7 6

7_5

7_7

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 7 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 6 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X5,12,6,13 X9,1,10,14 X13,11,14,10 X11,6,12,7 X7283
Gauss code	-1, 7, -2, 1, -3, 6, -7, 2, -4, 5, -6, 3, -5, 4
Dowker-Thistlethwaite code	4 8 12 2 14 6 10
Conway Notation	[2212]

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

[edit Notes on presentations of 7 6]

Knot 7_6.

A graph, knot 7_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["7 6"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X5,12,6,13 X9,1,10,14 X13,11,14,10 X11,6,12,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 2 14 6 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2212]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-1] Hyperbolic Volume 7.08493 A-Polynomial See Data:7 6/A-polynomial

[edit Notes for 7 6's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,6))$ Rasmussen s-Invariant -2

[edit Notes for 7 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}-t^{-2}+5t+5t^{-1}-7$
Conway polynomial	$-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, -2 }
Jones polynomial	$q-2+3q^{-1}-3q^{-2}+4q^{-3}-3q^{-4}+2q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{6}+2a^{4}z^{2}+2a^{4}-a^{2}z^{4}-2a^{2}z^{2}-a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$a^{7}z^{3}-a^{7}z+2a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{5}-a^{5}z^{3}+a^{4}z^{6}+2a^{4}z^{4}-4a^{4}z^{2}+2a^{4}+4a^{3}z^{5}-6a^{3}z^{3}+2a^{3}z+a^{2}z^{6}+a^{2}z^{4}-4a^{2}z^{2}+a^{2}+2az^{5}-4az^{3}+az+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^{6}-q^{4}+q^{2}+q^{-4}$
The G2 invariant	$q^{100}-q^{98}+2q^{96}-2q^{94}-3q^{88}+5q^{86}-6q^{84}+4q^{82}-4q^{80}-q^{78}+5q^{76}-8q^{74}+8q^{72}-5q^{70}+q^{68}+2q^{66}-5q^{64}+4q^{62}-2q^{58}+6q^{56}-5q^{54}+2q^{52}+6q^{50}-9q^{48}+11q^{46}-9q^{44}+5q^{42}+2q^{40}-6q^{38}+10q^{36}-9q^{34}+8q^{32}-3q^{30}-3q^{28}+5q^{26}-5q^{24}+3q^{22}-3q^{18}+5q^{16}-3q^{14}-q^{12}+5q^{10}-8q^{8}+8q^{6}-5q^{4}-q^{2}+5-6q^{-2}+8q^{-4}-4q^{-6}+2q^{-8}+q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 7_6.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+q^{11}-q^{9}+q^{7}+q^{5}+q-q^{-1}+q^{-3}$
2	$q^{36}-q^{34}-q^{32}+3q^{30}-2q^{28}-3q^{26}+4q^{24}-q^{22}-3q^{20}+2q^{18}+q^{16}-q^{12}+2q^{10}+q^{8}-3q^{6}+2q^{4}+3q^{2}-3+q^{-2}+3q^{-4}-2q^{-6}-q^{-8}+q^{-10}$
3	$-q^{69}+q^{67}+q^{65}-q^{63}-2q^{61}+2q^{59}+5q^{57}-3q^{55}-7q^{53}+2q^{51}+9q^{49}-q^{47}-11q^{45}-q^{43}+11q^{41}+2q^{39}-7q^{37}-4q^{35}+5q^{33}+3q^{31}-2q^{29}-4q^{27}-2q^{25}+3q^{23}+4q^{21}-3q^{19}-6q^{17}+5q^{15}+8q^{13}-2q^{11}-8q^{9}+2q^{7}+10q^{5}+q^{3}-9q-2q^{-1}+7q^{-3}+4q^{-5}-5q^{-7}-5q^{-9}+3q^{-11}+4q^{-13}-2q^{-17}-q^{-19}+q^{-21}$
