8 4

8_3

8_5

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 4 at Knotilus!

Somewhat symmetric representation

Knot presentations

Planar diagram presentation	X6271 X14,10,15,9 X10,3,11,4 X2,13,3,14 X12,5,13,6 X16,8,1,7 X4,11,5,12 X8,16,9,15
Gauss code	1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code	6 10 12 16 14 4 2 8
Conway Notation	[413]

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

[edit Notes on presentations of 8 4]

Knot 8_4.

A graph, knot 8_4.

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 4"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X14,10,15,9 X10,3,11,4 X2,13,3,14 X12,5,13,6 X16,8,1,7 X4,11,5,12 X8,16,9,15

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 12 16 14 4 2 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[413]

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]=

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

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[{2, 10}, {1, 3}, {4, 2}, {3, 5}, {9, 4}, {10, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 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 $\{3,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 5.50049 A-Polynomial See Data:8 4/A-polynomial

[edit Notes for 8 4'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 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+5t-5+5t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, -2 }
Jones polynomial	$q^{3}-q^{2}+2q-3+3q^{-1}-3q^{-2}+3q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-z^{4}-3z^{2}-2+z^{2}a^{-2}+2a^{-2}$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+z^{6}a^{-2}+3z^{6}+3a^{3}z^{5}-az^{5}-4z^{5}a^{-1}+3a^{4}z^{4}-3a^{2}z^{4}-5z^{4}a^{-2}-11z^{4}+2a^{5}z^{3}-5a^{3}z^{3}-3az^{3}+4z^{3}a^{-1}+a^{6}z^{2}-3a^{4}z^{2}-a^{2}z^{2}+7z^{2}a^{-2}+10z^{2}+2a^{3}z+az-za^{-1}+a^{4}-2a^{-2}-2$
The A2 invariant	$q^{16}+q^{10}+q^{6}-q^{4}-q^{2}-1-q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}$
The G2 invariant	$q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3q^{72}-2q^{70}+2q^{68}-2q^{66}+q^{62}-2q^{60}+3q^{58}-2q^{56}+q^{54}+q^{50}+2q^{48}-q^{46}+2q^{44}+2q^{38}-q^{36}+q^{34}+2q^{32}-2q^{30}+3q^{28}-3q^{26}+2q^{22}-6q^{20}+5q^{18}-5q^{16}+2q^{12}-4q^{10}+3q^{8}-4q^{6}+q^{4}-q^{2}-2+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}-q^{-14}+q^{-16}+3q^{-18}-4q^{-20}+4q^{-22}-2q^{-24}+q^{-26}+4q^{-28}-4q^{-30}+4q^{-32}-q^{-34}+q^{-36}+q^{-38}-2q^{-40}+2q^{-42}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 8_4.

A1 Invariants.

