9 12

9_11

9_13

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 12 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 10 16 14 2 18 8 6 12
Conway Notation	[4212]

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

[edit Notes on presentations of 9 12]

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["9 12"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 16 14 2 18 8 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4212]

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]=

${\textrm {BR}}(5,\{-1,-1,2,-1,-3,2,-3,-4,3,-4\})$

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]=

{ 5, 10, 5 }

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, 10}, {6, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 9}, {8, 6}, {4, 7}, {3, 5}, {11, 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 $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 8.83664 A-Polynomial See Data:9 12/A-polynomial

[edit Notes for 9 12'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 9 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$q-2+4q^{-1}-5q^{-2}+6q^{-3}-6q^{-4}+5q^{-5}-3q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+2z^{2}a^{6}+2a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+3z^{3}a^{7}-za^{7}+z^{8}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+4z^{7}a^{5}-11z^{5}a^{5}+13z^{3}a^{5}-4za^{5}+z^{8}a^{4}-z^{4}a^{4}+3z^{2}a^{4}-a^{4}+2z^{7}a^{3}-3z^{5}a^{3}+4z^{3}a^{3}-2za^{3}+2z^{6}a^{2}-z^{4}a^{2}-2z^{2}a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{26}-q^{24}+q^{22}+q^{18}+2q^{16}-q^{14}-q^{10}+q^{6}-q^{4}+2q^{2}+q^{-4}$
The G2 invariant	$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-2q^{118}-2q^{116}+7q^{114}-10q^{112}+10q^{110}-10q^{108}+4q^{106}+4q^{104}-15q^{102}+20q^{100}-20q^{98}+14q^{96}-q^{94}-12q^{92}+19q^{90}-18q^{88}+15q^{86}-3q^{84}-10q^{82}+14q^{80}-11q^{78}+2q^{76}+10q^{74}-16q^{72}+18q^{70}-8q^{68}-4q^{66}+17q^{64}-26q^{62}+29q^{60}-21q^{58}+6q^{56}+11q^{54}-23q^{52}+30q^{50}-25q^{48}+13q^{46}-13q^{42}+16q^{40}-14q^{38}+3q^{36}+8q^{34}-13q^{32}+10q^{30}-2q^{28}-9q^{26}+17q^{24}-19q^{22}+15q^{20}-6q^{18}-5q^{16}+14q^{14}-16q^{12}+17q^{10}-10q^{8}+5q^{6}+q^{4}-7q^{2}+9-8q^{-2}+7q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 9_12.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+q^{15}-q^{13}+2q^{11}-q^{9}+q^{5}-q^{3}+2q-q^{-1}+q^{-3}$
2	$q^{48}-q^{46}-q^{44}+3q^{42}-2q^{40}-4q^{38}+5q^{36}+q^{34}-6q^{32}+5q^{30}+3q^{28}-7q^{26}+2q^{24}+3q^{22}-3q^{20}-2q^{18}+2q^{16}+4q^{14}-5q^{12}+7q^{8}-5q^{6}-2q^{4}+7q^{2}-2-2q^{-2}+3q^{-4}-q^{-6}-q^{-8}+q^{-10}$