4	$q^{112}-q^{110}-q^{108}+q^{106}+2q^{102}-4q^{100}-3q^{98}+4q^{96}+4q^{94}+7q^{92}-10q^{90}-12q^{88}+4q^{86}+12q^{84}+18q^{82}-13q^{80}-25q^{78}-6q^{76}+15q^{74}+32q^{72}-5q^{70}-28q^{68}-15q^{66}+8q^{64}+31q^{62}+5q^{60}-16q^{58}-16q^{56}-3q^{54}+17q^{52}+9q^{50}-3q^{48}-10q^{46}-8q^{44}+9q^{40}+9q^{38}-5q^{36}-13q^{34}-10q^{32}+12q^{30}+19q^{28}-q^{26}-16q^{24}-18q^{22}+14q^{20}+26q^{18}+4q^{16}-16q^{14}-26q^{12}+7q^{10}+25q^{8}+13q^{6}-7q^{4}-29q^{2}-4+16q^{-2}+17q^{-4}+6q^{-6}-19q^{-8}-11q^{-10}+q^{-12}+11q^{-14}+12q^{-16}-6q^{-18}-7q^{-20}-5q^{-22}+q^{-24}+6q^{-26}+q^{-28}-2q^{-32}-q^{-34}+q^{-36}$
5	$-q^{165}+q^{163}+q^{161}-q^{159}+2q^{151}+2q^{149}-4q^{147}-6q^{145}-2q^{143}+5q^{141}+11q^{139}+7q^{137}-6q^{135}-21q^{133}-17q^{131}+9q^{129}+31q^{127}+31q^{125}-q^{123}-41q^{121}-52q^{119}-11q^{117}+47q^{115}+68q^{113}+30q^{111}-43q^{109}-81q^{107}-49q^{105}+31q^{103}+85q^{101}+61q^{99}-16q^{97}-78q^{95}-65q^{93}-q^{91}+60q^{89}+68q^{87}+12q^{85}-42q^{83}-52q^{81}-21q^{79}+21q^{77}+43q^{75}+25q^{73}-8q^{71}-27q^{69}-26q^{67}-8q^{65}+15q^{63}+26q^{61}+18q^{59}-7q^{57}-29q^{55}-29q^{53}-2q^{51}+34q^{49}+39q^{47}+4q^{45}-39q^{43}-48q^{41}-10q^{39}+44q^{37}+61q^{35}+15q^{33}-47q^{31}-67q^{29}-26q^{27}+46q^{25}+75q^{23}+36q^{21}-35q^{19}-77q^{17}-47q^{15}+23q^{13}+73q^{11}+58q^{9}-3q^{7}-58q^{5}-64q^{3}-14q+42q^{-1}+60q^{-3}+28q^{-5}-19q^{-7}-48q^{-9}-38q^{-11}+q^{-13}+32q^{-15}+35q^{-17}+12q^{-19}-14q^{-21}-26q^{-23}-19q^{-25}+3q^{-27}+16q^{-29}+15q^{-31}+4q^{-33}-5q^{-35}-9q^{-37}-7q^{-39}+q^{-41}+4q^{-43}+3q^{-45}+q^{-47}-2q^{-51}-q^{-53}+q^{-55}$
6	$q^{228}-q^{226}-q^{224}+q^{222}-2q^{216}+2q^{214}-q^{212}-2q^{210}+6q^{208}+4q^{206}-8q^{202}-4q^{200}-8q^{198}-3q^{196}+19q^{194}+21q^{192}+9q^{190}-19q^{188}-24q^{186}-38q^{184}-19q^{182}+41q^{180}+67q^{178}+57q^{176}-9q^{174}-58q^{172}-110q^{170}-86q^{168}+32q^{166}+126q^{164}+155q^{162}+67q^{160}-49q^{158}-188q^{156}-200q^{154}-51q^{152}+124q^{150}+237q^{148}+178q^{146}+35q^{144}-185q^{142}-273q^{140}-156q^{138}+43q^{136}+222q^{134}+233q^{132}+128q^{130}-98q^{128}-235q^{126}-194q^{124}-46q^{122}+123q^{120}+186q^{118}+156q^{116}-4q^{114}-128q^{112}-149q^{110}-83q^{108}+25q^{106}+93q^{104}+116q^{102}+47q^{100}-29q^{98}-77q^{96}-75q^{94}-33q^{92}+20q^{90}+65q^{88}+65q^{86}+35q^{84}-26q^{82}-66q^{80}-71q^{78}-24q^{76}+39q^{74}+88q^{72}+80q^{70}-6q^{68}-83q^{66}-118q^{64}-54q^{62}+43q^{60}+133q^{58}+132q^{56}+7q^{54}-115q^{52}-173q^{50}-92q^{48}+42q^{46}+178q^{44}+191q^{42}+43q^{40}-123q^{38}-220q^{36}-149q^{34}-q^{32