9 44

9_43

9_45

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 44's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 44 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
Gauss code	-1, 4, -3, 1, -2, -9, 5, 3, -4, 2, 7, -8, 9, -5, 6, -7, 8, -6
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -6 -18 -12
Conway Notation	[22,21,2-]

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

[edit Notes on presentations of 9 44]

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 44"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 -14 2 -16 -6 -18 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,21,2-]

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]=

[math]\displaystyle{ \textrm{BR}(4,\{-1,-1,-1,-2,1,1,3,-2,3\}) }[/math]

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, 3}, {1, 7}, {8, 4}, {7, 10}, {6, 9}, {3, 8}, {2, 5}, {4, 6}, {5, 1}, {9, 2}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,5\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-3] Hyperbolic Volume 7.40677 A-Polynomial See Data:9 44/A-polynomial

[edit Notes for 9 44's three dimensional invariants] 9_44 has girth 4. See arXiv:math.GT/0508590 and a forthcoming paper by the same authors.

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant 0

[edit Notes for 9 44's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2-4 t+7-4 t^{-1} + t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 17, 0 }
Jones polynomial	[math]\displaystyle{ q^2-2 q+3-3 q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+3 a^2-2 z^2-2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^7+a z^7+2 a^4 z^6+3 a^2 z^6+z^6+a^5 z^5-2 a^3 z^5-3 a z^5-7 a^4 z^4-10 a^2 z^4-3 z^4-3 a^5 z^3-a^3 z^3+4 a z^3+2 z^3 a^{-1} +5 a^4 z^2+10 a^2 z^2+z^2 a^{-2} +6 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+2 q^8+q^6+q^4-1- q^{-4} + q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+q^{72}-4 q^{68}+6 q^{66}-5 q^{64}+3 q^{62}-q^{60}-4 q^{58}+5 q^{56}-4 q^{54}+2 q^{50}-4 q^{48}+4 q^{46}-4 q^{42}+7 q^{40}-5 q^{38}+3 q^{36}-2 q^{32}+6 q^{30}-4 q^{28}+6 q^{26}-2 q^{24}+2 q^{22}+4 q^{20}-4 q^{18}+4 q^{16}-2 q^{14}+q^{12}+3 q^{10}-4 q^8+2 q^6-4 q^2+5-6 q^{-2} +2 q^{-6} -6 q^{-8} +5 q^{-10} -3 q^{-12} + q^{-14} + q^{-16} -3 q^{-18} +2 q^{-20} + q^{-24} + q^{-26} + q^{-32} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_44.