10 8

10_7

10_9

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 8 at Knotilus!

Knot presentations

Planar diagram presentation	X1627 X7,16,8,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X9,18,10,19 X11,20,12,1 X17,8,18,9 X19,10,20,11
Gauss code	-1, 6, -4, 5, -3, 1, -2, 9, -7, 10, -8, 3, -5, 4, -6, 2, -9, 7, -10, 8
Dowker-Thistlethwaite code	6 14 12 16 18 20 4 2 8 10
Conway Notation	[514]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{12, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {11, 9}, {10, 12}, {3, 5}, {4, 6}, {5, 7}, {6, 11}, {7, 1}]

[edit Notes on presentations of 10 8]

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["10 8"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X7,16,8,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X9,18,10,19 X11,20,12,1 X17,8,18,9 X19,10,20,11

In[5]:=

GaussCode[K]

Out[5]=

-1, 6, -4, 5, -3, 1, -2, 9, -7, 10, -8, 3, -5, 4, -6, 2, -9, 7, -10, 8

In[6]:=

DTCode[K]

Out[6]=

6 14 12 16 18 20 4 2 8 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[514]

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]=

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

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, 11, 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[{12, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {11, 9}, {10, 12}, {3, 5}, {4, 6}, {5, 7}, {6, 11}, {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 Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-11][-1] Hyperbolic Volume 6.08323 A-Polynomial See Data:10 8/A-polynomial

[edit Notes for 10 8'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 10 8's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+5t^{2}-5t+5-5t^{-1}+5t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-7z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, -4 }
Jones polynomial	$q^{2}-q+2-3q^{-1}+4q^{-2}-4q^{-3}+4q^{-4}-4q^{-5}+3q^{-6}-2q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{6}+3z^{2}a^{6}+a^{6}-z^{6}a^{4}-4z^{4}a^{4}-3z^{2}a^{4}-z^{6}a^{2}-5z^{4}a^{2}-7z^{2}a^{2}-3a^{2}+z^{4}+4z^{2}+3$
Kauffman polynomial (db, data sources)	$z^{2}a^{10}+2z^{3}a^{9}+3z^{4}a^{8}-2z^{2}a^{8}+4z^{5}a^{7}-7z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-10z^{4}a^{6}+5z^{2}a^{6}-a^{6}+3z^{7}a^{5}-8z^{5}a^{5}+2z^{3}a^{5}+za^{5}+2z^{8}a^{4}-6z^{6}a^{4}+z^{4}a^{4}+3z^{2}a^{4}+z^{9}a^{3}-3z^{7}a^{3}-z^{5}a^{3}+5z^{3}a^{3}-za^{3}+3z^{8}a^{2}-17z^{6}a^{2}+30z^{4}a^{2}-18z^{2}a^{2}+3a^{2}+z^{9}a-6z^{7}a+11z^{5}a-6z^{3}a+z^{8}-7z^{6}+16z^{4}-13z^{2}+3$
The A2 invariant	$q^{24}+q^{14}-q^{12}-q^{8}-q^{6}+1+q^{-2}+q^{-4}+q^{-6}$
The G2 invariant	$q^{134}-q^{132}+q^{130}-q^{128}-q^{122}+2q^{120}-2q^{118}+2q^{116}-2q^{114}+q^{110}-q^{108}+3q^{106}-3q^{104}+2q^{102}-q^{100}-q^{98}+2q^{96}-3q^{94}+3q^{92}-2q^{90}+q^{88}+q^{86}-q^{84}+2q^{82}+q^{78}+q^{74}+2q^{70}+q^{68}+q^{66}+q^{62}-q^{60}-3q^{54}+3q^{52}-5q^{50}+3q^{48}-q^{46}-5q^{44}+5q^{42}-7q^{40}+2q^{38}-2q^{36}-2q^{34}+2q^{32}-3q^{30}+3q^{28}-q^{26}-2q^{20}+q^{18}+q^{16}-q^{14}+2q^{12}-3q^{10}+3q^{8}+q^{6}-3q^{4}+6q^{2}-6+4q^{-2}+q^{-4}-3q^{-6}+6q^{-8}-4q^{-10}+4q^{-12}+2q^{-18}-2q^{-20}+2q^{-22}+q^{-26}$

Further Quantum Invariants

Further quantum knot invariants for 10_8.