Weight	Invariant
1	$q^{11}-q^{9}+q^{7}-q^{-1}+q^{-3}+q^{-7}$
2	$q^{30}-q^{28}+2q^{24}-2q^{22}+q^{18}-2q^{16}+q^{12}-q^{8}+q^{6}+2q^{4}+2q^{-2}-2q^{-6}+q^{-8}-2q^{-12}+q^{-14}+q^{-16}-q^{-18}+q^{-22}$
3	$q^{57}-q^{55}+q^{51}-q^{47}-q^{45}+q^{43}-q^{41}-q^{39}+2q^{37}+q^{35}-2q^{33}-q^{31}+3q^{29}+2q^{27}-q^{25}-2q^{23}-q^{21}+2q^{19}+2q^{17}-3q^{13}+3q^{9}-3q^{5}-q^{3}+2q+q^{-1}-q^{-3}-q^{-5}+2q^{-7}+2q^{-9}-2q^{-13}+q^{-15}+3q^{-17}+q^{-19}-3q^{-21}-2q^{-23}+2q^{-25}+2q^{-27}-q^{-29}-3q^{-31}+2q^{-35}+q^{-37}-q^{-39}-q^{-41}+q^{-45}$
4	$q^{92}-q^{90}+q^{86}-q^{84}+q^{82}-2q^{80}-2q^{74}+4q^{72}-2q^{66}-4q^{64}+5q^{62}+4q^{60}+2q^{58}-5q^{56}-8q^{54}+3q^{52}+7q^{50}+6q^{48}-2q^{46}-9q^{44}-3q^{42}+2q^{40}+5q^{38}+3q^{36}-2q^{34}-3q^{32}-5q^{30}+5q^{26}+4q^{24}-q^{22}-6q^{20}-3q^{18}+3q^{16}+6q^{14}+q^{12}-4q^{10}-2q^{8}+4q^{6}+6q^{4}-q^{2}-3-2q^{-2}+3q^{-4}+5q^{-6}-2q^{-8}-4q^{-10}-5q^{-12}+6q^{-16}+q^{-18}-5q^{-22}-4q^{-24}+4q^{-26}+4q^{-28}+5q^{-30}-2q^{-32}-6q^{-34}-2q^{-36}+q^{-38}+7q^{-40}+3q^{-42}-3q^{-44}-4q^{-46}-4q^{-48}+3q^{-50}+4q^{-52}+2q^{-54}-q^{-56}-5q^{-58}-q^{-60}+q^{-62}+2q^{-64}+2q^{-66}-q^{-68}-q^{-70}-q^{-72}+q^{-76}$
5	$q^{135}-q^{133}+q^{129}-q^{127}-q^{121}-q^{119}+q^{115}+2q^{113}+2q^{111}-q^{109}-5q^{107}-4q^{105}+4q^{103}+8q^{101}+5q^{99}-3q^{97}-12q^{95}-8q^{93}+5q^{91}+16q^{89}+11q^{87}-4q^{85}-17q^{83}-17q^{81}-q^{79}+18q^{77}+20q^{75}+4q^{73}-14q^{71}-21q^{69}-11q^{67}+7q^{65}+19q^{63}+14q^{61}-q^{59}-11q^{57}-13q^{55}-7q^{53}+4q^{51}+11q^{49}+12q^{47}+4q^{45}-3q^{43}-11q^{41}-11q^{39}+10q^{35}+11q^{33}+3q^{31}-9q^{29}-11q^{27}-3q^{25}+7q^{23}+9q^{21}+2q^{19}-7q^{17}-7q^{15}+8q^{11}+8q^{9}-2q^{7}-10q^{5}-10q^{3}+q+12q^{-1}+12q^{-3}-11q^{-7}-13q^{-9}-2q^{-11}+11q^{-13}+16q^{-15}+7q^{-17}-5q^{-19}-13q^{-21}-11q^{-23}+2q^{-25}+12q^{-27}+11q^{-29}+4q^{-31}-7q^{-33}-13q^{-35}-10q^{-37}+q^{-39}+9q^{-41}