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^9+q^5+ q^{-1} - q^{-3} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{32}-q^{30}-2 q^{28}+2 q^{26}+q^{24}-2 q^{22}+2 q^{18}-q^{16}-q^{14}+2 q^{12}-q^8+q^6+q^4-q^2-1+3 q^{-2} -2 q^{-6} +2 q^{-8} + q^{-10} - q^{-12} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{61}+2 q^{59}-3 q^{55}-3 q^{53}+3 q^{51}+4 q^{49}-4 q^{45}-3 q^{43}+3 q^{41}+5 q^{39}-q^{37}-6 q^{35}-3 q^{33}+6 q^{31}+4 q^{29}-4 q^{27}-4 q^{25}+4 q^{23}+5 q^{21}-3 q^{19}-4 q^{17}+2 q^{15}+3 q^{13}-2 q^{11}-3 q^9+3 q^5+3 q^3-2 q-4 q^{-1} +2 q^{-3} +7 q^{-5} +3 q^{-7} -7 q^{-9} -4 q^{-11} +5 q^{-13} +4 q^{-15} -2 q^{-17} -5 q^{-19} +3 q^{-23} + q^{-25} - q^{-29} }[/math]
4	[math]\displaystyle{ q^{104}-q^{102}-2 q^{100}+q^{96}+5 q^{94}+q^{92}-3 q^{90}-5 q^{88}-6 q^{86}+5 q^{84}+7 q^{82}+5 q^{80}-10 q^{76}-7 q^{74}-q^{72}+9 q^{70}+14 q^{68}+q^{66}-11 q^{64}-16 q^{62}-5 q^{60}+15 q^{58}+17 q^{56}+2 q^{54}-18 q^{52}-18 q^{50}+5 q^{48}+21 q^{46}+14 q^{44}-9 q^{42}-20 q^{40}-5 q^{38}+14 q^{36}+13 q^{34}-4 q^{32}-15 q^{30}-5 q^{28}+9 q^{26}+8 q^{24}-2 q^{22}-9 q^{20}-3 q^{18}+7 q^{16}+6 q^{14}+2 q^{12}-4 q^{10}-7 q^8-q^6+6 q^4+13 q^2+3-15 q^{-2} -15 q^{-4} - q^{-6} +22 q^{-8} +20 q^{-10} -9 q^{-12} -25 q^{-14} -15 q^{-16} +15 q^{-18} +26 q^{-20} +5 q^{-22} -14 q^{-24} -18 q^{-26} -2 q^{-28} +13 q^{-30} +9 q^{-32} -7 q^{-36} -5 q^{-38} + q^{-40} +2 q^{-42} +2 q^{-44} - q^{-48} }[/math]
5	[math]\displaystyle{ -q^{155}+q^{153}+2 q^{151}-q^{147}-3 q^{145}-3 q^{143}-q^{141}+5 q^{139}+7 q^{137}+4 q^{135}-q^{133}-7 q^{131}-11 q^{129}-8 q^{127}+5 q^{125}+11 q^{123}+13 q^{121}+9 q^{119}-4 q^{117}-16 q^{115}-20 q^{113}-10 q^{111}+6 q^{109}+24 q^{107}+29 q^{105}+13 q^{103}-16 q^{101}-37 q^{99}-34 q^{97}-8 q^{95}+32 q^{93}+50 q^{91}+32 q^{89}-13 q^{87}-54 q^{85}-54 q^{83}-12 q^{81}+43 q^{79}+67 q^{77}+38 q^{75}-25 q^{73}-67 q^{71}-53 q^{69}+3 q^{67}+59 q^{65}+62 q^{63}+13 q^{61}-46 q^{59}-62 q^{57}-22 q^{55}+32 q^{53}+54 q^{51}+25 q^{49}-23 q^{47}-45 q^{45}-23 q^{43}+18 q^{41}+34 q^{39}+17 q^{37}-13 q^{35}-26 q^{33}-10 q^{31}+12 q^{29}+19 q^{27}+8 q^{25}-9 q^{23}-16 q^{21}-10 q^{19}+4 q^{17}+17 q^{15}+17 q^{13}+6 q^{11}-14 q^9-29 q^7-21 q^5+9 q^3+41 q+40 q^{-1} +5 q^{-3} -44 q^{-5} -63 q^{-7} -27 q^{-9} +42 q^{-11} +81 q^{-13} +49 q^{-15} -23 q^{-17} -82 q^{-19} -70 q^{-21} + q^{-23} +69 q^{-25} +78 q^{-27} +19 q^{-29} -44 q^{-31} -66 q^{-33} -36 q^{-35} +18 q^{-37} +47 q^{-39} +36 q^{-41} + q^{-43} -23 q^{-45} -26 q^{-47} -11 q^{-49} +6 q^{-51} +14 q^{-53} +9 q^{-55} -3 q^{-59} -4 q^{-61} -2 q^{-63} + q^{-65} + q^{-67} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+2 q^8+q^6+q^4-1- q^{-4} + q^{-6} + q^{-8} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+4 