10 142

10_141

10_143

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 142's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 142 at Knotilus!
	10_142 is also known as the pretzel knot P(-4,3,3).

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X13,1,14,20 X19,13,20,12 X2,10,3,9
Gauss code	1, -10, 2, -1, -4, 5, -6, 7, 10, -2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8
Dowker-Thistlethwaite code	4 10 -14 -16 2 -18 -20 -8 -6 -12
Conway Notation	[31,3,3-]

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

[edit Notes on presentations of 10 142]

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["10 142"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X13,1,14,20 X19,13,20,12 X2,10,3,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 -14 -16 2 -18 -20 -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]=

[31,3,3-]

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,\{1,1,1,2,1,1,1,2,3,-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, 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[{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {9, 7}, {11, 6}, {10, 12}, {2, 11}, {12, 3}, {8, 2}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [5][-14] Hyperbolic Volume 6.77082 A-Polynomial See Data:10 142/A-polynomial

[edit Notes for 10 142's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 3 }[/math] Topological 4 genus [math]\displaystyle{ 3 }[/math] Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant -6

[edit Notes for 10 142's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+9 z^4+8 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature	{ 15, 6 }
Jones polynomial	[math]\displaystyle{ -2 q^{10}+2 q^9-2 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +7 z^2 a^{-8} -5 z^2 a^{-10} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -6 z^6 a^{-10} -4 z^5 a^{-7} -9 z^5 a^{-9} -5 z^5 a^{-11} -5 z^4 a^{-6} +9 z^4 a^{-8} +15 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-7} +12 z^3 a^{-9} +9 z^3 a^{-11} +6 z^2 a^{-6} -10 z^2 a^{-8} -17 z^2 a^{-10} -z^2 a^{-12} -6 z a^{-9} -4 z a^{-11} +2 z a^{-13} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} +2 q^{-22} +3 q^{-24} - q^{-28} -3 q^{-30} -2 q^{-32} - q^{-34} + q^{-38} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +4 q^{-68} - q^{-70} - q^{-72} +3 q^{-74} -2 q^{-76} +2 q^{-78} - q^{-82} +3 q^{-84} +3 q^{-90} -4 q^{-92} +4 q^{-94} -2 q^{-98} +5 q^{-100} -2 q^{-102} +4 q^{-104} + q^{-106} +3 q^{-108} +2 q^{-110} -2 q^{-112} + q^{-114} +3 q^{-118} - q^{-120} -3 q^{-122} - q^{-124} + q^{-126} - q^{-130} -9 q^{-132} + q^{-136} -4 q^{-138} -7 q^{-142} + q^{-144} +3 q^{-146} -3 q^{-148} -2 q^{-150} -2 q^{-152} + q^{-154} +3 q^{-156} - q^{-158} - q^{-160} +2 q^{-162} +2 q^{-166} +2 q^{-168} - q^{-170} + q^{-172} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_142.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-5} + q^{-9} + q^{-13} + q^{-15} -2 q^{-21} }[/math]