+10q^{-43}+6q^{-45}-4q^{-47}-11q^{-49}-10q^{-51}-q^{-53}+7q^{-55}+11q^{-57}+8q^{-59}-q^{-61}-9q^{-63}-10q^{-65}-4q^{-67}+4q^{-69}+9q^{-71}+8q^{-73}+q^{-75}-5q^{-77}-8q^{-79}-5q^{-81}+q^{-83}+5q^{-85}+6q^{-87}+3q^{-89}-2q^{-91}-5q^{-93}-3q^{-95}-q^{-97}+q^{-99}+3q^{-101}+2q^{-103}-q^{-107}-q^{-109}-q^{-111}+q^{-115}$
6	$q^{186}-q^{184}+q^{180}-q^{178}-q^{174}+q^{172}-2q^{170}-q^{168}+3q^{166}+2q^{162}+q^{160}-q^{158}-7q^{156}-3q^{154}+6q^{152}+5q^{150}+7q^{148}+q^{146}-7q^{144}-16q^{142}-5q^{140}+13q^{138}+15q^{136}+13q^{134}-4q^{132}-21q^{130}-28q^{128}-6q^{126}+24q^{124}+30q^{122}+22q^{120}-6q^{118}-35q^{116}-42q^{114}-15q^{112}+26q^{110}+43q^{108}+38q^{106}+8q^{104}-32q^{102}-52q^{100}-35q^{98}+5q^{96}+37q^{94}+49q^{92}+31q^{90}-5q^{88}-38q^{86}-44q^{84}-23q^{82}+3q^{80}+28q^{78}+35q^{76}+24q^{74}+q^{72}-19q^{70}-25q^{68}-24q^{66}-10q^{64}+9q^{62}+26q^{60}+25q^{58}+12q^{56}-8q^{54}-25q^{52}-25q^{50}-11q^{48}+13q^{46}+22q^{44}+18q^{42}-16q^{38}-17q^{36}-7q^{34}+12q^{32}+16q^{30}+7q^{28}-9q^{26}-16q^{24}-9q^{22}+5q^{20}+22q^{18}+20q^{16}+2q^{14}-19q^{12}-24q^{10}-15q^{8}+6q^{6}+28q^{4}+30q^{2}+10-18q^{-2}-30q^{-4}-28q^{-6}-6q^{-8}+24q^{-10}+38q^{-12}+24q^{-14}-6q^{-16}-27q^{-18}-37q^{-20}-23q^{-22}+7q^{-24}+34q^{-26}+35q^{-28}+13q^{-30}-9q^{-32}-31q^{-34}-33q^{-36}-15q^{-38}+15q^{-40}+32q^{-42}+28q^{-44}+17q^{-46}-9q^{-48}-27q^{-50}-30q^{-52}-12q^{-54}+7q^{-56}+19q^{-58}+28q^{-60}+16q^{-62}-q^{-64}-20q^{-66}-23q^{-68}-18q^{-70}-7q^{-72}+14q^{-74}+21q^{-76}+21q^{-78}+7q^{-80}-5q^{-82}-18q^{-84}-23q^{-86}-11q^{-88}+q^{-90}+15q^{-92}+17q^{-94}+17q^{-96}+4q^{-98}-11q^{-100}-15q^{-102}-15q^{-104}-6q^{-106}+2q^{-108}+14q^{-110}+14q^{-112}+7q^{-114}-7q^{-118}-10q^{-120}-11q^{-122}-q^{-124}+4q^{-126}+7q^{-128}+7q^{-130}+4q^{-132}-6q^{-136}-4q^{-138}-3q^{-140}-q^{-142}+q^{-144}+3q^{-146}+3q^{-148}-q^{-154}-q^{-156}-q^{-158}+q^{-162}$