3	$-q^{93}+q^{91}+q^{89}-q^{87}-2q^{85}+2q^{83}+5q^{81}-2q^{79}-8q^{77}+10q^{73}+3q^{71}-12q^{69}-10q^{67}+12q^{65}+15q^{63}-6q^{61}-19q^{59}+3q^{57}+21q^{55}+2q^{53}-21q^{51}-7q^{49}+19q^{47}+7q^{45}-14q^{43}-9q^{41}+9q^{39}+10q^{37}-5q^{35}-10q^{33}-3q^{31}+11q^{29}+10q^{27}-10q^{25}-16q^{23}+9q^{21}+20q^{19}-5q^{17}-21q^{15}+q^{13}+20q^{11}+4q^{9}-15q^{7}-5q^{5}+11q^{3}+5q-5q^{-1}-4q^{-3}+3q^{-5}+2q^{-7}-2q^{-9}+2q^{-13}-q^{-17}-q^{-19}+q^{-21}$
4	$q^{152}-q^{150}-q^{148}+q^{146}+2q^{142}-4q^{140}-3q^{138}+4q^{136}+3q^{134}+8q^{132}-8q^{130}-13q^{128}+2q^{126}+8q^{124}+23q^{122}-4q^{120}-25q^{118}-17q^{116}-4q^{114}+40q^{112}+23q^{110}-12q^{108}-36q^{106}-41q^{104}+28q^{102}+50q^{100}+33q^{98}-24q^{96}-77q^{94}-14q^{92}+44q^{90}+69q^{88}+12q^{86}-80q^{84}-49q^{82}+16q^{80}+74q^{78}+38q^{76}-56q^{74}-51q^{72}-5q^{70}+52q^{68}+39q^{66}-25q^{64}-39q^{62}-15q^{60}+29q^{58}+32q^{56}+3q^{54}-30q^{52}-29q^{50}+31q^{46}+43q^{44}-17q^{42}-50q^{40}-36q^{38}+24q^{36}+79q^{34}+13q^{32}-50q^{30}-68q^{28}-5q^{26}+84q^{24}+42q^{22}-18q^{20}-67q^{18}-38q^{16}+48q^{14}+42q^{12}+18q^{10}-33q^{8}-40q^{6}+10q^{4}+14q^{2}+24-q^{-2}-18q^{-4}-2q^{-6}-5q^{-8}+10q^{-10}+4q^{-12}-3q^{-14}+3q^{-16}-6q^{-18}+4q^{-26}-q^{-28}-q^{-32}-q^{-34}+q^{-36}$
5	$-q^{225}+q^{223}+q^{221}-q^{219}+2q^{211}+2q^{209}-4q^{207}-6q^{205}-q^{203}+4q^{201}+9q^{199}+8q^{197}-4q^{195}-19q^{193}-17q^{191}+3q^{189}+23q^{187}+33q^{185}+13q^{183}-24q^{181}-50q^{179}-36q^{177}+12q^{175}+56q^{173}+66q^{171}+23q^{169}-47q^{167}-91q^{165}-71q^{163}+6q^{161}+91q^{159}+119q^{157}+58q^{155}-56q^{153}-149q^{151}-138q^{149}-4q^{147}+147q^{145}+202q^{143}+99q^{141}-106q^{139}-247q^{137}-186q^{135}+40q^{133}+245q^{131}+256q^{129}+44q^{127}-220q^{125}-299q^{123}-113q^{121}+170q^{119}+300q^{117}+164q^{115}-109q^{113}-278q^{111}-188q^{109}+65q^{107}+231q^{105}+181q^{103}-19q^{101}-182q^{99}-164q^{97}-3q^{95}+138q^{93}+134q^{91}+17q^{89}-96q^{87}-114q^{85}-35q^{83}+68q^{81}+106q^{79}+51q^{77}-34q^{75}-103q^{73}-93q^{71}+q^{69}+108q^{67}+140q^{65}+48q^{63}-107q^{61}-193q^{59}-114q^{57}+89q^{55}+238q^{53}+187q^{51}-47q^{49}-263q^{47}-256q^{45}-19q^{43}+249q^{41}+307q^{39}+93q^{37}-201q^{35}-316q^{33}-160q^{31}+122q^{29}+287q^{27}+207q^{25}-35q^{23}-224q^{21}-210q^{19}-36q^{17}+139q^{15}+184q^{13}+84q^{11}-65q^{9}-135q^{7}-90q^{5}+7q^{3}+80q+81q^{-1}+26q^{-3}-39q^{-5}-57q^{-7}-32q^{-9}+8q^{-11}+33q^{-13}+28q^{-15}+7q^{-17}-15q^{-19}-20q^{-21}-8q^{-23}+3q^{-25}+9q^{-27}+9q^{-29}+4q^{-31}-5q^{-33}-6q^{-35}-2q^{-37}-2q^{-39}+2q^{-41}+4q^{-43}+q^{-45}-q^{-47}-q^{-51}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+q^{22}+q^{18}+2q^{16}-q^{14}-q^{10}+q^{6}-q^{4}+2q^{2}+q^{-4}$