A1 Invariants.

Weight	Invariant
1	$q^{17}-q^{15}+q^{13}-q^{11}+q^{3}-q+q^{-1}+q^{-5}$
2	$q^{46}-q^{44}+q^{40}-2q^{38}+q^{36}+q^{34}-q^{32}-q^{26}+q^{22}-q^{20}+2q^{16}+q^{14}-q^{12}+q^{8}-q^{6}-q^{4}+2q^{2}-q^{-2}+2q^{-4}-2q^{-8}+q^{-10}+q^{-12}-q^{-14}+q^{-18}$
3	$q^{87}-q^{85}+q^{73}-q^{71}-q^{69}-q^{67}+q^{65}+q^{63}-2q^{59}+4q^{55}+2q^{53}-3q^{51}-2q^{49}+q^{47}+2q^{45}+q^{43}-q^{41}-2q^{39}-q^{37}+3q^{35}+q^{33}-3q^{31}-3q^{29}+q^{27}+q^{25}-q^{23}-q^{21}+q^{19}+q^{17}+q^{15}+q^{11}+2q^{9}+q^{7}-q^{5}-q^{3}+q+2q^{-1}-3q^{-5}-2q^{-7}+2q^{-9}+3q^{-11}-3q^{-15}-q^{-17}+3q^{-19}+2q^{-21}-q^{-23}-3q^{-25}+2q^{-29}+q^{-31}-q^{-33}-q^{-35}+q^{-39}$
4	$q^{140}-q^{138}-q^{132}+2q^{130}-q^{128}+q^{126}-q^{124}-2q^{122}+q^{120}-2q^{118}+3q^{116}+2q^{114}-2q^{112}-2q^{110}-2q^{108}+6q^{106}+5q^{104}-2q^{102}-6q^{100}-7q^{98}+6q^{96}+9q^{94}+2q^{92}-6q^{90}-11q^{88}+q^{86}+8q^{84}+6q^{82}-q^{80}-10q^{78}-5q^{76}+2q^{74}+6q^{72}+8q^{70}-q^{68}-7q^{66}-7q^{64}+9q^{60}+7q^{58}-q^{56}-8q^{54}-6q^{52}+4q^{50}+8q^{48}+2q^{46}-4q^{44}-4q^{42}+2q^{40}+3q^{38}-2q^{36}-4q^{34}-2q^{32}+3q^{30}+5q^{28}-2q^{26}-5q^{24}-3q^{22}+2q^{20}+7q^{18}+2q^{16}-q^{14}-4q^{12}-5q^{10}+3q^{8}+4q^{6}+5q^{4}+q^{2}-7-3q^{-2}+6q^{-6}+7q^{-8}-2q^{-10}-4q^{-12}-6q^{-14}+7q^{-18}+4q^{-20}+q^{-22}-6q^{-24}-6q^{-26}+3q^{-30}+7q^{-32}+q^{-34}-4q^{-36}-4q^{-38}-3q^{-40}+4q^{-42}+4q^{-44}+2q^{-46}-q^{-48}-5q^{-50}-q^{-52}+q^{-54}+2q^{-56}+2q^{-58}-q^{-60}-q^{-62}-q^{-64}+q^{-68}$
5	$q^{205}-q^{203}-q^{197}+q^{195}+q^{193}-q^{189}-q^{187}-2q^{185}+2q^{181}+3q^{179}+q^{177}-2q^{175}-4q^{173}-2q^{171}+5q^{169}+8q^{167}+2q^{165}-8q^{163}-11q^{161}-3q^{159}+10q^{157}+16q^{155}+3q^{153}-15q^{151}-16q^{149}-4q^{147}+13q^{145}+19q^{143}+5q^{141}-15q^{139}-20q^{137}-5q^{135}+14q^{133}+21q^{131}+8q^{129}-12q^{127}-24q^{125}-12q^{123}+11q^{121}+22q^{119}+15q^{117}-3q^{115}-16q^{113}-18q^{111}-6q^{109}+10q^{107}+16q^{105}+13q^{103}+3q^{101}-11q^{99}-19q^{97}-15q^{95}+4q^{93}+17q^{91}+20q^{89}+8q^{87}-11q^{85}-20q^{83}-11q^{81}+5q^{79}+14q^{77}+10q^{75}-8q^{71}-7q^{69}+q^{67}+3q^{65}-5q^{61}-7q^{59}+q^{57}+11q^{55}+10q^{53}-13q^{49}-16q^{47}-3q^{45}+15q^{43}+20q^{41}+10q^{39}-9q^{37}-20q^{35}-14q^{33}+4q^{31}+17q^{29}+16q^{27}+3q^{25}-14q^{23}-17q^{21}-7q^{19}+7q^{17}+14q^{15}+10q^{13}+q^{11}-9q^{9}-11q^{7}-4q^{5}+3q^{3}+8q+10q^{-1}+5q^{-3}-4q^{-5}-9q^{-7}-9q^{-9}-4q^{-11}+4q^{-13}+12q^{-15}+11q^{-17}+q^{-19}-8q^{-21}-13q^{-23}-9q^{-25}+q^{-27}+12q^{-29}+14q^{-31}+6q^{-33}-4q^{-35}-12q^{-37}-12q^{-39}-4q^{-41}+7q^{-43}+12q^{-45}+9q^{-47}+2q^{-49}-7q^{-51}-11q^{-53}-8q^{-55}+7q^{-59}+9q^{-61}+6q^{-63}-q^{-65}-6q^{-67}-8q^{-69}-4q^{-71}+2q^{-73}+5q^{-75}+6q^{-77}+3q^{-79}-2q^{-81}-5q^{-83}-3q^{-85}-q^{-87}+q^{-89}+3q^{-91}+2q^{-93}-q^{-97}-q^{-99}-q^{-101}+q^{-105}$