q^{40}-8 q^{38}+11 q^{36}-12 q^{34}+12 q^{32}-12 q^{30}+4 q^{28}+2 q^{26}-8 q^{24}+16 q^{22}-19 q^{20}+22 q^{18}-20 q^{16}+20 q^{14}-14 q^{12}+10 q^{10}-2 q^8-4 q^6+8 q^4-14 q^2+14-12 q^{-2} +11 q^{-4} -4 q^{-6} +4 q^{-8} +2 q^{-10} - q^{-12} -2 q^{-16} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{38}-q^{36}+q^{32}-q^{30}-q^{28}+2 q^{18}+3 q^{16}+q^{14}+q^{12}-q^8-2 q^6-q^4-2 q^2+2 q^{-2} +3 q^{-4} +2 q^{-6} +2 q^{-10} -2 q^{-14} - q^{-16} + q^{-20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-q^{32}-2 q^{26}+q^{24}-q^{22}-q^{20}+q^{18}+q^{16}+2 q^{12}+2 q^{10}+q^8+q^6+q^2-1- q^{-2} + q^{-4} -2 q^{-6} +2 q^{-10} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}+2 q^{11}+2 q^9+2 q^7+q^5-q-2 q^{-1} - q^{-5} + q^{-7} + q^{-9} + q^{-11} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}+q^{38}-q^{36}-3 q^{34}-2 q^{32}-q^{30}-3 q^{28}-q^{26}+3 q^{24}+5 q^{22}+2 q^{20}+3 q^{18}+4 q^{16}-q^{14}-2 q^{12}-q^8-q^6+2 q^4+3 q^2+1+ q^{-4} -3 q^{-8} - q^{-10} + q^{-12} + q^{-18} + q^{-20} + q^{-22} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{22}-q^{20}+2 q^{14}+2 q^{12}+3 q^{10}+2 q^8+q^6-q^2-2-2 q^{-2} - q^{-6} + q^{-8} + q^{-10} + q^{-12} + q^{-14} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+q^{32}-2 q^{30}+2 q^{28}-2 q^{26}+q^{24}-q^{22}+q^{20}+q^{18}-q^{16}+4 q^{14}-2 q^{12}+4 q^{10}-3 q^8+3 q^6-2 q^4+q^2-1- q^{-2} + q^{-4} -2 q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-12} + q^{-16} }[/math]
1,0	[math]\displaystyle{ q^{56}-q^{52}-q^{50}+q^{48}+q^{46}-2 q^{44}-2 q^{42}+q^{40}+2 q^{38}-2 q^{34}-q^{32}+2 q^{30}+q^{28}-q^{24}+q^{22}+q^{20}+q^{18}-q^{16}+2 q^{12}+q^{10}-q^8-q^6+q^4+2 q^2-2 q^{-2} +2 q^{-6} -2 q^{-10} - q^{-12} + q^{-14} +2 q^{-16} - q^{-20} + q^{-26} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{46}-q^{44}+q^{42}-2 q^{40}+2 q^{38}-2 q^{36}-2 q^{32}-q^{28}-q^{26}+q^{24}-q^{22}+3 q^{20}+5 q^{16}+5 q^{12}-q^{10}+3 q^8-q^6+q^4-2 q^2-1- q^{-2} -2 q^{-4} + q^{-6} -2 q^{-8} + q^{-10} - q^{-12} +3 q^{-14} + q^{-18} + q^{-22} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+q^{72}-4 q^{68}+6 q^{66}-5 q^{64}+3 q^{62}-q^{60}-4 q^{58}+5 q^{56}-4 q^{54}+2 q^{50}-4 q^{48}+4 q^{46}-4 q^{42}+7 q^{40}-5 q^{38}+3 q^{36}-2 q^{32}+6 q^{30}-4 q^{28}+6 q^{26}-2 q^{24}+2 q^{22}+4 q^{20}-4 q^{18}+4 q^{16}-2 q^{14}+q^{12}+3 q^{10}-4 q^8+2 q^6-4 q^2+5-6 q^{-2} +2 q^{-6} -6 q^{-8} +5 q^{-10} -3 q^{-12} + q^{-14} + q^{-16} -3 q^{-18} +2 q^{-20} + q^{-24} + q^{-26} + q^{-32} + q^{-38} }[/math]