1,1	$q^{68}-2q^{66}+4q^{64}-8q^{62}+17q^{60}-24q^{58}+32q^{56}-48q^{54}+57q^{52}-64q^{50}+62q^{48}-58q^{46}+46q^{44}-18q^{42}-6q^{40}+38q^{38}-65q^{36}+90q^{34}-112q^{32}+116q^{30}-122q^{28}+110q^{26}-90q^{24}+68q^{22}-37q^{20}+10q^{18}+20q^{16}-36q^{14}+47q^{12}-54q^{10}+56q^{8}-48q^{6}+42q^{4}-36q^{2}+30-20q^{-2}+15q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{66}+q^{64}-2q^{60}-q^{58}+q^{56}-q^{54}-3q^{52}-q^{50}+4q^{48}+3q^{46}-2q^{44}+4q^{40}-5q^{36}-3q^{34}+2q^{32}-2q^{28}+2q^{26}+q^{24}-q^{22}+3q^{20}+2q^{18}-3q^{16}-q^{14}+4q^{12}+q^{10}-4q^{8}+6q^{4}-3+q^{-2}+2q^{-4}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-q^{52}+q^{48}-3q^{46}+q^{44}+2q^{42}-5q^{40}+2q^{38}+4q^{36}-6q^{34}+q^{32}+4q^{30}-3q^{28}+3q^{24}+q^{22}-q^{20}-q^{18}+4q^{16}-2q^{14}-4q^{12}+6q^{10}-q^{8}-5q^{6}+5q^{4}+q^{2}-2+3q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$-q^{35}-q^{33}-q^{31}+q^{29}+2q^{25}+q^{23}+2q^{21}-q^{19}-q^{15}-q^{13}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+q^{52}-2q^{50}+3q^{48}-5q^{46}+5q^{44}-6q^{42}+5q^{40}-4q^{38}+4q^{36}-q^{32}+6q^{30}-7q^{28}+10q^{26}-11q^{24}+11q^{22}-11q^{20}+7q^{18}-6q^{16}+2q^{14}-2q^{10}+5q^{8}-5q^{6}+7q^{4}-5q^{2}+4-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{88}-q^{84}-q^{82}+q^{80}+2q^{78}-q^{76}-4q^{74}-q^{72}+4q^{70}+4q^{68}-3q^{66}-6q^{64}+6q^{60}+4q^{58}-5q^{56}-5q^{54}+q^{52}+5q^{50}-4q^{46}-q^{44}+4q^{42}+2q^{40}-3q^{38}-2q^{36}+3q^{34}+4q^{32}-2q^{30}-4q^{28}+q^{26}+5q^{24}-5q^{20}-2q^{18}+5q^{16}+5q^{14}-3q^{12}-6q^{10}-q^{8}+6q^{6}+3q^{4}-2q^{2}-3+3q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-2q^{118}-2q^{116}+7q^{114}-10q^{112}+10q^{110}-10q^{108}+4q^{106}+4q^{104}-15q^{102}+20q^{100}-20q^{98}+14q^{96}-q^{94}-12q^{92}+19q^{90}-18q^{88}+15q^{86}-3q^{84}-10q^{82}+14q^{80}-11q^{78}+2q^{76}+10q^{74}-16q^{72}+18q^{70}-8q^{68}-4q^{66}+17q^{64}-26q^{62}+29q^{60}-21q^{58}+6q^{56}+11q^{54}-23q^{52}+30q^{50}-25q^{48}+13q^{46}-13q^{42}+16q^{40}-14q^{38}+3q^{36}+8q^{34}-13q^{32}+10q^{30}-2q^{28}-9q^{26}+17q^{24}-19q^{22}+15q^{20}-6q^{18}-5q^{16}+14q^{14}-16q^{12}+17q^{10}-10q^{8}+5q^{6}+q^{4}-7q^{2}+9-8q^{-2}+7q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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["9 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+2z^{2}a^{6}+2a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+3z^{3}a^{7}-za^{7}+z^{8}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+4z^{7}a^{5}-11z^{5}a^{5}+13z^{3}a^{5}-4za^{5}+z^{8}a^{4}-z^{4}a^{4}+3z^{2}a^{4}-a^{4}+2z^{7}a^{3}-3z^{5}a^{3}+4z^{3}a^{3}-2za^{3}+2z^{6}a^{2}-z^{4}a^{2}-2z^{2}a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n84,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n15,}

