132}+3q^{130}+10q^{128}+5q^{126}-5q^{124}-9q^{122}-7q^{120}+q^{118}+9q^{116}+14q^{114}+8q^{112}-8q^{110}-14q^{108}-10q^{106}+11q^{102}+16q^{100}+11q^{98}-5q^{96}-15q^{94}-14q^{92}-7q^{90}+6q^{88}+16q^{86}+16q^{84}+2q^{82}-12q^{80}-17q^{78}-15q^{76}-2q^{74}+13q^{72}+21q^{70}+13q^{68}-2q^{66}-14q^{64}-21q^{62}-14q^{60}+2q^{58}+17q^{56}+19q^{54}+10q^{52}-3q^{50}-17q^{48}-19q^{46}-10q^{44}+6q^{42}+14q^{40}+15q^{38}+10q^{36}-3q^{34}-10q^{32}-12q^{30}-5q^{28}+q^{26}+6q^{24}+10q^{22}+6q^{20}+3q^{18}-3q^{16}-3q^{14}-5q^{12}-5q^{10}-q^{8}+q^{6}+5q^{4}+4q^{2}+7+3q^{-2}-3q^{-4}-5q^{-6}-7q^{-8}-4q^{-10}-2q^{-12}+6q^{-14}+8q^{-16}+6q^{-18}+2q^{-20}-3q^{-22}-6q^{-24}-9q^{-26}-2q^{-28}+2q^{-30}+5q^{-32}+6q^{-34}+4q^{-36}+q^{-38}-5q^{-40}-4q^{-42}-3q^{-44}-q^{-46}+q^{-48}+3q^{-50}+3q^{-52}-q^{-58}-q^{-60}-q^{-62}+q^{-66}$

A2 Invariants.

Weight	Invariant
1,0	$q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^{6}+q^{4}+q^{2}+1$
1,1	$q^{68}-2q^{66}+2q^{64}-2q^{62}+3q^{60}-4q^{58}+2q^{56}+2q^{52}-2q^{50}-2q^{46}+q^{44}-2q^{42}+2q^{40}+2q^{38}+6q^{34}-6q^{32}+6q^{30}-11q^{28}+8q^{26}-10q^{24}+6q^{22}-6q^{20}+4q^{18}-2q^{14}+5q^{12}-4q^{10}+8q^{8}-6q^{6}+6q^{4}-2q^{2}+4+q^{-4}$
2,0	$q^{60}-q^{54}-q^{52}-q^{44}+q^{42}+q^{40}+q^{38}+2q^{34}+q^{32}-q^{30}-q^{28}-2q^{26}-2q^{24}-2q^{22}+q^{16}+2q^{14}+2q^{12}+q^{6}+q^{4}+1+q^{-2}+q^{-4}$
3,0	$q^{108}-q^{102}-q^{100}+q^{94}-2q^{92}-q^{90}+3q^{86}+q^{84}-q^{82}+2q^{78}+4q^{76}-3q^{72}-3q^{70}+q^{66}-3q^{62}-q^{60}+2q^{56}+q^{50}+q^{48}-q^{44}+q^{42}-q^{38}-2q^{36}+2q^{32}+q^{30}-2q^{28}-3q^{26}-q^{24}+2q^{22}+q^{20}-q^{16}+2q^{14}+4q^{12}+3q^{10}-q^{6}+2q^{2}+1-q^{-4}+q^{-8}+q^{-10}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{56}-q^{54}-q^{52}+q^{50}-q^{48}-q^{46}+q^{44}+q^{42}+2q^{38}+q^{36}-2q^{26}-q^{24}-q^{22}-3q^{20}-2q^{18}+2q^{12}+3q^{10}+2q^{8}+2q^{6}+2q^{4}+1$
1,0,0	$q^{31}+q^{27}-q^{25}-q^{21}-q^{19}-q^{17}-q^{15}+2q^{9}+q^{7}+2q^{5}+q^{3}+q$
1,0,1	$q^{90}-2q^{88}+q^{86}+q^{84}-2q^{82}+3q^{80}-2q^{78}-q^{76}+q^{74}+q^{70}-q^{68}+q^{62}-2q^{60}+2q^{58}-3q^{56}-2q^{54}+3q^{52}-4q^{50}+5q^{48}-q^{46}+4q^{44}+q^{40}+3q^{38}-2q^{36}+4q^{34}-5q^{32}+2q^{30}-5q^{28}-q^{26}-5q^{24}-4q^{22}-q^{20}-3q^{18}+4q^{16}-2q^{14}+6q^{12}+2q^{10}+5q^{8}+6q^{6}+2q^{4}+4q^{2}+q^{-2}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{70}-q^{66}-2q^{60}-2q^{58}+q^{52}+2q^{50}+4q^{48}+2q^{46}+2q^{44}+2q^{42}-q^{38}-q^{34}-4q^{32}-3q^{30}-4q^{28}-4q^{26}-4q^{24}-2q^{22}+2q^{18}+3q^{16}+4q^{14}+4q^{12}+4q^{10}+3q^{8}+2q^{6}+q^{4}+q^{2}$
1,0,0,0	$q^{38}+q^{34}-q^{26}-q^{24}-2q^{22}-q^{20}-2q^{18}+2q^{12}+2q^{10}+2q^{8}+2q^{6}+q^{4}+q^{2}$

