8 14

8_13

8_15

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 14's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 14 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X7,14,8,15 X11,16,12,1 X15,12,16,13 X13,6,14,7
Gauss code	-1, 4, -3, 1, -2, 8, -5, 3, -4, 2, -6, 7, -8, 5, -7, 6
Dowker-Thistlethwaite code	4 8 10 14 2 16 6 12
Conway Notation	[22112]

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

[edit Notes on presentations of 8 14]

Knot 8_14.

A graph, knot 8_14.

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 14"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X7,14,8,15 X11,16,12,1 X15,12,16,13 X13,6,14,7

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 16 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22112]

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[{10, 4}, {3, 8}, {9, 5}, {4, 6}, {8, 10}, {5, 2}, {1, 3}, {2, 7}, {6, 9}, {7, 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 $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-1] Hyperbolic Volume 9.2178 A-Polynomial See Data:8 14/A-polynomial

[edit Notes for 8 14'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 14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2 t^2+8 t-11+8 t^{-1} -2 t^{-2}$
Conway polynomial	$1-2 z^4$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, -2 }
Jones polynomial	$q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -5 q^{-4} +4 q^{-5} -3 q^{-6} + q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^2 a^6-z^4 a^4-z^2 a^4-z^4 a^2-z^2 a^2+z^2+1$
Kauffman polynomial (db, data sources)	$z^4 a^8-z^2 a^8+3 z^5 a^7-5 z^3 a^7+z a^7+3 z^6 a^6-4 z^4 a^6+z^2 a^6+z^7 a^5+4 z^5 a^5-8 z^3 a^5+3 z a^5+5 z^6 a^4-7 z^4 a^4+3 z^2 a^4+z^7 a^3+3 z^5 a^3-6 z^3 a^3+3 z a^3+2 z^6 a^2-z^4 a^2-z^2 a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1$
The A2 invariant	$q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^6-q^4+2 q^2+ q^{-4}$
The G2 invariant	$q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+3 q^{106}-6 q^{102}+14 q^{100}-16 q^{98}+17 q^{96}-11 q^{94}-4 q^{92}+17 q^{90}-25 q^{88}+25 q^{86}-17 q^{84}+4 q^{82}+11 q^{80}-17 q^{78}+17 q^{76}-9 q^{74}-3 q^{72}+13 q^{70}-16 q^{68}+5 q^{66}+7 q^{64}-19 q^{62}+28 q^{60}-26 q^{58}+15 q^{56}+q^{54}-21 q^{52}+33 q^{50}-37 q^{48}+28 q^{46}-11 q^{44}-7 q^{42}+21 q^{40}-23 q^{38}+19 q^{36}-7 q^{34}-6 q^{32}+13 q^{30}-12 q^{28}+2 q^{26}+12 q^{24}-19 q^{22}+22 q^{20}-13 q^{18}+12 q^{14}-21 q^{12}+24 q^{10}-18 q^8+8 q^6+2 q^4-9 q^2+12-10 q^{-2} +9 q^{-4} -3 q^{-6} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 8_14.

A1 Invariants.