}+184q^{30}+239q^{28}+111q^{26}-69q^{24}-213q^{22}-200q^{20}-88q^{18}+116q^{16}+230q^{14}+175q^{12}+32q^{10}-130q^{8}-192q^{6}-166q^{4}-q^{2}+137+170q^{-2}+114q^{-4}-6q^{-6}-101q^{-8}-160q^{-10}-84q^{-12}+16q^{-14}+86q^{-16}+108q^{-18}+69q^{-20}+3q^{-22}-78q^{-24}-76q^{-26}-47q^{-28}+39q^{-32}+56q^{-34}+43q^{-36}-8q^{-38}-23q^{-40}-32q^{-42}-23q^{-44}-6q^{-46}+14q^{-48}+23q^{-50}+9q^{-52}+4q^{-54}-5q^{-56}-8q^{-58}-9q^{-60}-q^{-62}+4q^{-64}+q^{-66}+3q^{-68}+q^{-70}-2q^{-74}-q^{-76}+q^{-78}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^{6}-q^{4}+q^{2}+q^{-4}$
1,1	$q^{52}-2q^{50}+4q^{48}-6q^{46}+11q^{44}-14q^{42}+16q^{40}-22q^{38}+19q^{36}-18q^{34}+10q^{32}-2q^{30}-7q^{28}+22q^{26}-28q^{24}+36q^{22}-39q^{20}+36q^{18}-34q^{16}+26q^{14}-16q^{12}+8q^{10}+6q^{8}-10q^{6}+19q^{4}-20q^{2}+22-18q^{-2}+13q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{50}+q^{48}-2q^{44}-q^{42}+q^{40}-q^{38}-2q^{36}+2q^{32}-2q^{28}+q^{26}+q^{24}-q^{22}+q^{20}+q^{18}+2q^{12}-q^{8}+q^{6}+3q^{4}-1+2q^{-2}+q^{-4}-q^{-6}-q^{-8}+q^{-12}$
3,0	$-q^{90}-q^{88}+q^{84}+3q^{82}-q^{78}-q^{76}+2q^{74}+5q^{72}-2q^{70}-5q^{68}-5q^{66}+3q^{64}+7q^{62}-q^{60}-7q^{58}-6q^{56}+4q^{54}+8q^{52}-5q^{48}-2q^{46}+5q^{44}+3q^{42}-3q^{40}-4q^{38}-q^{36}-4q^{32}-2q^{30}+q^{28}+4q^{26}+q^{24}-2q^{22}+3q^{20}+8q^{18}+5q^{16}-3q^{14}-6q^{12}+3q^{10}+7q^{8}+2q^{6}-6q^{4}-6q^{2}+4+6q^{-2}+2q^{-4}-4q^{-6}-4q^{-8}+2q^{-10}+3q^{-12}+3q^{-14}-q^{-16}-2q^{-18}-q^{-20}+q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{42}-q^{40}+2q^{36}-3q^{34}-2q^{32}+2q^{30}-3q^{28}-q^{26}+3q^{24}+q^{20}+q^{18}+2q^{16}-q^{12}+2q^{10}+q^{8}-3q^{6}+2q^{4}+2q^{2}-2+2q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$-q^{27}-q^{25}-q^{23}+q^{21}+2q^{17}+q^{15}+q^{13}-q^{5}+q^{3}+q^{-1}+q^{-5}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{56}+q^{54}-q^{50}-3q^{44}-4q^{42}-2q^{36}+q^{34}+4q^{32}+2q^{30}+2q^{26}+q^{24}-2q^{22}-q^{20}+3q^{18}-q^{16}+4q^{12}+2q^{10}-2q^{8}+2q^{4}-1+q^{-2}+2q^{-4}+q^{-10}$
1,0,0,0	$-q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+2q^{22}+2q^{20}+q^{18}+q^{16}-q^{10}-q^{6}+q^{4}+1+q^{-2}+q^{-6}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}+q^{40}-2q^{38}+2q^{36}-3q^{34}+2q^{32}-2q^{30}+q^{28}+q^{26}-q^{24}+4q^{22}-3q^{20}+5q^{18}-4q^{16}+4q^{14}-3q^{12}+2q^{10}-q^{8}-q^{6}+2q^{4}-2q^{2}+2-2q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{68}-q^{64}-q^{62}+q^{60}+2q^{58}-3q^{54}-2q^{52}+q^{50}+2q^{48}-q^{46}-3q^{44}+2q^{40}+q^{38}-2q^{36}+2q^{32}+2q^{30}-q^{28}-q^{26}+q^{24}+2q^{22}-q^{18}+2q^{14}+q^{12}-2q^{10}-2q^{8}+2q^{6}+3q^{4}-2-q^{-2}+2q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