2	[math]\displaystyle{ q^{-10} +2 q^{-16} + q^{-18} - q^{-20} + q^{-22} +2 q^{-24} - q^{-28} - q^{-34} + q^{-36} +2 q^{-38} - q^{-40} - q^{-46} -3 q^{-48} - q^{-50} - q^{-54} + q^{-56} + q^{-58} + q^{-60} }[/math]
3	[math]\displaystyle{ q^{-15} + q^{-21} +2 q^{-23} + q^{-25} - q^{-27} - q^{-29} +2 q^{-31} +3 q^{-33} + q^{-35} -2 q^{-37} -2 q^{-39} + q^{-41} +3 q^{-43} + q^{-45} -2 q^{-47} -3 q^{-49} + q^{-51} +4 q^{-53} + q^{-55} -4 q^{-57} -3 q^{-59} +2 q^{-61} +2 q^{-63} -3 q^{-65} -2 q^{-67} +3 q^{-69} + q^{-71} -2 q^{-73} - q^{-75} + q^{-77} - q^{-81} -3 q^{-83} -2 q^{-85} +3 q^{-89} -2 q^{-91} -4 q^{-93} +7 q^{-97} +4 q^{-99} -2 q^{-101} -2 q^{-103} + q^{-105} +4 q^{-107} -2 q^{-111} -2 q^{-113} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} +2 q^{-22} +3 q^{-24} - q^{-28} -3 q^{-30} -2 q^{-32} - q^{-34} + q^{-38} }[/math]
1,1	[math]\displaystyle{ q^{-20} +2 q^{-24} -2 q^{-26} +6 q^{-28} -2 q^{-30} +12 q^{-32} -2 q^{-34} +7 q^{-36} +6 q^{-42} -10 q^{-44} +10 q^{-46} -10 q^{-48} +8 q^{-50} -11 q^{-52} +2 q^{-54} -8 q^{-56} -4 q^{-58} -3 q^{-60} -6 q^{-62} +6 q^{-64} -4 q^{-66} +10 q^{-68} +2 q^{-72} +2 q^{-74} -2 q^{-76} -2 q^{-78} -4 q^{-80} -2 q^{-82} +4 q^{-84} +2 q^{-88} }[/math]
2,0	[math]\displaystyle{ q^{-20} + q^{-26} +2 q^{-28} + q^{-30} + q^{-32} +2 q^{-34} +3 q^{-36} +2 q^{-38} + q^{-40} + q^{-42} - q^{-46} + q^{-52} +2 q^{-54} +3 q^{-56} +2 q^{-58} + q^{-60} -4 q^{-62} -6 q^{-64} -10 q^{-66} -9 q^{-68} -5 q^{-70} - q^{-72} +3 q^{-74} +4 q^{-76} +6 q^{-78} +5 q^{-80} +4 q^{-82} -2 q^{-88} -2 q^{-90} - q^{-92} + q^{-96} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-20} + q^{-24} + q^{-26} + q^{-30} +2 q^{-32} +3 q^{-34} +5 q^{-36} +4 q^{-38} +4 q^{-40} +2 q^{-42} -3 q^{-46} -3 q^{-48} -5 q^{-50} -4 q^{-52} -4 q^{-54} -3 q^{-56} - q^{-58} - q^{-60} + q^{-62} + q^{-64} + q^{-66} + q^{-68} +2 q^{-70} }[/math]
1,0,0	[math]\displaystyle{ q^{-15} + q^{-19} + q^{-23} +2 q^{-27} +3 q^{-29} +3 q^{-31} +3 q^{-33} - q^{-37} -4 q^{-39} -3 q^{-41} -3 q^{-43} - q^{-45} + q^{-49} + q^{-51} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-30} + q^{-34} + q^{-36} + q^{-38} +2 q^{-42} +2 q^{-44} +4 q^{-46} +5 q^{-48} +5 q^{-50} +7 q^{-52} +7 q^{-54} +5 q^{-56} +4 q^{-58} +4 q^{-60} +2 q^{-62} -2 q^{-64} -4 q^{-66} -6 q^{-68} -10 q^{-70} -14 q^{-72} -12 q^{-74} -12 q^{-76} -10 q^{-78} -2 q^{-80} +3 q^{-82} +6 q^{-84} +8 q^{-86} +10 q^{-88} +6 q^{-90} +2 