Weight	Invariant
1	$q^{15}-2 q^{13}+q^{11}-q^9+q^7+q^5-q^3+2 q- q^{-1} + q^{-3}$
2	$q^{42}-2 q^{40}-2 q^{38}+6 q^{36}-q^{34}-6 q^{32}+7 q^{30}+q^{28}-8 q^{26}+4 q^{24}+2 q^{22}-5 q^{20}+3 q^{16}+2 q^{14}-5 q^{12}+2 q^{10}+7 q^8-7 q^6-q^4+8 q^2-4-2 q^{-2} +4 q^{-4} - q^{-6} - q^{-8} + q^{-10}$
3	$q^{81}-2 q^{79}-2 q^{77}+3 q^{75}+6 q^{73}-q^{71}-13 q^{69}-q^{67}+16 q^{65}+7 q^{63}-19 q^{61}-15 q^{59}+19 q^{57}+22 q^{55}-16 q^{53}-25 q^{51}+12 q^{49}+26 q^{47}-5 q^{45}-25 q^{43}+2 q^{41}+19 q^{39}+3 q^{37}-15 q^{35}-8 q^{33}+6 q^{31}+13 q^{29}-18 q^{25}-7 q^{23}+20 q^{21}+16 q^{19}-20 q^{17}-21 q^{15}+19 q^{13}+25 q^{11}-12 q^9-25 q^7+6 q^5+23 q^3-q-16 q^{-1} -2 q^{-3} +11 q^{-5} +3 q^{-7} -6 q^{-9} -2 q^{-11} +3 q^{-13} + q^{-15} - q^{-17} - q^{-19} + q^{-21}$
4	$q^{132}-2 q^{130}-2 q^{128}+3 q^{126}+3 q^{124}+6 q^{122}-8 q^{120}-13 q^{118}-q^{116}+8 q^{114}+31 q^{112}-2 q^{110}-33 q^{108}-27 q^{106}-2 q^{104}+65 q^{102}+34 q^{100}-30 q^{98}-68 q^{96}-50 q^{94}+74 q^{92}+84 q^{90}+9 q^{88}-84 q^{86}-101 q^{84}+43 q^{82}+102 q^{80}+57 q^{78}-60 q^{76}-116 q^{74}+3 q^{72}+81 q^{70}+73 q^{68}-23 q^{66}-91 q^{64}-24 q^{62}+41 q^{60}+62 q^{58}+8 q^{56}-49 q^{54}-42 q^{52}-q^{50}+45 q^{48}+42 q^{46}-q^{44}-65 q^{42}-46 q^{40}+28 q^{38}+75 q^{36}+49 q^{34}-74 q^{32}-88 q^{30}-6 q^{28}+90 q^{26}+99 q^{24}-53 q^{22}-104 q^{20}-48 q^{18}+64 q^{16}+115 q^{14}-7 q^{12}-73 q^{10}-68 q^8+14 q^6+85 q^4+22 q^2-24-49 q^{-2} -15 q^{-4} +37 q^{-6} +17 q^{-8} +2 q^{-10} -19 q^{-12} -12 q^{-14} +11 q^{-16} +4 q^{-18} +4 q^{-20} -4 q^{-22} -4 q^{-24} +3 q^{-26} + q^{-30} - q^{-32} - q^{-34} + q^{-36}$
5	$q^{195}-2 q^{193}-2 q^{191}+3 q^{189}+3 q^{187}+3 q^{185}-q^{183}-8 q^{181}-13 q^{179}-q^{177}+17 q^{175}+23 q^{173}+13 q^{171}-16 q^{169}-43 q^{167}-44 q^{165}+10 q^{163}+70 q^{161}+78 q^{159}+23 q^{157}-76 q^{155}-138 q^{153}-86 q^{151}+67 q^{149}+189 q^{147}+165 q^{145}-8 q^{143}-220 q^{141}-266 q^{139}-76 q^{137}+215 q^{135}+352 q^{133}+184 q^{131}-165 q^{129}-404 q^{127}-295 q^{125}+85 q^{123}+414 q^{121}+381 q^{119}+7 q^{117}-379 q^{115}-432 q^{113}-94 q^{111}+316 q^{109}+431 q^{107}+164 q^{105}-241 q^{103}-402 q^{101}-194 q^{99}+161 q^{97}+341 q^{95}+211 q^{93}-89 q^{91}-279 q^{89}-199 q^{87}+30 q^{85}+203 q^{83}+196 q^{81}+28 q^{79}-146 q^{77}-184 q^{75}-83 q^{73}+84 q^{71}+187 q^{69}+147 q^{67}-27 q^{65}-200 q^{63}-214 q^{61}-37 q^{59}+202 q^{57}+287 q^{55}+114 q^{53}-199 q^{51}-361 q^{49}-193 q^{47}+170 q^{45}+411 q^{43}+284 q^{41}-114 q^{39}-432 q^{37}-367 q^{35}+39 q^{33}+407 q^{31}+416 q^{29}+57 q^{27}-339 q^{25}-427 q^{23}-145 q^{21}+245 q^{19}+392 q^{17}+203 q^{15}-130 q^{13}-316 q^{11}-226 q^9+33 q^7+228 q^5+205 q^3+31 q-132 q^{-1} -161 q^{-3} -62 q^{-5} +63 q^{-7} +109 q^{-9} +60 q^{-11} -19 q^{-13} -61 q^{-15} -44 q^{-17} - q^{-19} +29 q^{-21} +28 q^{-23} +5 q^{-25} -13 q^{-27} -13 q^{-29} -3 q^{-31} +2 q^{-33} +6 q^{-35} +4 q^{-37} -3 q^{-39} -2 q^{-41} + q^{-43} + q^{-49} - q^{-51} - q^{-53} + q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^6-q^4+2 q^2+ q^{-4}$