A2 Invariants.

Weight	Invariant
1,0	$q^{16}+q^{10}+q^{6}-q^{4}-q^{2}-1-q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}$
1,1	$q^{44}-2q^{42}+2q^{40}-2q^{38}+5q^{36}-4q^{34}+4q^{32}-4q^{30}+5q^{28}-4q^{26}+2q^{24}-4q^{22}+2q^{20}-2q^{18}-2q^{16}+2q^{14}-6q^{12}+10q^{10}-10q^{8}+16q^{6}-10q^{4}+16q^{2}-8+8q^{-2}-7q^{-4}-6q^{-10}+5q^{-12}-8q^{-14}+10q^{-16}-8q^{-18}+6q^{-20}-4q^{-22}+4q^{-24}+q^{-28}$
2,0	$q^{40}+q^{34}+q^{32}-q^{28}-q^{24}-3q^{22}+q^{18}-q^{14}+2q^{6}+3q^{4}+3q^{2}+2+3q^{-2}-2q^{-6}-2q^{-8}-2q^{-10}-2q^{-12}-2q^{-14}+q^{-18}+q^{-20}+q^{-24}+q^{-26}+q^{-28}$
3,0	$q^{72}+q^{66}+q^{64}+q^{62}-2q^{60}-q^{58}-q^{54}-2q^{52}-3q^{50}+q^{48}+3q^{46}-4q^{42}-4q^{40}+3q^{38}+6q^{36}+4q^{34}-q^{32}+3q^{28}+3q^{26}+q^{24}-2q^{22}+q^{18}-q^{14}-2q^{8}-5q^{6}-3q^{4}-q^{2}-1-3q^{-2}-3q^{-4}+2q^{-6}+5q^{-8}+6q^{-10}+2q^{-12}+3q^{-14}+5q^{-16}+6q^{-18}+3q^{-20}-2q^{-22}-4q^{-24}-3q^{-26}-q^{-28}-2q^{-32}-3q^{-34}-2q^{-36}+2q^{-40}+q^{-42}-q^{-46}+q^{-50}+q^{-52}+q^{-54}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{36}-q^{34}-q^{32}+2q^{30}-q^{26}+2q^{24}-q^{20}+q^{18}+q^{16}+q^{12}+q^{10}-2q^{6}-q^{4}+q^{2}-2+q^{-4}-q^{-6}+q^{-10}+q^{-14}+2q^{-16}+q^{-20}$
1,0,0	$q^{21}+q^{17}+q^{13}+q^{9}-q^{5}-q^{3}-2q-q^{-1}-q^{-3}+q^{-5}+q^{-7}+2q^{-9}+q^{-11}+q^{-13}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{46}-q^{42}+q^{38}+q^{36}+q^{32}+2q^{30}-q^{26}-2q^{20}+q^{10}+2q^{8}+2q^{4}+3q^{2}+1-2q^{-2}-q^{-4}-2q^{-6}-4q^{-8}-3q^{-10}-q^{-12}+q^{-14}+q^{-16}+3q^{-18}+3q^{-20}+2q^{-22}+q^{-24}+q^{-26}$
1,0,0,0	$q^{26}+q^{22}+q^{20}+q^{16}+q^{12}-q^{6}-q^{4}-2q^{2}-2-q^{-2}-q^{-4}+q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+q^{-14}+q^{-16}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{50}-q^{48}-q^{44}+2q^{42}-q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}+q^{24}+2q^{20}-q^{18}+3q^{16}-2q^{14}+3q^{12}-3q^{10}+q^{8}-3q^{6}-3q^{2}-q^{-2}-q^{-4}-q^{-8}+2q^{-10}+2q^{-14}+3q^{-18}+2q^{-22}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3q^{72}-2q^{70}+2q^{68}-2q^{66}+q^{62}-2q^{60}+3q^{58}-2q^{56}+q^{54}+q^{50}+2q^{48}-q^{46}+2q^{44}+2q^{38}-q^{36}+q^{34}+2q^{32}-2q^{30}+3q^{28}-3q^{26}+2q^{22}-6q^{20}+5q^{18}-5q^{16}+2q^{12}-4q^{10}+3q^{8}-4q^{6}+q^{4}-q^{2}-2+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}-q^{-14}+q^{-16}+3q^{-18}-4q^{-20}+4q^{-22}-2q^{-24}+q^{-26}+4q^{-28}-4q^{-30}+4q^{-32}-q^{-34}+q^{-36}+q^{-38}-2q^{-40}+2q^{-42}+q^{-46}$

.

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 4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{4}-3z^{2}+1$

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-z^{4}-3z^{2}-2+z^{2}a^{-2}+2a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+z^{6}a^{-2}+3z^{6}+3a^{3}z^{5}-az^{5}-4z^{5}a^{-1}+3a^{4}z^{4}-3a^{2}z^{4}-5z^{4}a^{-2}-11z^{4}+2a^{5}z^{3}-5a^{3}z^{3}-3az^{3}+4z^{3}a^{-1}+a^{6}z^{2}-3a^{4}z^{2}-a^{2}z^{2}+7z^{2}a^{-2}+10z^{2}+2a^{3}z+az-za^{-1}+a^{4}-2a^{-2}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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 4"];

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]=

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

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]=

{}

Vassiliev invariants

V2 and V3:

(-3, 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 $-12$ $8$ $72$ $114$ $54$ $-96$ $-{\frac {496}{3}}$ $-{\frac {160}{3}}$ $-24$ $-288$ $32$ $-1368$ $-648$ $-{\frac {11871}{10}}$ ${\frac {2182}{5}}$ $-{\frac {20902}{15}}$ ${\frac {1055}{6}}$ $-{\frac {3391}{10}}$

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 8 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{10}-q^{9}-q^{8}+3q^{7}-q^{6}-4q^{5}+5q^{4}-7q^{2}+7q+2-9q^{-1}+7q^{-2}+4q^{-3}-10q^{-4}+5q^{-5}+5q^{-6}-9q^{-7}+4q^{-8}+3q^{-9}-6q^{-10}+3q^{-11}+q^{-12}-2q^{-13}+q^{-14}$
3	$q^{21}-q^{20}-q^{19}+3q^{17}-3q^{15}-3q^{14}+5q^{13}+3q^{12}-3q^{11}-7q^{10}+4q^{9}+7q^{8}-q^{7}-9q^{6}+q^{5}+9q^{4}+q^{3}-9q^{2}-2q+9+3q^{-1}-8q^{-2}-5q^{-3}+7q^{-4}+6q^{-5}-5q^{-6}-8q^{-7}+4q^{-8}+9q^{-9}-3q^{-10}-8q^{-11}+q^{-12}+8q^{-13}-2q^{-14}-5q^{-15}+2q^{-16}+4q^{-17}-3q^{-18}-2q^{-19}+3q^{-20}+q^{-21}-3q^{-22}+q^{-24}+q^{-25}-2q^{-26}+q^{-27}$
4	$q^{36}-q^{35}-q^{34}+4q^{31}-q^{30}-2q^{29}-2q^{28}-4q^{27}+8q^{26}+2q^{25}-3q^{23}-11q^{22}+8q^{21}+3q^{20}+6q^{19}+q^{18}-17q^{17}+5q^{16}-q^{15}+10q^{14}+8q^{13}-18q^{12}+5q^{11}-9q^{10}+9q^{9}+13q^{8}-17q^{7}+10q^{6}-15q^{5}+4q^{4}+14q^{3}-15q^{2}+17q-17-q^{-1}+13q^{-2}-13q^{-3}+24q^{-4}-19q^{-5}-7q^{-6}+11q^{-7}-8q^{-8}+29q^{-9}-22q^{-10}-13q^{-11}+8q^{-12}-3q^{-13}+34q^{-14}-21q^{-15}-18q^{-16}+3q^{-17}-q^{-18}+35q^{-19}-16q^{-20}-16q^{-21}-6q^{-23}+29q^{-24}-9q^{-25}-8q^{-26}+q^{-27}-10q^{-28}+18q^{-29}-6q^{-30}-q^{-31}+3q^{-32}-9q^{-33}+9q^{-34}-4q^{-35}+q^{-36}+3q^{-37}-5q^{-38}+3q^{-39}-2q^{-40}+q^{-41}+q^{-42}-2q^{-43}+q^{-44}$
5	$q^{55}-q^{54}-q^{53}+q^{50}+3q^{49}-3q^{47}-2q^{46}-2q^{45}-q^{44}+6q^{43}+5q^{42}-3q^{40}-6q^{39}-7q^{38}+3q^{37}+8q^{36}+6q^{35}+4q^{34}-5q^{33}-12q^{32}-5q^{31}+2q^{30}+7q^{29}+12q^{28}+4q^{27}-9q^{26}-9q^{25}-6q^{24}-2q^{23}+11q^{22}+11q^{21}+q^{20}-5q^{19}-7q^{18}-10q^{17}+8q^{15}+7q^{14}+6q^{13}-9q^{11}-10q^{10}-5q^{9}+5q^{8}+14q^{7}+12q^{6}-15q^{4}-18q^{3}-6q^{2}+16q+23+12q^{-1}-15q^{-2}-29q^{-3}-17q^{-4}+16q^{-5}+31q^{-6}+22q^{-7}-15q^{-8}-37q^{-9}-24q^{-10}+16q^{-11