q^{-92} -2 q^{-98} -2 q^{-100} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-20} + q^{-24} + q^{-28} + q^{-32} +2 q^{-34} +3 q^{-36} +4 q^{-38} +3 q^{-40} +3 q^{-42} - q^{-46} -4 q^{-48} -4 q^{-50} -4 q^{-52} -3 q^{-54} - q^{-56} + q^{-60} + q^{-62} + q^{-64} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-20} + q^{-24} - q^{-26} +2 q^{-28} - q^{-30} +2 q^{-32} + q^{-34} + q^{-36} +2 q^{-38} +2 q^{-42} -2 q^{-44} +3 q^{-46} -3 q^{-48} + q^{-50} -2 q^{-52} - q^{-56} - q^{-58} + q^{-60} - q^{-62} + q^{-64} - q^{-66} + q^{-68} -2 q^{-70} }[/math]
1,0	[math]\displaystyle{ q^{-30} + q^{-38} + q^{-40} - q^{-44} + q^{-46} +2 q^{-48} + q^{-50} - q^{-52} +2 q^{-54} +3 q^{-56} +3 q^{-58} + q^{-60} + q^{-62} + q^{-64} +2 q^{-66} - q^{-70} - q^{-72} - q^{-76} -2 q^{-78} -2 q^{-80} - q^{-82} -2 q^{-86} -3 q^{-88} -2 q^{-90} - q^{-94} -2 q^{-96} - q^{-98} + q^{-100} + q^{-102} + q^{-108} + q^{-110} +2 q^{-112} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{-30} + q^{-34} +2 q^{-38} - q^{-40} +2 q^{-42} +3 q^{-46} +3 q^{-48} +3 q^{-50} +5 q^{-52} +5 q^{-54} +6 q^{-56} +3 q^{-58} +4 q^{-60} -2 q^{-62} -6 q^{-66} -4 q^{-68} -8 q^{-70} -5 q^{-72} -6 q^{-74} -3 q^{-76} -2 q^{-78} - q^{-80} + q^{-82} +2 q^{-86} +2 q^{-90} + q^{-94} +2 q^{-98} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +4 q^{-68} - q^{-70} - q^{-72} +3 q^{-74} -2 q^{-76} +2 q^{-78} - q^{-82} +3 q^{-84} +3 q^{-90} -4 q^{-92} +4 q^{-94} -2 q^{-98} +5 q^{-100} -2 q^{-102} +4 q^{-104} + q^{-106} +3 q^{-108} +2 q^{-110} -2 q^{-112} + q^{-114} +3 q^{-118} - q^{-120} -3 q^{-122} - q^{-124} + q^{-126} - q^{-130} -9 q^{-132} + q^{-136} -4 q^{-138} -7 q^{-142} + q^{-144} +3 q^{-146} -3 q^{-148} -2 q^{-150} -2 q^{-152} + q^{-154} +3 q^{-156} - q^{-158} - q^{-160} +2 q^{-162} +2 q^{-166} +2 q^{-168} - q^{-170} + q^{-172} }[/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["10 142"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+9 z^4+8 z^2+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{ \left\{2,t^2+t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 15, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -2 q^{10}+2 q^9-2 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +7 z^2 a^{-8} -5 z^2 a^{-10} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -6 z^6 a^{-10} -4 z^5 a^{-7} -9 z^5 a^{-9} -5 z^5 a^{-11} -5 z^4 a^{-6} +9 z^4 a^{-8} +15 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-7} +12 z^3 a^{-9} +9 z^3 a^{-11} +6 z^2 a^{-6} -10 z^2 a^{-8} -17 z^2 a^{-10} -z^2 a^{-12} -6 z a^{-9} -4 z a^{-11} +2 z a^{-13} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12} }[/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["10 142"];