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["9 12"];

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]=

{ $-2t^{2}+9t-13+9t^{-1}-2t^{-2}$, $q-2+4q^{-1}-5q^{-2}+6q^{-3}-6q^{-4}+5q^{-5}-3q^{-6}+2q^{-7}-q^{-8}$ }

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]=

{K11n84,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n15,}

Vassiliev invariants

V2 and V3:

(1, -3)

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 $4$ $-24$ $8$ ${\frac {302}{3}}$ ${\frac {58}{3}}$ $-96$ $-400$ $-32$ $-120$ ${\frac {32}{3}}$ $288$ ${\frac {1208}{3}}$ ${\frac {232}{3}}$ ${\frac {49471}{30}}$ $-{\frac {5942}{15}}$ ${\frac {49622}{45}}$ ${\frac {1793}{18}}$ ${\frac {4831}{30}}$

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

\   r    \    j   \ -7 -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         3 3         0 -9       2 3           -1 -11     1 3             2 -13   1 2               -1 -15   1                 1 -17 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{4}-2q^{3}+5q-7+14q^{-2}-16q^{-3}-3q^{-4}+26q^{-5}-23q^{-6}-8q^{-7}+35q^{-8}-25q^{-9}-12q^{-10}+34q^{-11}-19q^{-12}-13q^{-13}+25q^{-14}-9q^{-15}-11q^{-16}+14q^{-17}-2q^{-18}-7q^{-19}+5q^{-20}-2q^{-22}+q^{-23}$
3	$q^{9}-2q^{8}+q^{6}+3q^{5}-4q^{4}-2q^{3}+5q^{2}+4q-11-3q^{-1}+15q^{-2}+10q^{-3}-27q^{-4}-13q^{-5}+34q^{-6}+26q^{-7}-46q^{-8}-35q^{-9}+50q^{-10}+51q^{-11}-57q^{-12}-60q^{-13}+56q^{-14}+71q^{-15}-56q^{-16}-74q^{-17}+49q^{-18}+76q^{-19}-41q^{-20}-75q^{-21}+31q^{-22}+71q^{-23}-20q^{-24}-63q^{-25}+5q^{-26}+57q^{-27}+3q^{-28}-44q^{-29}-13q^{-30}+35q^{-31}+16q^{-32}-23q^{-33}-16q^{-34}+13q^{-35}+14q^{-36}-8q^{-37}-9q^{-38}+3q^{-39}+6q^{-40}-2q^{-41}-2q^{-42}+2q^{-44}-q^{-45}$
4	$q^{16}-2q^{15}+q^{13}-q^{12}+6q^{11}-6q^{10}+q^{8}-7q^{7}+15q^{6}-12q^{5}+7q^{4}+7q^{3}-22q^{2}+18q-28+24q^{-1}+32q^{-2}-32q^{-3}+14q^{-4}-78q^{-5}+31q^{-6}+83q^{-7}-8q^{-8}+20q^{-9}-164q^{-10}+2q^{-11}+132q^{-12}+52q^{-13}+62q^{-14}-253q^{-15}-61q^{-16}+150q^{-17}+115q^{-18}+128q^{-19}-308q^{-20}-121q^{-21}+136q^{-22}+148q^{-23}+188q^{-24}-320q^{-25}-152q^{-26}+107q^{-27}+147q^{-28}+221q^{-29}-291q^{-30}-155q^{-31}+63q^{-32}+123q^{-33}+235q^{-34}-227q^{-35}-