A2 Invariants.

Weight	Invariant
1,0	$q^{24}+q^{14}-q^{12}-q^{8}-q^{6}+1+q^{-2}+q^{-4}+q^{-6}$
1,1	$q^{68}-2q^{66}+2q^{64}-2q^{62}+3q^{60}-4q^{58}+4q^{56}-2q^{54}+5q^{52}-6q^{50}+6q^{48}-6q^{46}+5q^{44}-8q^{42}+2q^{40}-2q^{38}+2q^{34}-2q^{32}+4q^{30}-2q^{28}+8q^{26}-6q^{24}+10q^{22}-14q^{20}+16q^{18}-18q^{16}+16q^{14}-15q^{12}+16q^{10}-10q^{8}+6q^{6}-3q^{4}-2q^{2}+6-12q^{-2}+13q^{-4}-14q^{-6}+14q^{-8}-10q^{-10}+8q^{-12}-4q^{-14}+4q^{-16}+q^{-20}$
2,0	$q^{60}+q^{50}-q^{46}-q^{44}-2q^{38}-q^{36}+q^{34}-2q^{30}+2q^{26}+q^{24}+2q^{22}+2q^{20}+2q^{18}+q^{16}+q^{14}-2q^{10}-q^{8}-q^{4}-2q^{2}+q^{-2}-q^{-6}+q^{-10}+q^{-12}+q^{-16}+q^{-18}+q^{-20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{56}-q^{54}-q^{52}+2q^{50}-2q^{46}+q^{44}-2q^{40}+q^{38}+q^{36}+q^{32}+2q^{30}+q^{28}+q^{24}+q^{22}-q^{20}-2q^{18}-2q^{14}-2q^{12}-2q^{8}+q^{4}+1+2q^{-2}+q^{-4}+q^{-6}+2q^{-8}+q^{-12}$
1,0,0	$q^{31}+q^{27}-q^{25}+q^{23}+q^{19}-q^{13}-2q^{11}-q^{9}-2q^{7}+2q+q^{-1}+2q^{-3}+q^{-5}+q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{70}-q^{66}+q^{62}-q^{58}+q^{54}-2q^{50}-q^{44}+q^{40}+2q^{38}+2q^{36}+3q^{34}+2q^{32}+q^{30}+q^{26}-q^{24}-3q^{22}-q^{16}-2q^{14}-q^{12}-2q^{10}-3q^{8}-3q^{6}-2q^{4}+1+3q^{-2}+3q^{-4}+4q^{-6}+3q^{-8}+2q^{-10}+q^{-12}+q^{-14}$
1,0,0,0	$q^{38}+q^{34}+q^{28}+q^{24}+q^{20}-q^{18}-q^{16}-2q^{14}-2q^{12}-2q^{10}-2q^{8}+2q^{2}+2+2q^{-2}+2q^{-4}+q^{-6}+q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$q^{56}-q^{54}+q^{52}-2q^{50}+2q^{48}-2q^{46}+3q^{44}-2q^{42}+2q^{40}-q^{38}+q^{36}-q^{32}+2q^{30}-3q^{28}+4q^{26}-5q^{24}+5q^{22}-5q^{20}+4q^{18}-4q^{16}+2q^{14}-2q^{12}-2q^{6}+3q^{4}-2q^{2}+3-2q^{-2}+3q^{-4}-q^{-6}+2q^{-8}+q^{-12}$
1,0	$q^{90}-q^{86}-q^{84}+2q^{80}+q^{78}-q^{76}-2q^{74}-q^{72}+2q^{70}+q^{68}-q^{66}-2q^{64}+2q^{60}+q^{58}-q^{56}-q^{54}+q^{52}+2q^{50}+q^{48}-q^{46}-q^{44}+q^{42}+2q^{40}-q^{36}+q^{32}-q^{30}-2q^{28}+2q^{24}-3q^{20}-2q^{18}+q^{16}+2q^{14}-3q^{10}-q^{8}+2q^{6}+2q^{4}-2+2q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-12}+2q^{-14}+q^{-22}$

