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