142q^{-36}+6q^{-37}+77q^{-38}+230q^{-39}-133q^{-40}-106q^{-41}-52q^{-42}+12q^{-43}+199q^{-44}-41q^{-45}-49q^{-46}-77q^{-47}-46q^{-48}+136q^{-49}+12q^{-50}+8q^{-51}-60q^{-52}-68q^{-53}+67q^{-54}+17q^{-55}+32q^{-56}-25q^{-57}-51q^{-58}+23q^{-59}+4q^{-60}+24q^{-61}-4q^{-62}-24q^{-63}+8q^{-64}-2q^{-65}+9q^{-66}+q^{-67}-8q^{-68}+3q^{-69}-q^{-70}+2q^{-71}-2q^{-73}+q^{-74}$
5	$q^{25}-2q^{24}+q^{22}-q^{21}+2q^{20}+4q^{19}-4q^{18}-4q^{17}+q^{16}-5q^{15}+3q^{14}+13q^{13}+q^{12}-4q^{11}-5q^{10}-16q^{9}-9q^{8}+18q^{7}+23q^{6}+17q^{5}-41q^{3}-49q^{2}-7q+41+82q^{-1}+55q^{-2}-42q^{-3}-122q^{-4}-104q^{-5}-4q^{-6}+152q^{-7}+204q^{-8}+58q^{-9}-167q^{-10}-279q^{-11}-178q^{-12}+138q^{-13}+393q^{-14}+300q^{-15}-87q^{-16}-444q^{-17}-460q^{-18}-18q^{-19}+508q^{-20}+594q^{-21}+127q^{-22}-502q^{-23}-728q^{-24}-255q^{-25}+501q^{-26}+810q^{-27}+361q^{-28}-451q^{-29}-877q^{-30}-458q^{-31}+422q^{-32}+896q^{-33}+516q^{-34}-359q^{-35}-909q^{-36}-565q^{-37}+328q^{-38}+886q^{-39}+585q^{-40}-274q^{-41}-854q^{-42}-603q^{-43}+225q^{-44}+807q^{-45}+603q^{-46}-161q^{-47}-737q^{-48}-599q^{-49}+84q^{-50}+646q^{-51}+585q^{-52}+2q^{-53}-537q^{-54}-549q^{-55}-82q^{-56}+393q^{-57}+495q^{-58}+171q^{-59}-264q^{-60}-413q^{-61}-212q^{-62}+110q^{-63}+309q^{-64}+250q^{-65}-201q^{-67}-223q^{-68}-95q^{-69}+83q^{-70}+189q^{-71}+141q^{-72}+4q^{-73}-120q^{-74}-150q^{-75}-68q^{-76}+55q^{-77}+130q^{-78}+97q^{-79}-6q^{-80}-89q^{-81}-96q^{-82}-30q^{-83}+53q^{-84}+77q^{-85}+38q^{-86}-19q^{-87}-53q^{-88}-40q^{-89}+9q^{-90}+29q^{-91}+24q^{-92}+7q^{-93}-16q^{-94}-20q^{-95}-q^{-96}+9q^{-97}+4q^{-98}+5q^{-99}-q^{-100}-8q^{-101}+4q^{-103}-q^{-104}+q^{-106}-2q^{-107}+2q^{-109}-q^{-110}$
6	$q^{36}-2q^{35}+q^{33}-q^{32}+2q^{31}+6q^{29}-8q^{28}-4q^{27}+3q^{26}-6q^{25}+4q^{24}+4q^{23}+23q^{22}-14q^{21}-10q^{20}+q^{19}-23q^{18}-5q^{17}+7q^{16}+63q^{15}-8q^{14}-6q^{13}+q^{12}-66q^{11}-48q^{10}-9q^{9}+128q^{8}+34q^{7}+41q^{6}+30q^{5}-140q^{4}-167q^{3}-103q^{2}+179q+123+203q^{-1}+184q^{-2}-185q^{-3}-384q^{-4}-388q^{-5}+74q^{-6}+164q^{-7}+514q^{-8}+622q^{-9}+3q^{-10}-566q^{-11}-904q^{-12}-390q^{-13}-102q^{-14}+801q^{-15}+1362q^{-16}+649q^{-17}-403q^{-18}-1420q^{-19}-1200q^{-20}-893q^{-21}+729q^{-22}+2109q