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

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

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

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

. </div></div>

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]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"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]=

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

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 $t^rq^j$ 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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

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

<p>

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