1,1	$q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+32 q^{52}-48 q^{50}+66 q^{48}-78 q^{46}+83 q^{44}-78 q^{42}+64 q^{40}-36 q^{38}-q^{36}+42 q^{34}-84 q^{32}+116 q^{30}-145 q^{28}+156 q^{26}-154 q^{24}+138 q^{22}-105 q^{20}+68 q^{18}-26 q^{16}-12 q^{14}+47 q^{12}-68 q^{10}+78 q^8-76 q^6+70 q^4-56 q^2+42-28 q^{-2} +19 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12}$
2,0	$q^{56}-q^{54}-2 q^{52}+3 q^{48}+2 q^{46}-4 q^{44}+4 q^{40}+q^{38}-4 q^{36}-q^{34}+3 q^{32}-2 q^{30}-4 q^{28}+q^{24}-2 q^{22}+3 q^{20}+3 q^{18}-q^{16}+q^{14}+4 q^{12}-5 q^8+5 q^4-q^2-3+2 q^{-2} +3 q^{-4} - q^{-8} + q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-2 q^{46}+3 q^{42}-5 q^{40}+2 q^{38}+5 q^{36}-6 q^{34}+q^{32}+5 q^{30}-4 q^{28}-q^{26}+3 q^{24}-q^{22}-2 q^{20}-2 q^{18}+3 q^{16}-q^{14}-4 q^{12}+7 q^{10}+q^8-5 q^6+6 q^4+q^2-3+3 q^{-2} + q^{-4} - q^{-6} + q^{-8}$
1,0,0	$q^{29}-q^{27}-q^{23}+q^{21}-q^{19}+q^{17}+q^7-q^5+2 q^3+ q^{-1} + q^{-5}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-2 q^{46}+4 q^{44}-5 q^{42}+5 q^{40}-6 q^{38}+5 q^{36}-4 q^{34}+q^{32}+q^{30}-4 q^{28}+7 q^{26}-9 q^{24}+11 q^{22}-10 q^{20}+10 q^{18}-7 q^{16}+5 q^{14}-2 q^{12}-q^{10}+3 q^8-5 q^6+6 q^4-5 q^2+5-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8}$
1,0	$q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+4 q^{68}-q^{66}-5 q^{64}-2 q^{62}+6 q^{60}+5 q^{58}-3 q^{56}-6 q^{54}+5 q^{50}+2 q^{48}-4 q^{46}-3 q^{44}+3 q^{42}+3 q^{40}-2 q^{38}-4 q^{36}+q^{34}+4 q^{32}-5 q^{28}-q^{26}+4 q^{24}+2 q^{22}-4 q^{20}-3 q^{18}+4 q^{16}+6 q^{14}-q^{12}-6 q^{10}-q^8+6 q^6+4 q^4-2 q^2-4+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-2 q^{64}+2 q^{62}-3 q^{60}+4 q^{58}-5 q^{56}+4 q^{54}-4 q^{52}+5 q^{50}-3 q^{48}+q^{46}+q^{42}+3 q^{40}-5 q^{38}+5 q^{36}-6 q^{34}+8 q^{32}-9 q^{30}+6 q^{28}-9 q^{26}+7 q^{24}-5 q^{22}+3 q^{20}-3 q^{18}+2 q^{16}+3 q^{14}-q^{12}+3 q^{10}-4 q^8+6 q^6-3 q^4+4 q^2-3+4 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} + q^{-10}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+3 q^{106}-6 q^{102}+14 q^{100}-16 q^{98}+17 q^{96}-11 q^{94}-4 q^{92}+17 q^{90}-25 q^{88}+25 q^{86}-17 q^{84}+4 q^{82}+11 q^{80}-17 q^{78}+17 q^{76}-9 q^{74}-3 q^{72}+13 q^{70}-16 q^{68}+5 q^{66}+7 q^{64}-19 q^{62}+28 q^{60}-26 q^{58}+15 q^{56}+q^{54}-21 q^{52}+33 q^{50}-37 q^{48}+28 q^{46}-11 q^{44}-7 q^{42}+21 q^{40}-23 q^{38}+19 q^{36}-7 q^{34}-6 q^{32}+13 q^{30}-12 q^{28}+2 q^{26}+12 q^{24}-19 q^{22}+22 q^{20}-13 q^{18}+12 q^{14}-21 q^{12}+24 q^{10}-18 q^8+8 q^6+2 q^4-9 q^2+12-10 q^{-2} +9 q^{-4} -3 q^{-6} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18}$