B2 Invariants.

Weight	Invariant
0,1	$q^{56}-q^{54}+q^{52}-q^{50}+q^{48}-q^{46}+q^{44}-q^{42}-q^{36}+2q^{34}-2q^{32}+2q^{30}-2q^{28}+2q^{26}-3q^{24}+q^{22}-q^{20}+2q^{12}-q^{10}+2q^{8}+2q^{4}+1$
1,0	$q^{90}-q^{86}-q^{84}+q^{80}-q^{76}-q^{74}+q^{70}+q^{68}+q^{62}+2q^{60}-q^{56}+q^{52}-q^{48}-q^{46}-q^{40}-q^{38}-q^{32}-2q^{30}-q^{28}+q^{26}+q^{24}-q^{20}+q^{18}+2q^{16}+2q^{14}+q^{8}+2q^{6}+q^{-2}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{78}-q^{76}-q^{72}+q^{70}-q^{68}-q^{64}+q^{62}+q^{58}+q^{54}+q^{52}+2q^{48}-q^{46}+2q^{44}-q^{42}+2q^{40}-2q^{38}+q^{36}-3q^{34}-q^{32}-4q^{30}-2q^{28}-3q^{26}-2q^{24}-q^{22}+3q^{18}+2q^{16}+4q^{14}+2q^{12}+4q^{10}+q^{8}+2q^{6}+q^{2}$

G2 Invariants.

Weight	Invariant
1,0	$q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2q^{120}-2q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2q^{92}-q^{84}+q^{82}-2q^{78}+q^{76}-q^{74}+q^{70}-4q^{68}+2q^{66}-3q^{64}-3q^{58}+3q^{56}-2q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2q^{30}-2q^{28}+3q^{26}-q^{24}+q^{22}+3q^{20}-3q^{18}+4q^{16}+q^{12}+q^{10}-q^{8}+2q^{6}+q^{2}$

.

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 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-3z^{4}+1$

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]=

{ 17, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-q^{-1}+2q^{-2}-2q^{-3}+3q^{-4}-3q^{-5}+2q^{-6}-2q^{-7}+q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{2}a^{10}+2z^{3}a^{9}-za^{9}+2z^{4}a^{8}-z^{2}a^{8}+2z^{5}a^{7}-2z^{3}a^{7}-za^{7}+2z^{6}a^{6}-5z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{7}a^{5}-2z^{5}a^{5}-z^{3}a^{5}+za^{5}+3z^{6}a^{4}-12z^{4}a^{4}+12z^{2}a^{4}-3a^{4}+z^{7}a^{3}-4z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}-3a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n6,}

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 2"];

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]=

{ $-t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}$, $1-q^{-1}+2q^{-2}-2q^{-3}+3q^{-4}-3q^{-5}+2q^{-6}-2q^{-7}+q^{-8}$ }

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]=

{K11n6,}

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:

(0, 1)