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{ 2 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -2 q^{10}+2 q^9-2 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/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:

(8, 21)

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{ 32 }[/math] [math]\displaystyle{ 168 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3520}{3} }[/math] [math]\displaystyle{ \frac{536}{3} }[/math] [math]\displaystyle{ 5376 }[/math] [math]\displaystyle{ 9072 }[/math] [math]\displaystyle{ 1600 }[/math] [math]\displaystyle{ 1160 }[/math] [math]\displaystyle{ \frac{16384}{3} }[/math] [math]\displaystyle{ 14112 }[/math] [math]\displaystyle{ \frac{112640}{3} }[/math] [math]\displaystyle{ \frac{17152}{3} }[/math] [math]\displaystyle{ \frac{1074964}{15} }[/math] [math]\displaystyle{ \frac{12488}{5} }[/math] [math]\displaystyle{ \frac{1215256}{45} }[/math] [math]\displaystyle{ \frac{3644}{9} }[/math] [math]\displaystyle{ \frac{52564}{15} }[/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]6 is the signature of 10 142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 χ 21               2 -2 19                 0 17           2 2   0 15         1       1 13       1 2       1 11     1 1         0 9     1           1 7 1 1             0 5 1               1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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^{29}-q^{26}+q^{25}-q^{24}-3 q^{23}+3 q^{22}-3 q^{20}+2 q^{19}+3 q^{18}-4 q^{17}+4 q^{15}-4 q^{14}-q^{13}+5 q^{12}-2 q^{11}-2 q^{10}+3 q^9-q^7+q^6 }[/math]
3	[math]\displaystyle{ -2 q^{55}+2 q^{53}+4 q^{52}-5 q^{51}-3 q^{50}+2 q^{49}+10 q^{48}-2 q^{47}-10 q^{46}-2 q^{45}+12 q^{44}+3 q^{43}-13 q^{42}-4 q^{41}+11 q^{40}+5 q^{39}-12 q^{38}-3 q^{37}+9 q^{36}+4 q^{35}-9 q^{34}-q^{33}+4 q^{32}+3 q^{31}-4 q^{30}-q^{29}-q^{28}+2 q^{27}+q^{26}+2 q^{25}-4 q^{24}-2 q^{23}+2 q^{22}+5 q^{21}-2 q^{20}-4 q^{19}-q^{18}+5 q^{17}+q^{16}-2 q^{15}-2 q^{14}+2 q^{13}+q^{12}-q^{10}+q^9 }[/math]
4	[math]\displaystyle{ q^{90}+2 q^{88}-2 q^{87}-4 q^{86}-q^{85}-q^{84}+10 q^{83}+5 q^{82}-9 q^{81}-11 q^{80}-11 q^{79}+19 q^{78}+23 q^{77}-4 q^{76}-20 q^{75}-30 q^{74}+19 q^{73}+36 q^{72}+8 q^{71}-22 q^{70}-42 q^{69}+12 q^{68}+39 q^{67}+15 q^{66}-19 q^{65}-45 q^{64}+9 q^{63}+37 q^{62}+13 q^{61}-14 q^{60}-41 q^{59}+6 q^{58}+34 q^{57}+12 q^{56}-9 q^{55}-36 q^{54}-q^{53}+29 q^{52}+11 q^{51}-q^{50}-28 q^{49}-10 q^{48}+18 q^{47}+9 q^{46}+9 q^{45}-14 q^{44}-13 q^{43}+6 q^{42}+q^{41}+12 q^{40}-q^{39}-8 q^{38}+2 q^{37}-8 q^{36}+5 q^{35}+3 q^{34}+7 q^{32}-9 q^{31}-q^{30}-2 q^{29}-q^{28}+10 q^{27}-2 q^{26}-4 q^{24}-4 q^{23}+6 q^{22}+2 q^{20}-q^{19}-3 q^{18}+2 q^{17}+q^{15}-q^{13}+q^{12} }[/math]