. </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["8 14"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2 t^2+8 t-11+8 t^{-1} -2 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-2 z^4$

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]=

{ 31, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^4 a^8-z^2 a^8+3 z^5 a^7-5 z^3 a^7+z a^7+3 z^6 a^6-4 z^4 a^6+z^2 a^6+z^7 a^5+4 z^5 a^5-8 z^3 a^5+3 z a^5+5 z^6 a^4-7 z^4 a^4+3 z^2 a^4+z^7 a^3+3 z^5 a^3-6 z^3 a^3+3 z a^3+2 z^6 a^2-z^4 a^2-z^2 a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_8, 10_131,}

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 14"];

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]=

{ $-2 t^2+8 t-11+8 t^{-1} -2 t^{-2}$, $q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -5 q^{-4} +4 q^{-5} -3 q^{-6} + q^{-7}$ }

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]=

{9_8, 10_131,}

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, 0)

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 $0$ $0$ $0$ $16$ $16$ $0$ $-64$ $-32$ $-32$ $0$ $0$ $0$ $0$ $136$ $\frac{32}{3}$ $\frac{320}{3}$ $\frac{56}{3}$ $8$

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 $s=$-2 is the signature of 8 14. 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 3                 1 1 1               1   -1 -1             3 1   2 -3           3 2     -1 -5         3 2       1 -7       2 3         1 -9     2 3           -1 -11   1 2             1 -13   2               -2 -15 1                 1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