}+40q^{-12}+29q^{-13}-17q^{-14}-47q^{-15}-32q^{-16}+18q^{-17}+52q^{-18}+37q^{-19}-18q^{-20}-57q^{-21}-43q^{-22}+18q^{-23}+60q^{-24}+44q^{-25}-10q^{-26}-58q^{-27}-50q^{-28}+7q^{-29}+54q^{-30}+46q^{-31}-43q^{-33}-45q^{-34}-5q^{-35}+36q^{-36}+36q^{-37}+7q^{-38}-25q^{-39}-29q^{-40}-7q^{-41}+17q^{-42}+20q^{-43}+7q^{-44}-12q^{-45}-14q^{-46}-2q^{-47}+6q^{-48}+7q^{-49}+3q^{-50}-3q^{-51}-6q^{-52}+q^{-53}+2q^{-54}-q^{-55}+2q^{-56}+q^{-57}-3q^{-58}+q^{-59}+q^{-60}-2q^{-61}+q^{-62}+q^{-63}-2q^{-64}+q^{-65}$
6	$q^{78}-q^{77}-q^{76}+q^{73}+4q^{71}-q^{70}-3q^{69}-2q^{68}-2q^{67}-2q^{65}+10q^{64}+3q^{63}-2q^{61}-5q^{60}-5q^{59}-12q^{58}+11q^{57}+6q^{56}+7q^{55}+5q^{54}+2q^{53}-5q^{52}-24q^{51}+3q^{50}-3q^{49}+7q^{48}+9q^{47}+17q^{46}+8q^{45}-24q^{44}+q^{43}-17q^{42}-5q^{41}-3q^{40}+22q^{39}+21q^{38}-12q^{37}+15q^{36}-17q^{35}-12q^{34}-24q^{33}+11q^{32}+16q^{31}-9q^{30}+34q^{29}-33q^{26}-q^{25}-3q^{24}-27q^{23}+37q^{22}+18q^{21}+26q^{20}-22q^{19}+3q^{18}-20q^{17}-57q^{16}+19q^{15}+20q^{14}+48q^{13}+25q^{11}-21q^{10}-84q^{9}-11q^{8}+6q^{7}+58q^{6}+21q^{5}+55q^{4}-7q^{3}-98q^{2}-41q-16+56q^{-1}+33q^{-2}+83q^{-3}+13q^{-4}-100q^{-5}-63q^{-6}-37q^{-7}+47q^{-8}+38q^{-9}+104q^{-10}+31q^{-11}-98q^{-12}-80q^{-13}-51q^{-14}+40q^{-15}+45q^{-16}+120q^{-17}+40q^{-18}-102q^{-19}-99q^{-20}-61q^{-21}+41q^{-22}+61q^{-23}+138q^{-24}+44q^{-25}-111q^{-26}-123q^{-27}-75q^{-28}+41q^{-29}+78q^{-30}+158q^{-31}+57q^{-32}-110q^{-33}-140q^{-34}-94q^{-35}+27q^{-36}+77q^{-37}+164q^{-38}+77q^{-39}-87q^{-40}-129q^{-41}-101q^{-42}+2q^{-43}+51q^{-44}+143q^{-45}+83q^{-46}-54q^{-47}-93q^{-48}-83q^{-49}-10q^{-50}+19q^{-51}+103q^{-52}+66q^{-53}-34q^{-54}-53q^{-55}-53q^{-56}-5q^{-57}+2q^{-58}+62q^{-59}+39q^{-60}-27q^{-61}-24q^{-62}-25q^{-63}+3q^{-64}-4q^{-65}+32q^{-66}+17q^{-67}-20q^{-68}-7q^{-69}-8q^{-70}+5q^{-71}-6q^{-72}+14q^{-73}+6q^{-74}-11q^{-75}+q^{-76}-2q^{-77}+3q^{-78}-5q^{-79}+5q^{-80}+2q^{-81}-5q^{-82}+3q^{-83}-q^{-84}+q^{-85}-2q^{-86}+q^{-87}+q^{-88}-2q^{-89}+q^{-90}$