^{-23}+1675q^{-24}+284q^{-25}-1581q^{-26}-2006q^{-27}-2038q^{-28}+171q^{-29}+2492q^{-30}+2666q^{-31}+1258q^{-32}-1295q^{-33}-2446q^{-34}-3078q^{-35}-598q^{-36}+2439q^{-37}+3266q^{-38}+2081q^{-39}-819q^{-40}-2475q^{-41}-3703q^{-42}-1212q^{-43}+2177q^{-44}+3457q^{-45}+2535q^{-46}-430q^{-47}-2300q^{-48}-3934q^{-49}-1547q^{-50}+1910q^{-51}+3419q^{-52}+2693q^{-53}-174q^{-54}-2078q^{-55}-3930q^{-56}-1709q^{-57}+1630q^{-58}+3254q^{-59}+2709q^{-60}+79q^{-61}-1770q^{-62}-3764q^{-63}-1836q^{-64}+1206q^{-65}+2904q^{-66}+2635q^{-67}+435q^{-68}-1252q^{-69}-3366q^{-70}-1943q^{-71}+568q^{-72}+2263q^{-73}+2379q^{-74}+848q^{-75}-500q^{-76}-2642q^{-77}-1884q^{-78}-154q^{-79}+1349q^{-80}+1810q^{-81}+1092q^{-82}+303q^{-83}-1635q^{-84}-1485q^{-85}-652q^{-86}+403q^{-87}+959q^{-88}+938q^{-89}+810q^{-90}-626q^{-91}-773q^{-92}-670q^{-93}-204q^{-94}+121q^{-95}+428q^{-96}+794q^{-97}+6q^{-98}-73q^{-99}-295q^{-100}-283q^{-101}-336q^{-102}-82q^{-103}+412q^{-104}+124q^{-105}+265q^{-106}+83q^{-107}-43q^{-108}-333q^{-109}-281q^{-110}+63q^{-111}-25q^{-112}+223q^{-113}+196q^{-114}+145q^{-115}-138q^{-116}-201q^{-117}-49q^{-118}-120q^{-119}+69q^{-120}+115q^{-121}+152q^{-122}-17q^{-123}-75q^{-124}-19q^{-125}-94q^{-126}-6q^{-127}+28q^{-128}+82q^{-129}+6q^{-130}-20q^{-131}+10q^{-132}-40q^{-133}-12q^{-134}-q^{-135}+31q^{-136}+q^{-137}-8q^{-138}+11q^{-139}-11q^{-140}-4q^{-141}-3q^{-142}+10q^{-143}-q^{-144}-5q^{-145}+5q^{-146}-2q^{-147}-q^{-149}+2q^{-150}-2q^{-152}+q^{-153}$
7	$q^{49}-2q^{48}+q^{46}-q^{45}+2q^{44}+2q^{42}+2q^{41}-8q^{40}-2q^{39}+2q^{38}-5q^{37}+6q^{36}+2q^{35}+10q^{34}+14q^{33}-20q^{32}-8q^{31}-3q^{30}-18q^{29}+5q^{28}+q^{27}+27q^{26}+47q^{25}-22q^{24}-15q^{23}-17q^{22}-52q^{21}-3q^{20}-13q^{19}+47q^{18}+115q^{17}+2q^{16}-10q^{15}-54q^{14}-129q^{13}-46q^{12}-45q^{11}+87q^{10}+245q^{9}+105q^{8}+56q^{7}-107q^{6}-328q^{5}-232q^{4}-190q^{3}+107q^{2}+497q+437+394q^{-1}-21q^{-2}-633q^{-3}-751q^{-4}-785q^{-5}-213q^{-6}+719q^{-7}+1108q^{-8}+1367q^{-9}+734q^{-10}-604q^{-11}-1499q^{-12}-2163q^{-13}-1532q^{-14}+211q^{-15}+1679q^{-16}+3035q^{-17}+2740q^{-18}+666q^{-19}-1634q^{-20}-3948q^{-21}-4153q^{-22}-1911q^{-23}+1075q^{-24}+4556q^{-25}+5758q^{-26}+3676q^{-27}-81q^