.

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 44"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^2-4 t+7-4 t^{-1} + t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^4+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^2-2 q+3-3 q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+3 a^2-2 z^2-2+ a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^7+a z^7+2 a^4 z^6+3 a^2 z^6+z^6+a^5 z^5-2 a^3 z^5-3 a z^5-7 a^4 z^4-10 a^2 z^4-3 z^4-3 a^5 z^3-a^3 z^3+4 a z^3+2 z^3 a^{-1} +5 a^4 z^2+10 a^2 z^2+z^2 a^{-2} +6 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

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 44"];

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]=

{ [math]\displaystyle{ t^2-4 t+7-4 t^{-1} + t^{-2} }[/math], [math]\displaystyle{ q^2-2 q+3-3 q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} - q^{-5} }[/math] }

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:

(0, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ -\frac{56}{3} }[/math] [math]\displaystyle{ \frac{104}{3} }[/math] [math]\displaystyle{ -\frac{16}{3} }[/math] [math]\displaystyle{ 0 }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 9 44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ -q^5+2 q^4+q^3-5 q^2+4 q+4-9 q^{-1} +4 q^{-2} +6 q^{-3} -9 q^{-4} +2 q^{-5} +7 q^{-6} -7 q^{-7} - q^{-8} +7 q^{-9} -4 q^{-10} -3 q^{-11} +5 q^{-12} - q^{-13} -2 q^{-14} + q^{-15} }[/math]
3	[math]\displaystyle{ -q^{13}+q^{12}+q^{11}+2 q^{10}-4 q^9-4 q^8+4 q^7+8 q^6-3 q^5-13 q^4+q^3+18 q^2+q-18-5 q^{-1} +20 q^{-2} +6 q^{-3} -18 q^{-4} -8 q^{-5} +17 q^{-6} +7 q^{-7} -13 q^{-8} -9 q^{-9} +11 q^{-10} +8 q^{-11} -5 q^{-12} -10 q^{-13} +3 q^{-14} +8 q^{-15} +3 q^{-16} -8 q^{-17} -6 q^{-18} +5 q^{-19} +8 q^{-20} -2 q^{-21} -8 q^{-22} - q^{-23} +7 q^{-24} +2 q^{-25} -4 q^{-26} -2 q^{-27} + q^{-28} +2 q^{-29} - q^{-30} }[/math]
4	[math]\displaystyle{ -q^{22}+q^{21}+2 q^{20}-q^{18}-7 q^{17}-q^{16}+9 q^{15}+9 q^{14}+3 q^{13}-22 q^{12}-17 q^{11}+13 q^{10}+28 q^9+24 q^8-33 q^7-47 q^6+3 q^5+44 q^4+53 q^3-31 q^2-70 q-11+44 q^{-1} +71 q^{-2} -21 q^{-3} -77 q^{-4} -18 q^{-5} +38 q^{-6} +74 q^{-7} -15 q^{-8} -73 q^{-9} -17 q^{-10} +28 q^{-11} +68 q^{-12} -8 q^{-13} -63 q^{-14} -16 q^{-15} +14 q^{-16} +58 q^{-17} +3 q^{-18} -46 q^{-19} -15 q^{-20} -5 q^{-21} +43 q^{-22} +14 q^{-23} -23 q^{-24} -8 q^{-25} -21 q^{-26} +20 q^{-27} +14 q^{-28} -3 q^{-29} +7 q^{-30} -23 q^{-31} +3 q^{-33} +2 q^{-34} +19 q^{-35} -10 q^{-36} -5 q^{-37} -7 q^{-38} -4 q^{-39} +16 q^{-40} -5 q^{-43} -6 q^{-44} +5 q^{-45} + q^{-46} +2 q^{-47} - q^{-48} -2 q^{-49} + q^{-50} }[/math]