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

.

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["7 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}-t^{-2}+5t+5t^{-1}-7$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 19, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{6}+2a^{4}z^{2}+2a^{4}-a^{2}z^{4}-2a^{2}z^{2}-a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{7}z^{3}-a^{7}z+2a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{5}-a^{5}z^{3}+a^{4}z^{6}+2a^{4}z^{4}-4a^{4}z^{2}+2a^{4}+4a^{3}z^{5}-6a^{3}z^{3}+2a^{3}z+a^{2}z^{6}+a^{2}z^{4}-4a^{2}z^{2}+a^{2}+2az^{5}-4az^{3}+az+z^{4}-2z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_133,}

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["7 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]=

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

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]=

{10_133,}

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:

(1, -2)

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 $4$ $-16$ $8$ ${\frac {158}{3}}$ ${\frac {34}{3}}$ $-64$ $-{\frac {544}{3}}$ $-{\frac {64}{3}}$ $-48$ ${\frac {32}{3}}$ $128$ ${\frac {632}{3}}$ ${\frac {136}{3}}$ ${\frac {19471}{30}}$ $-{\frac {474}{5}}$ ${\frac {17342}{45}}$ ${\frac {401}{18}}$ ${\frac {1711}{30}}$

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 $s=$-2 is the signature of 7 6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 χ 3               1 1 1             1   -1 -1           2 1   1 -3         2 2     0 -5       2 1       1 -7     1 2         1 -9   1 2           -1 -11   1             1 -13 1               -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{4}-2q^{3}-q^{2}+6q-4-5q^{-1}+12q^{-2}-5q^{-3}-10q^{-4}+16q^{-5}-4q^{-6}-13q^{-7}+17q^{-8}-3q^{-9}-12q^{-10}+12q^{-11}-q^{-12}-7q^{-13}+5q^{-14}-2q^{-16}+q^{-17}$
3	$q^{9}-2q^{8}-q^{7}+2q^{6}+5q^{5}-3q^{4}-9q^{3}+2q^{2}+14q-18q^{-1}-5q^{-2}+24q^{-3}+9q^{-4}-26q^{-5}-15q^{-6}+30q^{-7}+19q^{-8}-29q^{-9}-26q^{-10}+33q^{-11}+26q^{-12}-30q^{-13}-31q^{-14}+31q^{-15}+28q^{-16}-25q^{-17}-29q^{-18}+22q^{-19}+25q^{-20}-16q^{-21}-20q^{-22}+10q^{-23}+15q^{-24}-6q^{-25}-10q^{-26}+3q^{-27}+6q^{-28}-2q^{-29}-2q^{-30}+2q^{-32}-q^{-33}$