<p>

$n$	$J_n$
2	$q^4-2 q^3+6 q-8-2 q^{-1} +18 q^{-2} -17 q^{-3} -8 q^{-4} +32 q^{-5} -22 q^{-6} -15 q^{-7} +39 q^{-8} -21 q^{-9} -18 q^{-10} +34 q^{-11} -14 q^{-12} -16 q^{-13} +22 q^{-14} -5 q^{-15} -10 q^{-16} +9 q^{-17} -3 q^{-19} + q^{-20}$
3	$q^9-2 q^8+2 q^6+3 q^5-7 q^4-4 q^3+11 q^2+11 q-20-18 q^{-1} +26 q^{-2} +35 q^{-3} -37 q^{-4} -49 q^{-5} +39 q^{-6} +72 q^{-7} -43 q^{-8} -89 q^{-9} +40 q^{-10} +108 q^{-11} -39 q^{-12} -116 q^{-13} +29 q^{-14} +126 q^{-15} -26 q^{-16} -123 q^{-17} +15 q^{-18} +119 q^{-19} -8 q^{-20} -107 q^{-21} -2 q^{-22} +92 q^{-23} +12 q^{-24} -76 q^{-25} -16 q^{-26} +55 q^{-27} +21 q^{-28} -38 q^{-29} -19 q^{-30} +21 q^{-31} +17 q^{-32} -12 q^{-33} -10 q^{-34} +4 q^{-35} +5 q^{-36} -3 q^{-38} + q^{-39}$
4	$q^{16}-2 q^{15}+2 q^{13}-q^{12}+4 q^{11}-9 q^{10}+10 q^8-q^7+11 q^6-32 q^5-7 q^4+31 q^3+14 q^2+31 q-84-41 q^{-1} +56 q^{-2} +60 q^{-3} +94 q^{-4} -155 q^{-5} -123 q^{-6} +51 q^{-7} +126 q^{-8} +216 q^{-9} -206 q^{-10} -235 q^{-11} -5 q^{-12} +177 q^{-13} +368 q^{-14} -215 q^{-15} -331 q^{-16} -87 q^{-17} +191 q^{-18} +491 q^{-19} -189 q^{-20} -378 q^{-21} -161 q^{-22} +172 q^{-23} +555 q^{-24} -146 q^{-25} -375 q^{-26} -207 q^{-27} +131 q^{-28} +548 q^{-29} -89 q^{-30} -321 q^{-31} -228 q^{-32} +66 q^{-33} +481 q^{-34} -21 q^{-35} -225 q^{-36} -220 q^{-37} -12 q^{-38} +362 q^{-39} +35 q^{-40} -108 q^{-41} -175 q^{-42} -71 q^{-43} +218 q^{-44} +52 q^{-45} -15 q^{-46} -100 q^{-47} -81 q^{-48} +94 q^{-49} +34 q^{-50} +23 q^{-51} -36 q^{-52} -50 q^{-53} +27 q^{-54} +9 q^{-55} +17 q^{-56} -5 q^{-57} -17 q^{-58} +4 q^{-59} +5 q^{-61} -3 q^{-63} + q^{-64}$
5	$q^{25}-2 q^{24}+2 q^{22}-q^{21}+2 q^{19}-5 q^{18}-q^{17}+9 q^{16}+q^{15}-4 q^{14}-3 q^{13}-15 q^{12}-q^{11}+27 q^{10}+24 q^9-3 q^8-33 q^7-58 q^6-18 q^5+69 q^4+103 q^3+46 q^2-79 q-183-117 q^{-1} +98 q^{-2} +266 q^{-3} +220 q^{-4} -56 q^{-5} -378 q^{-6} -376 q^{-7} +8 q^{-8} +452 q^{-9} +553 q^{-10} +133 q^{-11} -525 q^{-12} -766 q^{-13} -274 q^{-14} +540 q^{-15} +949 q^{-16} +492 q^{-17} -534 q^{-18} -1134 q^{-19} -680 q^{-20} +475 q^{-21} +1267 q^{-22} +890 q^{-23} -407 q^{-24} -1375 q^{-25} -1043 q^{-26} +307 q^{-27} +1429 q^{-28} +1203 q^{-29} -234 q^{-30} -1460 q^{-31} -1282 q^{-32} +130 q^{-33} +1443 q^{-34} +1376 q^{-35} -60 q^{-36} -1420 q^{-37} -1385 q^{-38} -37 q^{-39} +1342 q^{-40} +1414 q^{-41} +114 q^{-42} -1252 q^{-43} -1378 q^{-44} -210 q^{-45} +1113 q^{-46} +1334 q^{-47} +304 q^{-48} -952 q^{-49} -1248 q^{-50} -390 q^{-51} +758 q^{-52} +1126 q^{-53} +465 q^{-54} -547 q^{-55} -981 q^{-56} -505 q^{-57} +348 q^{-58} +788 q^{-59} +518 q^{-60} -161 q^{-61} -607 q^{-62} -472 q^{-63} +19 q^{-64} +408 q^{-65} +409 q^{-66} +78 q^{-67} -258 q^{-68} -304 q^{-69} -118 q^{-70} +117 q^{-71} +219 q^{-72} +124 q^{-73} -46 q^{-74} -131 q^{-75} -94 q^{-76} -5 q^{-77} +66 q^{-78} +72 q^{-79} +16 q^{-80} -32 q^{-81} -39 q^{-82} -13 q^{-83} +6 q^{-84} +18 q^{-85} +17 q^{-86} -5 q^{-87} -10 q^{-88} -3 q^{-89} +5 q^{-92} -3 q^{-94} + q^{-95}$