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{134}-q^{132}+q^{130}-q^{128}-q^{122}+2q^{120}-2q^{118}+2q^{116}-2q^{114}+q^{110}-q^{108}+3q^{106}-3q^{104}+2q^{102}-q^{100}-q^{98}+2q^{96}-3q^{94}+3q^{92}-2q^{90}+q^{88}+q^{86}-q^{84}+2q^{82}+q^{78}+q^{74}+2q^{70}+q^{68}+q^{66}+q^{62}-q^{60}-3q^{54}+3q^{52}-5q^{50}+3q^{48}-q^{46}-5q^{44}+5q^{42}-7q^{40}+2q^{38}-2q^{36}-2q^{34}+2q^{32}-3q^{30}+3q^{28}-q^{26}-2q^{20}+q^{18}+q^{16}-q^{14}+2q^{12}-3q^{10}+3q^{8}+q^{6}-3q^{4}+6q^{2}-6+4q^{-2}+q^{-4}-3q^{-6}+6q^{-8}-4q^{-10}+4q^{-12}+2q^{-18}-2q^{-20}+2q^{-22}+q^{-26}$

.

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["10 8"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+5t^{2}-5t+5-5t^{-1}+5t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{2}a^{10}+2z^{3}a^{9}+3z^{4}a^{8}-2z^{2}a^{8}+4z^{5}a^{7}-7z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-10z^{4}a^{6}+5z^{2}a^{6}-a^{6}+3z^{7}a^{5}-8z^{5}a^{5}+2z^{3}a^{5}+za^{5}+2z^{8}a^{4}-6z^{6}a^{4}+z^{4}a^{4}+3z^{2}a^{4}+z^{9}a^{3}-3z^{7}a^{3}-z^{5}a^{3}+5z^{3}a^{3}-za^{3}+3z^{8}a^{2}-17z^{6}a^{2}+30z^{4}a^{2}-18z^{2}a^{2}+3a^{2}+z^{9}a-6z^{7}a+11z^{5}a-6z^{3}a+z^{8}-7z^{6}+16z^{4}-13z^{2}+3$

"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["10 8"];

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]=

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

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]=

{}

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, 4)

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 $-12$ $32$ $72$ $114$ $94$ $-384$ $-{\frac {2368}{3}}$ $-{\frac {736}{3}}$ $-224$ $-288$ $512$ $-1368$ $-1128$ ${\frac {3969}{10}}$ ${\frac {16346}{15}}$ $-{\frac {8314}{5}}$ ${\frac {949}{2}}$ $-{\frac {6111}{10}}$

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 $s=$-4 is the signature of 10 8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                     1 1 3                       0 1                 2 1   1 -1               1       -1 -3             3 2       1 -5           2 2         0 -7         2 2           0 -9       2 2             0 -11     1 2               -1 -13   1 2                 1 -15   1                   -1 -17 1                     1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

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

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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