4	$q^{16}-2q^{15}-q^{14}+2q^{13}+q^{12}+6q^{11}-7q^{10}-7q^{9}+2q^{7}+24q^{6}-8q^{5}-17q^{4}-12q^{3}-6q^{2}+49q+3-18q^{-1}-32q^{-2}-31q^{-3}+71q^{-4}+23q^{-5}-6q^{-6}-50q^{-7}-64q^{-8}+81q^{-9}+43q^{-10}+16q^{-11}-62q^{-12}-96q^{-13}+83q^{-14}+58q^{-15}+36q^{-16}-69q^{-17}-118q^{-18}+80q^{-19}+66q^{-20}+50q^{-21}-69q^{-22}-127q^{-23}+72q^{-24}+64q^{-25}+57q^{-26}-57q^{-27}-119q^{-28}+52q^{-29}+51q^{-30}+57q^{-31}-36q^{-32}-93q^{-33}+29q^{-34}+28q^{-35}+44q^{-36}-13q^{-37}-56q^{-38}+12q^{-39}+7q^{-40}+25q^{-41}-q^{-42}-25q^{-43}+6q^{-44}-q^{-45}+9q^{-46}+q^{-47}-8q^{-48}+3q^{-49}-q^{-50}+2q^{-51}-2q^{-53}+q^{-54}$
5	$q^{25}-2q^{24}-q^{23}+2q^{22}+q^{21}+2q^{20}+2q^{19}-5q^{18}-9q^{17}+5q^{15}+11q^{14}+13q^{13}-4q^{12}-22q^{11}-22q^{10}-2q^{9}+23q^{8}+39q^{7}+19q^{6}-25q^{5}-53q^{4}-41q^{3}+13q^{2}+68q+66+7q^{-1}-71q^{-2}-97q^{-3}-37q^{-4}+74q^{-5}+121q^{-6}+68q^{-7}-56q^{-8}-147q^{-9}-107q^{-10}+44q^{-11}+163q^{-12}+139q^{-13}-17q^{-14}-176q^{-15}-179q^{-16}+3q^{-17}+183q^{-18}+201q^{-19}+29q^{-20}-193q^{-21}-233q^{-22}-35q^{-23}+192q^{-24}+244q^{-25}+64q^{-26}-198q^{-27}-269q^{-28}-62q^{-29}+192q^{-30}+266q^{-31}+89q^{-32}-190q^{-33}-280q^{-34}-85q^{-35}+174q^{-36}+265q^{-37}+108q^{-38}-157q^{-39}-262q^{-40}-107q^{-41}+132q^{-42}+234q^{-43}+118q^{-44}-103q^{-45}-206q^{-46}-115q^{-47}+71q^{-48}+170q^{-49}+105q^{-50}-41q^{-51}-129q^{-52}-91q^{-53}+17q^{-54}+90q^{-55}+73q^{-56}-3q^{-57}-56q^{-58}-53q^{-59}-4q^{-60}+32q^{-61}+32q^{-62}+8q^{-63}-16q^{-64}-21q^{-65}-4q^{-66}+10q^{-67}+6q^{-68}+4q^{-69}-q^{-70}-8q^{-71}+4q^{-73}-q^{-74}+q^{-76}-2q^{-77}+2q^{-79}-q^{-80}$
6	$q^{36}-2q^{35}-q^{34}+2q^{33}+q^{32}+2q^{31}-2q^{30}+4q^{29}-7q^{28}-9q^{27}+3q^{26}+4q^{25}+11q^{24}+3q^{23}+18q^{22}-16q^{21}-29q^{20}-14q^{19}-5q^{18}+20q^{17}+18q^{16}+69q^{15}-3q^{14}-46q^{13}-53q^{12}-52q^{11}-9q^{10}+16q^{9}+150q^{8}+63q^{7}-7q^{6}-75q^{5}-122q^{4}-109q^{3}-60q^{2}+209q+158+113q^{-1}-19q^{-2}-155q^{-3}-247q^{-4}-225q^{-5}+183q^{-6}+220q^{-7}+275q^{-8}+124q^{-9}-100q^{-10}-361q^{-11}-429q^{-12}+71q^{-13}+207q^{-14}+419q^{-15}+304q^{-16}+28q^{-17}-416q^{-18}-614q^{-19}-77q^{-20}+136q^{-21}+516q^{-22}+470q^{-23}+176q^{-24}-429q^{-25}-750q^{-26}-211q^{-27}+55q^{-28}+574q^{-29}+592q^{-30}+301q^{-31}-428q^{-32}-840q^{-33}-308q^{-34}-9q^{-35}+609q^{-36}+669q^{-37}+387q^{-38}-420q^{-39}-889q^{-40}-371q^{-41}-56q^{-42}+614q^{-43}+709q^{-44}+448q^{-45}-390q^{-46}-889q^{-47}-416q^{-48}-109q^{-49}+572q^{-50}+707q^{-51}+496q^{-52}-314q^{-53}-820q^{-54}-439q^{-55}-177q^{-56}+464q^{-57}+641q^{-58}+517q^{-59}-190q^{-60}-660q^{-61}-409q^{-62}-240q^{-63}+295q^{-64}+493q^{-65}+476q^{-66}-53q^{-67}-434q^{-68}-304q^{-69}-251q^{-70}+116q^{-71}+294q^{-72}+359q^{-73}+35q^{-74}-214q^{-75}-161q^{-76}-192q^{-77}+3q^{-78}+119q^{-79}+210q^{-80}+47q^{-81}-75q^{-82}-45q^{-83}-104q^{-84}-26q^{-85}+25q^{-86}+92q^{-87}+23q^{-88}-23q^{-89}+4q^{-90}-38q^{-91}-16q^{-92}-q^{-93}+32q^{-94}+4q^{-95}-9q^{-96}+9q^{-97}-10q^{-98}-4q^{-99}-3q^{-100}+10q^{-101}-q^{-102}-5q^{-103}+5q^{-104}-2q^{-105}-q^{-107}+2q^{-108}-2q^{-110}+q^{-111}$