7	$q^{105}-q^{104}-q^{103}+q^{100}+q^{98}+3q^{97}-q^{96}-3q^{95}-2q^{94}-3q^{93}+q^{92}+9q^{89}+4q^{88}-2q^{86}-8q^{85}-3q^{84}-6q^{83}-8q^{82}+9q^{81}+9q^{80}+9q^{79}+10q^{78}-5q^{77}-9q^{75}-22q^{74}-4q^{73}-2q^{72}+7q^{71}+20q^{70}+7q^{69}+16q^{68}+6q^{67}-21q^{66}-10q^{65}-21q^{64}-16q^{63}+10q^{62}+q^{61}+25q^{60}+27q^{59}+10q^{57}-16q^{56}-30q^{55}-9q^{54}-26q^{53}+5q^{52}+24q^{51}+6q^{50}+37q^{49}+14q^{48}-9q^{47}-41q^{45}-23q^{44}-5q^{43}-22q^{42}+28q^{41}+31q^{40}+24q^{39}+43q^{38}-17q^{37}-21q^{36}-21q^{35}-67q^{34}-15q^{33}+3q^{32}+26q^{31}+80q^{30}+32q^{29}+21q^{28}-85q^{26}-59q^{25}-52q^{24}-20q^{23}+81q^{22}+67q^{21}+76q^{20}+55q^{19}-63q^{18}-71q^{17}-102q^{16}-85q^{15}+40q^{14}+67q^{13}+114q^{12}+116q^{11}-13q^{10}-51q^{9}-122q^{8}-143q^{7}-17q^{6}+34q^{5}+123q^{4}+163q^{3}+44q^{2}-9q-115-180q^{-1}-74q^{-2}-14q^{-3}+108q^{-4}+190q^{-5}+95q^{-6}+41q^{-7}-92q^{-8}-200q^{-9}-120q^{-10}-61q^{-11}+82q^{-12}+201q^{-13}+136q^{-14}+84q^{-15}-68q^{-16}-210q^{-17}-153q^{-18}-94q^{-19}+62q^{-20}+210q^{-21}+166q^{-22}+108q^{-23}-60q^{-24}-223q^{-25}-178q^{-26}-110q^{-27}+63q^{-28}+232q^{-29}+194q^{-30}+118q^{-31}-72q^{-32}-250q^{-33}-214q^{-34}-122q^{-35}+80q^{-36}+267q^{-37}+233q^{-38}+134q^{-39}-77q^{-40}-281q^{-41}-258q^{-42}-151q^{-43}+74q^{-44}+284q^{-45}+268q^{-46}+171q^{-47}-50q^{-48}-270q^{-49}-278q^{-50}-191q^{-51}+25q^{-52}+248q^{-53}+265q^{-54}+196q^{-55}+4q^{-56}-205q^{-57}-239q^{-58}-200q^{-59}-29q^{-60}+171q^{-61}+205q^{-62}+179q^{-63}+37q^{-64}-128q^{-65}-161q^{-66}-154q^{-67}-43q^{-68}+101q^{-69}+123q^{-70}+122q^{-71}+32q^{-72}-79q^{-73}-87q^{-74}-88q^{-75}-19q^{-76}+61q^{-77}+56q^{-78}+62q^{-79}+12q^{-80}-53q^{-81}-39q^{-82}-36q^{-83}+3q^{-84}+40q^{-85}+16q^{-86}+23q^{-87}-2q^{-88}-32q^{-89}-10q^{-90}-14q^{-91}+9q^{-92}+23q^{-93}+q^{-94}+5q^{-95}-4q^{-96}-14q^{-97}+q^{-98}-6q^{-99}+4q^{-100}+12q^{-101}-4q^{-102}+q^{-103}-q^{-104}-4q^{-105}+3q^{-106}-4q^{-107}+q^{-108}+5q^{-109}-4q^{-110}+q^{-111}+q^{-112}-q^{-113}+q^{-114}-2q^{-115}+q^{-116}+q^{-117}-2q^{-118}+q^{-119}$

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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