6	$q^{36}-2 q^{35}+2 q^{33}-q^{32}-2 q^{30}+6 q^{29}-6 q^{28}-2 q^{27}+11 q^{26}-3 q^{25}-4 q^{24}-12 q^{23}+14 q^{22}-13 q^{21}-q^{20}+40 q^{19}+5 q^{18}-14 q^{17}-51 q^{16}+7 q^{15}-45 q^{14}+8 q^{13}+129 q^{12}+70 q^{11}+2 q^{10}-142 q^9-77 q^8-188 q^7-27 q^6+310 q^5+305 q^4+181 q^3-206 q^2-279 q-616-306 q^{-1} +459 q^{-2} +767 q^{-3} +751 q^{-4} +33 q^{-5} -420 q^{-6} -1391 q^{-7} -1106 q^{-8} +223 q^{-9} +1230 q^{-10} +1757 q^{-11} +882 q^{-12} -107 q^{-13} -2225 q^{-14} -2424 q^{-15} -703 q^{-16} +1269 q^{-17} +2849 q^{-18} +2286 q^{-19} +909 q^{-20} -2673 q^{-21} -3841 q^{-22} -2180 q^{-23} +676 q^{-24} +3567 q^{-25} +3777 q^{-26} +2400 q^{-27} -2542 q^{-28} -4884 q^{-29} -3703 q^{-30} -313 q^{-31} +3737 q^{-32} +4898 q^{-33} +3853 q^{-34} -2036 q^{-35} -5373 q^{-36} -4845 q^{-37} -1279 q^{-38} +3519 q^{-39} +5492 q^{-40} +4912 q^{-41} -1457 q^{-42} -5422 q^{-43} -5490 q^{-44} -1996 q^{-45} +3131 q^{-46} +5656 q^{-47} +5517 q^{-48} -923 q^{-49} -5177 q^{-50} -5734 q^{-51} -2486 q^{-52} +2625 q^{-53} +5496 q^{-54} +5772 q^{-55} -364 q^{-56} -4641 q^{-57} -5654 q^{-58} -2868 q^{-59} +1907 q^{-60} +4985 q^{-61} +5739 q^{-62} +325 q^{-63} -3718 q^{-64} -5196 q^{-65} -3161 q^{-66} +902 q^{-67} +4022 q^{-68} +5343 q^{-69} +1087 q^{-70} -2391 q^{-71} -4253 q^{-72} -3204 q^{-73} -237 q^{-74} +2629 q^{-75} +4445 q^{-76} +1641 q^{-77} -917 q^{-78} -2861 q^{-79} -2767 q^{-80} -1116 q^{-81} +1119 q^{-82} +3080 q^{-83} +1669 q^{-84} +228 q^{-85} -1373 q^{-86} -1851 q^{-87} -1365 q^{-88} -12 q^{-89} +1623 q^{-90} +1162 q^{-91} +686 q^{-92} -288 q^{-93} -834 q^{-94} -1018 q^{-95} -458 q^{-96} +564 q^{-97} +505 q^{-98} +552 q^{-99} +154 q^{-100} -156 q^{-101} -495 q^{-102} -377 q^{-103} +91 q^{-104} +85 q^{-105} +247 q^{-106} +155 q^{-107} +75 q^{-108} -151 q^{-109} -169 q^{-110} -5 q^{-111} -35 q^{-112} +59 q^{-113} +58 q^{-114} +67 q^{-115} -28 q^{-116} -46 q^{-117} -2 q^{-118} -25 q^{-119} +6 q^{-120} +9 q^{-121} +26 q^{-122} -5 q^{-123} -10 q^{-124} +4 q^{-125} -7 q^{-126} +5 q^{-129} -3 q^{-131} + q^{-132}$