7	$q^{49}-2q^{48}-q^{47}+2q^{46}+q^{45}+2q^{44}-2q^{43}+2q^{41}-7q^{40}-6q^{39}+2q^{38}+4q^{37}+13q^{36}+4q^{35}+q^{34}+9q^{33}-20q^{32}-25q^{31}-17q^{30}-7q^{29}+30q^{28}+29q^{27}+29q^{26}+44q^{25}-15q^{24}-51q^{23}-67q^{22}-82q^{21}+3q^{20}+41q^{19}+80q^{18}+146q^{17}+66q^{16}-10q^{15}-103q^{14}-210q^{13}-137q^{12}-61q^{11}+61q^{10}+264q^{9}+243q^{8}+177q^{7}+14q^{6}-281q^{5}-334q^{4}-328q^{3}-151q^{2}+244q+402+483q^{-1}+344q^{-2}-133q^{-3}-423q^{-4}-643q^{-5}-568q^{-6}-19q^{-7}+374q^{-8}+753q^{-9}+804q^{-10}+251q^{-11}-270q^{-12}-838q^{-13}-1033q^{-14}-485q^{-15}+115q^{-16}+846q^{-17}+1231q^{-18}+752q^{-19}+88q^{-20}-830q^{-21}-1410q^{-22}-986q^{-23}-293q^{-24}+764q^{-25}+1525q^{-26}+1227q^{-27}+515q^{-28}-695q^{-29}-1640q^{-30}-1409q^{-31}-698q^{-32}+593q^{-33}+1702q^{-34}+1594q^{-35}+882q^{-36}-533q^{-37}-1773q^{-38}-1707q^{-39}-1014q^{-40}+442q^{-41}+1802q^{-42}+1843q^{-43}+1142q^{-44}-411q^{-45}-1852q^{-46}-1903q^{-47}-1215q^{-48}+335q^{-49}+1857q^{-50}+1997q^{-51}+1310q^{-52}-320q^{-53}-1889q^{-54}-2017q^{-55}-1348q^{-56}+242q^{-57}+1855q^{-58}+2076q^{-59}+1433q^{-60}-210q^{-61}-1843q^{-62}-2062q^{-63}-1459q^{-64}+103q^{-65}+1748q^{-66}+2069q^{-67}+1529q^{-68}-21q^{-69}-1657q^{-70}-1998q^{-71}-1541q^{-72}-114q^{-73}+1477q^{-74}+1907q^{-75}+1561q^{-76}+239q^{-77}-1278q^{-78}-1748q^{-79}-1517q^{-80}-374q^{-81}+1031q^{-82}+1536q^{-83}+1432q^{-84}+486q^{-85}-768q^{-86}-1277q^{-87}-1301q^{-88}-551q^{-89}+514q^{-90}+994q^{-91}+1110q^{-92}+571q^{-93}-285q^{-94}-715q^{-95}-892q^{-96}-545q^{-97}+117q^{-98}+465q^{-99}+671q^{-100}+465q^{-101}-9q^{-102}-259q^{-103}-464q^{-104}-366q^{-105}-50q^{-106}+120q^{-107}+296q^{-108}+265q^{-109}+61q^{-110}-41q^{-111}-170q^{-112}-167q^{-113}-48q^{-114}-13q^{-115}+89q^{-116}+110q^{-117}+34q^{-118}+12q^{-119}-49q^{-120}-48q^{-121}-11q^{-122}-26q^{-123}+14q^{-124}+37q^{-125}+10q^{-126}+8q^{-127}-15q^{-128}-9q^{-129}+7q^{-130}-14q^{-131}-q^{-132}+10q^{-133}+2q^{-134}+3q^{-135}-6q^{-136}-q^{-137}+6q^{-138}-4q^{-139}-2q^{-140}+2q^{-141}+q^{-143}-2q^{-144}+2q^{-146}-q^{-147}$

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.

7_5 7_7