{-28}-4912q^{-29}-7247q^{-30}-5570q^{-31}-1417q^{-32}+4701q^{-33}+8508q^{-34}+7619q^{-35}+3201q^{-36}-4138q^{-37}-9366q^{-38}-9411q^{-39}-5108q^{-40}+3157q^{-41}+9801q^{-42}+10930q^{-43}+6913q^{-44}-2041q^{-45}-9850q^{-46}-11991q^{-47}-8469q^{-48}+875q^{-49}+9628q^{-50}+12691q^{-51}+9659q^{-52}+131q^{-53}-9240q^{-54}-13006q^{-55}-10516q^{-56}-995q^{-57}+8840q^{-58}+13141q^{-59}+11041q^{-60}+1579q^{-61}-8434q^{-62}-13063q^{-63}-11373q^{-64}-2057q^{-65}+8099q^{-66}+12985q^{-67}+11532q^{-68}+2363q^{-69}-7775q^{-70}-12798q^{-71}-11617q^{-72}-2689q^{-73}+7427q^{-74}+12593q^{-75}+11682q^{-76}+3005q^{-77}-7008q^{-78}-12276q^{-79}-11681q^{-80}-3415q^{-81}+6397q^{-82}+11827q^{-83}+11667q^{-84}+3922q^{-85}-5618q^{-86}-11181q^{-87}-11530q^{-88}-4511q^{-89}+4590q^{-90}+10267q^{-91}+11255q^{-92}+5150q^{-93}-3337q^{-94}-9083q^{-95}-10777q^{-96}-5738q^{-97}+1963q^{-98}+7609q^{-99}+9957q^{-100}+6175q^{-101}-463q^{-102}-5890q^{-103}-8898q^{-104}-6365q^{-105}-848q^{-106}+4064q^{-107}+7429q^{-108}+6152q^{-109}+2011q^{-110}-2213q^{-111}-5819q^{-112}-5594q^{-113}-2696q^{-114}+633q^{-115}+4038q^{-116}+4596q^{-117}+2962q^{-118}+657q^{-119}-2385q^{-120}-3432q^{-121}-2731q^{-122}-1386q^{-123}+993q^{-124}+2130q^{-125}+2124q^{-126}+1672q^{-127}-4q^{-128}-999q^{-129}-1338q^{-130}-1489q^{-131}-512q^{-132}+135q^{-133}+529q^{-134}+1043q^{-135}+632q^{-136}+370q^{-137}+107q^{-138}-522q^{-139}-459q^{-140}-520q^{-141}-489q^{-142}+56q^{-143}+163q^{-144}+444q^{-145}+608q^{-146}+232q^{-147}+118q^{-148}-241q^{-149}-546q^{-150}-343q^{-151}-278q^{-152}+30q^{-153}+369q^{-154}+315q^{-155}+358q^{-156}+105q^{-157}-224q^{-158}-217q^{-159}-294q^{-160}-160q^{-161}+65q^{-162}+109q^{-163}+245q^{-164}+166q^{-165}-26q^{-166}-45q^{-167}-134q^{-168}-113q^{-169}-24q^{-170}-19q^{-171}+92q^{-172}+93q^{-173}+11q^{-174}+10q^{-175}-42q^{-176}-35q^{-177}-10q^{-178}-32q^{-179}+18q^{-180}+36q^{-181}+6q^{-182}+8q^{-183}-14q^{-184}-6q^{-185}+6q^{-186}-16q^{-187}+10q^{-189}+2q^{-190}+3q^{-191}-6q^{-192}-q^{-193}+6q^{-194}-4q^{-195}-2q^{-196}+2q^{-197}+q^{-199}-2q^{-200}+2q^{-202}-q^{-203}$

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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