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 $0$ $8$ $0$ $0$ $24$ $0$ $-{\frac {304}{3}}$ $-{\frac {160}{3}}$ $-88$ $0$ $32$ $0$ $0$ $432$ $-104$ $472$ $80$ $32$

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 $t^{r}q^{j}$ 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 $s=$-4 is the signature of 8 2. 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 1                 1 1 -1                   0 -3             2 1   1 -5           1 1     0 -7         2 1       1 -9       1 1         0 -11     1 2           -1 -13   1 1             0 -15   1               -1 -17 1                 1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{2}-q-1+3q^{-1}-q^{-2}-3q^{-3}+5q^{-4}-5q^{-6}+5q^{-7}+q^{-8}-7q^{-9}+5q^{-10}+2q^{-11}-7q^{-12}+4q^{-13}+3q^{-14}-6q^{-15}+3q^{-16}+2q^{-17}-4q^{-18}+3q^{-19}-2q^{-21}+q^{-22}$
3	$q^{6}-q^{5}-q^{4}+3q^{2}-3-2q^{-1}+5q^{-2}+2q^{-3}-3q^{-4}-5q^{-5}+5q^{-6}+4q^{-7}-2q^{-8}-6q^{-9}+3q^{-10}+4q^{-11}-6q^{-13}+2q^{-14}+3q^{-15}-4q^{-17}+q^{-18}+q^{-19}+q^{-20}-q^{-21}-q^{-22}+q^{-24}+2q^{-25}-2q^{-26}-q^{-27}+2q^{-29}-q^{-30}+q^{-31}-2q^{-32}+q^{-34}+2q^{-35}-2q^{-36}-2q^{-37}+2q^{-38}+q^{-39}-2q^{-41}+q^{-42}$
4	$q^{12}-q^{11}-q^{10}+4q^{7}-q^{6}-2q^{5}-2q^{4}-3q^{3}+8q^{2}+q-1-3q^{-1}-8q^{-2}+9q^{-3}+2q^{-4}+2q^{-5}-q^{-6}-12q^{-7}+9q^{-8}+3q^{-10}+2q^{-11}-12q^{-12}+10q^{-13}-3q^{-14}+q^{-15}+3q^{-16}-12q^{-17}+14q^{-18}-4q^{-19}-3q^{-20}+q^{-21}-11q^{-22}+20q^{-23}-3q^{-24}-7q^{-25}-2q^{-26}-11q^{-27}+25q^{-28}-2q^{-29}-11q^{-30}-4q^{-31}-10q^{-32}+29q^{-33}-q^{-34}-15q^{-35}-6q^{-36}-8q^{-37}+32q^{-38}+q^{-39}-18q^{-40}-9q^{-41}-7q^{-42}+31q^{-43}+3q^{-44}-14q^{-45}-10q^{-46}-10q^{-47}+25q^{-48}+5q^{-49}-8q^{-50}-7q^{-51}-11q^{-52}+17q^{-53}+3q^{-54}-3q^{-55}-2q^{-56}-10q^{-57}+10q^{-58}+q^{-59}-6q^{-62}+4q^{-63}+q^{-65}-2q^{-67}+q^{-68}$
5	$q^{20}-q^{19}-q^{18}+q^{15}+3q^{14}-3q^{12}-2q^{11}-2q^{10}+6q^{8}+4q^{7}-q^{6}-4q^{5}-5q^{4}-4q^{3}+5q^{2}+7q+3-q^{-1}-6q^{-2}-7q^{-3}+2q^{-4}+5q^{-5}+4q^{-6}+2q^{-7}-3q^{-8}-7q^{-9}+3q^{-10}+3q^{-11}+q^{-12}-3q^{-14}-5q^{-15}+6q^{-16}+7q^{-17}-5q^{-19}-8q^{-20}-8q^{-21}+11q^{-22}+14q^{-23}+5q^{-24}-7q^{-25}-18q^{-26}-14q^{-27}