5	[math]\displaystyle{ q^{31}-3 q^{29}-2 q^{28}+q^{27}+3 q^{26}+10 q^{25}+5 q^{24}-11 q^{23}-19 q^{22}-14 q^{21}+6 q^{20}+34 q^{19}+40 q^{18}-48 q^{16}-68 q^{15}-24 q^{14}+56 q^{13}+103 q^{12}+59 q^{11}-57 q^{10}-136 q^9-95 q^8+44 q^7+162 q^6+131 q^5-25 q^4-175 q^3-164 q^2+8 q+181+180 q^{-1} +10 q^{-2} -174 q^{-3} -196 q^{-4} -22 q^{-5} +173 q^{-6} +195 q^{-7} +30 q^{-8} -163 q^{-9} -196 q^{-10} -35 q^{-11} +159 q^{-12} +189 q^{-13} +37 q^{-14} -146 q^{-15} -185 q^{-16} -42 q^{-17} +137 q^{-18} +173 q^{-19} +50 q^{-20} -116 q^{-21} -168 q^{-22} -58 q^{-23} +96 q^{-24} +151 q^{-25} +72 q^{-26} -68 q^{-27} -139 q^{-28} -80 q^{-29} +42 q^{-30} +111 q^{-31} +88 q^{-32} -9 q^{-33} -90 q^{-34} -83 q^{-35} -14 q^{-36} +55 q^{-37} +74 q^{-38} +33 q^{-39} -27 q^{-40} -54 q^{-41} -38 q^{-42} +32 q^{-44} +33 q^{-45} +14 q^{-46} -9 q^{-47} -20 q^{-48} -18 q^{-49} -8 q^{-50} +7 q^{-51} +11 q^{-52} +12 q^{-53} +9 q^{-54} -2 q^{-55} -13 q^{-56} -11 q^{-57} -5 q^{-58} +2 q^{-59} +13 q^{-60} +10 q^{-61} -7 q^{-63} -7 q^{-64} -4 q^{-65} +7 q^{-67} +4 q^{-68} - q^{-69} -2 q^{-70} - q^{-71} -2 q^{-72} + q^{-73} +2 q^{-74} - q^{-75} }[/math]
6	[math]\displaystyle{ q^{47}-q^{46}-q^{45}-q^{42}-q^{41}+8 q^{40}+3 q^{39}-2 q^{37}-8 q^{36}-17 q^{35}-16 q^{34}+17 q^{33}+26 q^{32}+31 q^{31}+25 q^{30}-6 q^{29}-65 q^{28}-88 q^{27}-25 q^{26}+31 q^{25}+100 q^{24}+134 q^{23}+82 q^{22}-83 q^{21}-213 q^{20}-173 q^{19}-66 q^{18}+128 q^{17}+297 q^{16}+285 q^{15}+9 q^{14}-288 q^{13}-365 q^{12}-266 q^{11}+47 q^{10}+399 q^9+506 q^8+182 q^7-259 q^6-478 q^5-452 q^4-93 q^3+397 q^2+625 q+328-179 q^{-1} -490 q^{-2} -539 q^{-3} -199 q^{-4} +349 q^{-5} +645 q^{-6} +390 q^{-7} -120 q^{-8} -460 q^{-9} -549 q^{-10} -240 q^{-11} +310 q^{-12} +624 q^{-13} +399 q^{-14} -96 q^{-15} -428 q^{-16} -530 q^{-17} -248 q^{-18} +279 q^{-19} +587 q^{-20} +395 q^{-21} -71 q^{-22} -384 q^{-23} -500 q^{-24} -261 q^{-25} +221 q^{-26} +524 q^{-27} +395 q^{-28} -12 q^{-29} -301 q^{-30} -454 q^{-31} -296 q^{-32} +115 q^{-33} +420 q^{-34} +391 q^{-35} +84 q^{-36} -168 q^{-37} -372 q^{-38} -330 q^{-39} -28 q^{-40} +264 q^{-41} +347 q^{-42} +175 q^{-43} - q^{-44} -234 q^{-45} -311 q^{-46} -151 q^{-47} +75 q^{-48} +231 q^{-49} +193 q^{-50} +135 q^{-51} -59 q^{-52} -203 q^{-53} -182 q^{-54} -74 q^{-55} +73 q^{-56} +109 q^{-57} +163 q^{-58} +67 q^{-59} -51 q^{-60} -105 q^{-61} -105 q^{-62} -29 q^{-63} -11 q^{-64} +85 q^{-65} +76 q^{-66} +36 q^{-67} -7 q^{-68} -43 q^{-69} -22 q^{-70} -58 q^{-71} +2 q^{-72} +16 q^{-73} +25 q^{-74} +18 q^{-75} +8 q^{-76} +25 q^{-77} -28 q^{-78} -10 q^{-79} -16 q^{-80} -6 q^{-81} -7 q^{-82} +2 q^{-83} +33 q^{-84} +7 q^{-86} -5 q^{-87} -6 q^{-88} -15 q^{-89} -10 q^{-90} +12 q^{-91} + q^{-92} +8 q^{-93} +3 q^{-94} +3 q^{-95} -7 q^{-96} -6 q^{-97} +3 q^{-98} -2 q^{-99} +2 q^{-100} + q^{-101} +2 q^{-102} - q^{-103} -2 q^{-104} + q^{-105} }[/math]