5	[math]\displaystyle{ -2 q^{132}+2 q^{129}+2 q^{128}+4 q^{127}-q^{126}-3 q^{125}-7 q^{124}-5 q^{123}+2 q^{122}+10 q^{121}+14 q^{120}+10 q^{119}-13 q^{118}-27 q^{117}-23 q^{116}+37 q^{114}+48 q^{113}+11 q^{112}-45 q^{111}-63 q^{110}-33 q^{109}+41 q^{108}+83 q^{107}+51 q^{106}-37 q^{105}-89 q^{104}-64 q^{103}+26 q^{102}+92 q^{101}+74 q^{100}-19 q^{99}-90 q^{98}-76 q^{97}+15 q^{96}+86 q^{95}+75 q^{94}-12 q^{93}-85 q^{92}-71 q^{91}+11 q^{90}+81 q^{89}+71 q^{88}-12 q^{87}-79 q^{86}-65 q^{85}+8 q^{84}+72 q^{83}+66 q^{82}-2 q^{81}-66 q^{80}-63 q^{79}-5 q^{78}+53 q^{77}+63 q^{76}+17 q^{75}-44 q^{74}-59 q^{73}-24 q^{72}+24 q^{71}+53 q^{70}+36 q^{69}-13 q^{68}-42 q^{67}-36 q^{66}-6 q^{65}+27 q^{64}+38 q^{63}+16 q^{62}-13 q^{61}-27 q^{60}-24 q^{59}-3 q^{58}+17 q^{57}+20 q^{56}+14 q^{55}-4 q^{54}-15 q^{53}-14 q^{52}-5 q^{51}+2 q^{50}+12 q^{49}+9 q^{48}+3 q^{47}-3 q^{46}-5 q^{45}-8 q^{44}-3 q^{43}+q^{42}+5 q^{41}+4 q^{40}+5 q^{39}-q^{38}-3 q^{37}-5 q^{36}-4 q^{35}+5 q^{33}+3 q^{32}+3 q^{31}-q^{30}-4 q^{29}-3 q^{28}+2 q^{27}+2 q^{25}+2 q^{24}-q^{23}-2 q^{22}+q^{21}+q^{18}-q^{16}+q^{15} }[/math]
6	[math]\displaystyle{ q^{183}+2 q^{181}-2 q^{179}-4 q^{178}-4 q^{177}-3 q^{176}+q^{175}+10 q^{174}+9 q^{173}+8 q^{172}-q^{171}-7 q^{170}-25 q^{169}-23 q^{168}-q^{167}+14 q^{166}+37 q^{165}+38 q^{164}+33 q^{163}-28 q^{162}-65 q^{161}-63 q^{160}-45 q^{159}+27 q^{158}+88 q^{157}+136 q^{156}+42 q^{155}-58 q^{154}-131 q^{153}-156 q^{152}-60 q^{151}+83 q^{150}+227 q^{149}+151 q^{148}+16 q^{147}-138 q^{146}-234 q^{145}-161 q^{144}+27 q^{143}+247 q^{142}+211 q^{141}+87 q^{140}-101 q^{139}-245 q^{138}-204 q^{137}-20 q^{136}+228 q^{135}+215 q^{134}+111 q^{133}-76 q^{132}-230 q^{131}-205 q^{130}-29 q^{129}+216 q^{128}+203 q^{127}+106 q^{126}-73 q^{125}-220 q^{124}-197 q^{123}-25 q^{122}+211 q^{121}+192 q^{120}+103 q^{119}-68 q^{118}-208 q^{117}-190 q^{116}-33 q^{115}+193 q^{114}+178 q^{113}+110 q^{112}-43 q^{111}-180 q^{110}-182 q^{109}-60 q^{108}+145 q^{107}+154 q^{106}+126 q^{105}+5 q^{104}-129 q^{103}-167 q^{102}-100 q^{101}+70 q^{100}+110 q^{99}+135 q^{98}+65 q^{97}-52 q^{96}-127 q^{95}-126 q^{94}-12 q^{93}+37 q^{92}+110 q^{91}+102 q^{90}+30 q^{89}-50 q^{88}-104 q^{87}-62 q^{86}-41 q^{85}+38 q^{84}+80 q^{83}+69 q^{82}+28 q^{81}-32 q^{80}-43 q^{79}-68 q^{78}-29 q^{77}+12 q^{76}+37 q^{75}+49 q^{74}+23 q^{73}+14 q^{72}-29 q^{71}-33 q^{70}-28 q^{69}-16 q^{68}+14 q^{67}+14 q^{66}+32 q^{65}+13 q^{64}+4 q^{63}-11 q^{62}-22 q^{61}-8 q^{60}-17 q^{59}+7 q^{58}+8 q^{57}+15 q^{56}+8 q^{55}+6 q^{53}-16 q^{52}-4 q^{51}-7 q^{50}+q^{49}-q^{48}+2 q^{47}+14 q^{46}-3 q^{45}+4 q^{44}-3 q^{43}-q^{42}-8 q^{41}-5 q^{40}+7 q^{39}-2 q^{38}+5 q^{37}+2 q^{36}+3 q^{35}-4 q^{34}-4 q^{33}+3 q^{32}-3 q^{31}+q^{30}+q^{29}+3 q^{28}-q^{27}-2 q^{26}+2 q^{25}-q^{24}+q^{21}-q^{19}+q^{18} }[/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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