+12q^{-28}+22q^{-29}+12q^{-30}-6q^{-31}-25q^{-32}-22q^{-33}+9q^{-34}+29q^{-35}+20q^{-36}-3q^{-37}-29q^{-38}-29q^{-39}+4q^{-40}+33q^{-41}+27q^{-42}-32q^{-44}-33q^{-45}+35q^{-47}+34q^{-48}+q^{-49}-36q^{-50}-37q^{-51}-2q^{-52}+39q^{-53}+41q^{-54}+q^{-55}-40q^{-56}-42q^{-57}-4q^{-58}+39q^{-59}+46q^{-60}+5q^{-61}-39q^{-62}-43q^{-63}-10q^{-64}+33q^{-65}+45q^{-66}+9q^{-67}-29q^{-68}-36q^{-69}-13q^{-70}+23q^{-71}+36q^{-72}+7q^{-73}-19q^{-74}-24q^{-75}-9q^{-76}+14q^{-77}+22q^{-78}+5q^{-79}-13q^{-80}-14q^{-81}-3q^{-82}+8q^{-83}+10q^{-84}+3q^{-85}-6q^{-86}-8q^{-87}-q^{-88}+5q^{-89}+2q^{-90}+3q^{-91}-2q^{-92}-4q^{-93}+2q^{-95}+q^{-97}-2q^{-99}+q^{-100}$
6	$q^{30}-q^{29}-q^{28}+q^{25}+4q^{23}-q^{22}-3q^{21}-2q^{20}-2q^{19}-q^{17}+10q^{16}+2q^{15}-q^{14}-3q^{13}-5q^{12}-4q^{11}-8q^{10}+13q^{9}+5q^{8}+4q^{7}+q^{6}-3q^{5}-6q^{4}-16q^{3}+11q^{2}+2q+6+3q^{-1}+3q^{-2}-2q^{-3}-19q^{-4}+12q^{-5}-2q^{-6}+4q^{-7}-q^{-8}+3q^{-9}-19q^{-11}+18q^{-12}+q^{-13}+8q^{-14}-5q^{-15}-2q^{-16}-6q^{-17}-26q^{-18}+20q^{-19}+8q^{-20}+21q^{-21}-3q^{-23}-14q^{-24}-42q^{-25}+11q^{-26}+10q^{-27}+35q^{-28}+13q^{-29}+6q^{-30}-16q^{-31}-57q^{-32}-5q^{-33}+3q^{-34}+42q^{-35}+25q^{-36}+21q^{-37}-8q^{-38}-65q^{-39}-20q^{-40}-10q^{-41}+40q^{-42}+30q^{-43}+35q^{-44}+6q^{-45}-65q^{-46}-30q^{-47}-23q^{-48}+33q^{-49}+29q^{-50}+45q^{-51}+22q^{-52}-60q^{-53}-35q^{-54}-34q^{-55}+23q^{-56}+25q^{-57}+51q^{-58}+38q^{-59}-54q^{-60}-40q^{-61}-42q^{-62}+15q^{-63}+23q^{-64}+55q^{-65}+48q^{-66}-49q^{-67}-47q^{-68}-49q^{-69}+11q^{-70}+25q^{-71}+62q^{-72}+56q^{-73}-49q^{-74}-56q^{-75}-57q^{-76}+7q^{-77}+29q^{-78}+70q^{-79}+64q^{-80}-45q^{-81}-60q^{-82}-64q^{-83}-q^{-84}+25q^{-85}+70q^{-86}+68q^{-87}-33q^{-88}-50q^{-89}-61q^{-90}-8q^{-91}+13q^{-92}+56q^{-93}+59q^{-94}-23q^{-95}-32q^{-96}-45q^{-97}-4q^{-98}+3q^{-99}+37q^{-100}+40q^{-101}-23q^{-102}-18q^{-103}-27q^{-104}+7q^{-105}+3q^{-106}+22q^{-107}+22q^{-108}-26q^{-109}-9q^{-110}-13q^{-111}+11q^{-112}+4q^{-113}+13q^{-114}+10q^{-115}-22q^{-116}-3q^{-117}-7q^{-118}+9q^{-119}+2q^{-120}+8q^{-121}+4q^{-122}-14q^{-123}+q^{-124}-4q^{-125}+5q^{-126}+4q^{-128}+q^{-129}-6q^{-130}+2q^{-131}-2q^{-132}+2q^{-133}+q^{-135}-2q^{-137}+q^{-138}$