7	[math]\displaystyle{ q^{63}-q^{62}-q^{61}-q^{60}+2 q^{58}+q^{57}+2 q^{56}+5 q^{55}+q^{54}-5 q^{53}-11 q^{52}-14 q^{51}-3 q^{50}+q^{49}+14 q^{48}+34 q^{47}+36 q^{46}+16 q^{45}-23 q^{44}-66 q^{43}-74 q^{42}-62 q^{41}-14 q^{40}+83 q^{39}+152 q^{38}+166 q^{37}+84 q^{36}-76 q^{35}-217 q^{34}-294 q^{33}-243 q^{32}-15 q^{31}+257 q^{30}+461 q^{29}+466 q^{28}+182 q^{27}-225 q^{26}-602 q^{25}-725 q^{24}-446 q^{23}+93 q^{22}+683 q^{21}+1000 q^{20}+770 q^{19}+109 q^{18}-684 q^{17}-1216 q^{16}-1089 q^{15}-386 q^{14}+595 q^{13}+1362 q^{12}+1380 q^{11}+672 q^{10}-460 q^9-1421 q^8-1593 q^7-911 q^6+284 q^5+1403 q^4+1723 q^3+1113 q^2-125 q-1358-1784 q^{-1} -1228 q^{-2} +3 q^{-3} +1281 q^{-4} +1788 q^{-5} +1300 q^{-6} +89 q^{-7} -1221 q^{-8} -1778 q^{-9} -1318 q^{-10} -135 q^{-11} +1171 q^{-12} +1745 q^{-13} +1319 q^{-14} +163 q^{-15} -1133 q^{-16} -1720 q^{-17} -1309 q^{-18} -173 q^{-19} +1103 q^{-20} +1689 q^{-21} +1291 q^{-22} +187 q^{-23} -1064 q^{-24} -1655 q^{-25} -1278 q^{-26} -208 q^{-27} +1010 q^{-28} +1608 q^{-29} +1266 q^{-30} +250 q^{-31} -934 q^{-32} -1544 q^{-33} -1252 q^{-34} -313 q^{-35} +818 q^{-36} +1456 q^{-37} +1250 q^{-38} +398 q^{-39} -679 q^{-40} -1339 q^{-41} -1222 q^{-42} -503 q^{-43} +484 q^{-44} +1189 q^{-45} +1197 q^{-46} +617 q^{-47} -285 q^{-48} -1000 q^{-49} -1117 q^{-50} -717 q^{-51} +45 q^{-52} +762 q^{-53} +1020 q^{-54} +791 q^{-55} +171 q^{-56} -512 q^{-57} -842 q^{-58} -798 q^{-59} -377 q^{-60} +236 q^{-61} +635 q^{-62} +751 q^{-63} +500 q^{-64} +10 q^{-65} -380 q^{-66} -617 q^{-67} -557 q^{-68} -209 q^{-69} +139 q^{-70} +428 q^{-71} +514 q^{-72} +321 q^{-73} +77 q^{-74} -220 q^{-75} -402 q^{-76} -337 q^{-77} -211 q^{-78} +28 q^{-79} +238 q^{-80} +277 q^{-81} +260 q^{-82} +108 q^{-83} -88 q^{-84} -164 q^{-85} -227 q^{-86} -163 q^{-87} -23 q^{-88} +42 q^{-89} +146 q^{-90} +154 q^{-91} +79 q^{-92} +34 q^{-93} -63 q^{-94} -94 q^{-95} -63 q^{-96} -76 q^{-97} -12 q^{-98} +41 q^{-99} +37 q^{-100} +64 q^{-101} +22 q^{-102} +4 q^{-103} +15 q^{-104} -33 q^{-105} -32 q^{-106} -18 q^{-107} -23 q^{-108} +8 q^{-109} +2 q^{-110} +4 q^{-111} +37 q^{-112} +12 q^{-113} +6 q^{-114} -20 q^{-116} -7 q^{-117} -13 q^{-118} -15 q^{-119} +7 q^{-120} +9 q^{-121} +12 q^{-122} +12 q^{-123} -5 q^{-124} + q^{-125} -3 q^{-126} -10 q^{-127} -3 q^{-128} -2 q^{-129} +4 q^{-130} +6 q^{-131} - q^{-132} +2 q^{-134} -2 q^{-135} - q^{-136} -2 q^{-137} + q^{-138} +2 q^{-139} - q^{-140} }[/math]

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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