7	$q^{49}-2 q^{48}+2 q^{46}-q^{45}-2 q^{43}+2 q^{42}+5 q^{41}-7 q^{40}+7 q^{38}-3 q^{37}-2 q^{36}-12 q^{35}+q^{34}+20 q^{33}-11 q^{32}+6 q^{31}+21 q^{30}-4 q^{29}-8 q^{28}-51 q^{27}-25 q^{26}+35 q^{25}+52 q^{23}+83 q^{22}+19 q^{21}-14 q^{20}-157 q^{19}-164 q^{18}-29 q^{17}-7 q^{16}+199 q^{15}+318 q^{14}+214 q^{13}+87 q^{12}-336 q^{11}-566 q^{10}-439 q^9-289 q^8+340 q^7+874 q^6+925 q^5+725 q^4-269 q^3-1203 q^2-1538 q-1475-152 q^{-1} +1434 q^{-2} +2356 q^{-3} +2577 q^{-4} +949 q^{-5} -1355 q^{-6} -3130 q^{-7} -4081 q^{-8} -2342 q^{-9} +847 q^{-10} +3813 q^{-11} +5765 q^{-12} +4216 q^{-13} +336 q^{-14} -4003 q^{-15} -7550 q^{-16} -6681 q^{-17} -2147 q^{-18} +3744 q^{-19} +9111 q^{-20} +9280 q^{-21} +4616 q^{-22} -2657 q^{-23} -10290 q^{-24} -12076 q^{-25} -7532 q^{-26} +1100 q^{-27} +10901 q^{-28} +14515 q^{-29} +10638 q^{-30} +1136 q^{-31} -10925 q^{-32} -16688 q^{-33} -13698 q^{-34} -3553 q^{-35} +10439 q^{-36} +18233 q^{-37} +16479 q^{-38} +6129 q^{-39} -9558 q^{-40} -19338 q^{-41} -18830 q^{-42} -8479 q^{-43} +8428 q^{-44} +19907 q^{-45} +20697 q^{-46} +10636 q^{-47} -7272 q^{-48} -20178 q^{-49} -22038 q^{-50} -12343 q^{-51} +6107 q^{-52} +20069 q^{-53} +23014 q^{-54} +13796 q^{-55} -5089 q^{-56} -19909 q^{-57} -23577 q^{-58} -14790 q^{-59} +4117 q^{-60} +19454 q^{-61} +23913 q^{-62} +15709 q^{-63} -3258 q^{-64} -19044 q^{-65} -23985 q^{-66} -16270 q^{-67} +2364 q^{-68} +18291 q^{-69} +23889 q^{-70} +16880 q^{-71} -1413 q^{-72} -17493 q^{-73} -23547 q^{-74} -17258 q^{-75} +318 q^{-76} +16274 q^{-77} +22936 q^{-78} +17653 q^{-79} +962 q^{-80} -14797 q^{-81} -21998 q^{-82} -17841 q^{-83} -2380 q^{-84} +12872 q^{-85} +20621 q^{-86} +17836 q^{-87} +3934 q^{-88} -10582 q^{-89} -18810 q^{-90} -17494 q^{-91} -5449 q^{-92} +8006 q^{-93} +16494 q^{-94} +16696 q^{-95} +6798 q^{-96} -5250 q^{-97} -13766 q^{-98} -15435 q^{-99} -7771 q^{-100} +2596 q^{-101} +10761 q^{-102} +13609 q^{-103} +8222 q^{-104} -179 q^{-105} -7700 q^{-106} -11405 q^{-107} -8068 q^{-108} -1650 q^{-109} +4794 q^{-110} +8892 q^{-111} +7352 q^{-112} +2891 q^{-113} -2354 q^{-114} -6441 q^{-115} -6116 q^{-116} -3371 q^{-117} +463 q^{-118} +4126 q^{-119} +4701 q^{-120} +3337 q^{-121} +685 q^{-122} -2330 q^{-123} -3209 q^{-124} -2765 q^{-125} -1283 q^{-126} +960 q^{-127} +1949 q^{-128} +2102 q^{-129} +1361 q^{-130} -212 q^{-131} -977 q^{-132} -1342 q^{-133} -1140 q^{-134} -211 q^{-135} +339 q^{-136} +772 q^{-137} +850 q^{-138} +295 q^{-139} -45 q^{-140} -353 q^{-141} -509 q^{-142} -244 q^{-143} -123 q^{-144} +110 q^{-145} +308 q^{-146} +175 q^{-147} +108 q^{-148} -28 q^{-149} -134 q^{-150} -71 q^{-151} -92 q^{-152} -42 q^{-153} +67 q^{-154} +53 q^{-155} +53 q^{-156} +12 q^{-157} -31 q^{-158} +2 q^{-159} -25 q^{-160} -25 q^{-161} +6 q^{-162} +9 q^{-163} +17 q^{-164} +4 q^{-165} -10 q^{-166} +4 q^{-167} -7 q^{-169} +5 q^{-172} -3 q^{-174} + q^{-175}$

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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