7	$q^{42}-q^{41}-q^{40}+q^{37}+q^{35}+3q^{34}-q^{33}-3q^{32}-2q^{31}-3q^{30}+q^{29}+q^{27}+9q^{26}+3q^{25}-q^{24}-3q^{23}-8q^{22}-3q^{21}-4q^{20}-4q^{19}+11q^{18}+8q^{17}+6q^{16}+5q^{15}-9q^{14}-4q^{13}-8q^{12}-13q^{11}+6q^{10}+5q^{9}+8q^{8}+13q^{7}-4q^{6}-4q^{4}-16q^{3}+4q^{2}-2q+2+14q^{-1}-5q^{-2}+q^{-3}-q^{-4}-14q^{-5}+10q^{-6}+2q^{-7}+q^{-8}+16q^{-9}-9q^{-10}-6q^{-11}-8q^{-12}-23q^{-13}+12q^{-14}+8q^{-15}+12q^{-16}+30q^{-17}-q^{-18}-7q^{-19}-17q^{-20}-43q^{-21}-7q^{-22}+2q^{-23}+17q^{-24}+50q^{-25}+20q^{-26}+10q^{-27}-11q^{-28}-58q^{-29}-32q^{-30}-22q^{-31}+3q^{-32}+55q^{-33}+41q^{-34}+36q^{-35}+11q^{-36}-52q^{-37}-44q^{-38}-46q^{-39}-25q^{-40}+40q^{-41}+44q^{-42}+54q^{-43}+36q^{-44}-30q^{-45}-35q^{-46}-54q^{-47}-49q^{-48}+14q^{-49}+28q^{-50}+54q^{-51}+52q^{-52}-5q^{-53}-12q^{-54}-44q^{-55}-57q^{-56}-9q^{-57}+q^{-58}+37q^{-59}+55q^{-60}+16q^{-61}+12q^{-62}-22q^{-63}-52q^{-64}-25q^{-65}-27q^{-66}+13q^{-67}+49q^{-68}+30q^{-69}+36q^{-70}+q^{-71}-42q^{-72}-36q^{-73}-49q^{-74}-11q^{-75}+38q^{-76}+41q^{-77}+57q^{-78}+21q^{-79}-33q^{-80}-43q^{-81}-67q^{-82}-31q^{-83}+30q^{-84}+50q^{-85}+73q^{-86}+33q^{-87}-28q^{-88}-52q^{-89}-80q^{-90}-40q^{-91}+30q^{-92}+62q^{-93}+85q^{-94}+38q^{-95}-32q^{-96}-68q^{-97}-90q^{-98}-41q^{-99}+36q^{-100}+76q^{-101}+97q^{-102}+45q^{-103}-37q^{-104}-84q^{-105}-103q^{-106}-49q^{-107}+33q^{-108}+84q^{-109}+108q^{-110}+59q^{-111}-25q^{-112}-80q^{-113}-109q^{-114}-66q^{-115}+13q^{-116}+67q^{-117}+103q^{-118}+70q^{-119}+2q^{-120}-50q^{-121}-92q^{-122}-70q^{-123}-12q^{-124}+35q^{-125}+73q^{-126}+59q^{-127}+17q^{-128}-14q^{-129}-54q^{-130}-49q^{-131}-16q^{-132}+4q^{-133}+36q^{-134}+30q^{-135}+12q^{-136}+6q^{-137}-19q^{-138}-16q^{-139}-7q^{-140}-11q^{-141}+9q^{-142}+6q^{-143}-2q^{-144}+11q^{-145}+5q^{-147}+4q^{-148}-15q^{-149}-q^{-150}-7q^{-151}-7q^{-152}+11q^{-153}+3q^{-154}+10q^{-155}+8q^{-156}-10q^{-157}-4q^{-158}-8q^{-159}-6q^{-160}+8q^{-161}+2q^{-162}+5q^{-163}+6q^{-164}-6q^{-165}-q^{-166}-4q^{-167}-4q^{-168}+5q^{-169}+2q^{-171}+2q^{-172}-3q^{-173}-2q^{-176}+2q^{-177}+q^{-179}-2q^{-181}+q^{-182}$

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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