10 125

10_124

10_126

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 125's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 125 at Knotilus!
	10_125 is also known as the pretzel knot P(5,-3,2).

Knot presentations

Planar diagram presentation	X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283
Gauss code	-1, 10, -2, 1, -3, 9, -10, 2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4
Dowker-Thistlethwaite code	4 8 14 2 -16 -18 6 -20 -10 -12
Conway Notation	[5,21,2-]

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

[edit Notes on presentations of 10 125]

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["10 125"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 -16 -18 6 -20 -10 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[5,21,2-]

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]=

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

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]=

{ 3, 10, 3 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-6] Hyperbolic Volume 4.61196 A-Polynomial See Data:10 125/A-polynomial

[edit Notes for 10 125's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 125's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-2t^{2}+2t-1+2t^{-1}-2t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 11, 2 }
Jones polynomial	$-q^{4}+q^{3}-q^{2}+2q-1+2q^{-1}-q^{-2}+q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-4a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}-3a^{2}-3a^{-2}+7$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+a^{3}z^{7}+2az^{7}+z^{7}a^{-1}-6a^{2}z^{6}-6z^{6}-6a^{3}z^{5}-11az^{5}-5z^{5}a^{-1}+11a^{2}z^{4}+2z^{4}a^{-2}+13z^{4}+10a^{3}z^{3}+17az^{3}+8z^{3}a^{-1}+z^{3}a^{-3}-8a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-15z^{2}-4a^{3}z-8az-6za^{-1}-za^{-3}+za^{-5}+3a^{2}+3a^{-2}+7$
The A2 invariant	$-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+3+2q^{-2}+q^{-4}-q^{-8}-q^{-10}-q^{-12}$
The G2 invariant	$q^{60}+q^{56}-q^{54}-q^{48}-2q^{44}-q^{40}-2q^{38}-q^{36}-q^{34}-2q^{32}-2q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2q^{14}+q^{12}+2q^{10}+2q^{8}+2q^{6}+2q^{4}+3q^{2}+1+2q^{-2}+2q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-14}+2q^{-16}-q^{-18}+q^{-20}-q^{-24}+q^{-26}-q^{-28}-q^{-30}-q^{-34}-q^{-36}-q^{-38}-2q^{-40}-q^{-44}-q^{-46}-q^{-50}-q^{-56}+q^{-72}$

Further Quantum Invariants

Further quantum knot invariants for 10_125.

A1 Invariants.

Weight	Invariant
1	$-q^{9}+q^{3}+q+q^{-1}+q^{-3}-q^{-9}$
2	$q^{28}-q^{24}-q^{18}-q^{16}+q^{6}+q^{4}+q^{2}+2+q^{-2}+q^{-6}-q^{-12}-q^{-18}-q^{-20}+q^{-24}$
3	$-q^{57}+q^{53}+q^{51}-q^{47}+q^{43}+q^{41}-q^{37}-q^{35}-q^{27}-q^{25}-q^{23}-q^{17}-q^{15}+q^{13}+2q^{11}+2q^{9}+q^{5}+2q^{3}+q+q^{-3}+q^{-5}+q^{-13}-q^{-17}-2q^{-19}-q^{-21}+q^{-23}+2q^{-25}-q^{-27}-2q^{-29}-q^{-31}+2q^{-33}-q^{-37}+q^{-41}+q^{-43}-q^{-45}$
5	$-q^{145}+q^{141}+q^{139}+q^{137}-q^{133}-2q^{131}-q^{129}+q^{125}+2q^{123}+2q^{121}-2q^{117}-2q^{115}-2q^{113}-q^{111}+q^{109}+2q^{107}+2q^{105}+q^{103}-2q^{99}-2q^{97}-q^{95}+q^{91}+2q^{89}+2q^{87}+q^{85}-2q^{81}-2q^{79}-q^{77}+q^{75}+4q^{73}+4q^{71}+2q^{69}-q^{67}-4q^{65}-5q^{63}-2q^{61}+2q^{59}+4q^{57}+3q^{55}-2q^{53}-5q^{51}-6q^{49}-3q^{47}+2q^{45}+4q^{43}+3q^{41}-4q^{37}-4q^{35}-2q^{33}+2q^{31}+4q^{29}+4q^{27}+q^{25}-2q^{23}-3q^{21}+3q^{17}+5q^{15}+3q^{13}-q^{11}-3q^{9}-2q^{7}+2q^{5}+4q^{3}+3q-3q^{-3}-2q^{-5}+q^{-7}+3q^{-9}+2q^{-11}-q^{-13}-q^{-15}+q^{-19}-2q^{-23}-3q^{-25}-q^{-27}+q^{-29}+3q^{-31}+2q^{-33}+q^{-35}-q^{-37}-3q^{-39}-3q^{-41}-q^{-43}+2q^{-45}+5q^{-47}+5q^{-49}+q^{-51}-5q^{-53}-7q^{-55}-5q^{-57}+q^{-59}+6q^{-61}+8q^{-63}+2q^{-65}-5q^{-67}-7q^{-69}-5q^{-71}+q^{-73}+4q^{-75}+3q^{-77}+q^{-79}-q^{-81}-q^{-83}+q^{-89}+q^{-91}+2q^{-93}-q^{-97}-2q^{-99}-q^{-101}+q^{-107}$
6	$q^{204}-q^{200}-q^{198}-q^{196}+2q^{190}+2q^{188}+q^{186}-q^{182}-2q^{180}-3q^{178}-q^{176}+2q^{172}+3q^{170}+3q^{168}+2q^{166}-q^{164}-2q^{162}-3q^{160}-3q^{158}-2q^{156}+2q^{152}+3q^{150}+3q^{148}+2q^{146}-2q^{142}-3q^{140}-3q^{138}-2q^{136}-q^{134}+q^{132}+2q^{130}+4q^{128}+3q^{126}+q^{124}-q^{122}-4q^{120}-5q^{118}-5q^{116}-q^{114}+3q^{112}+6q^{110}+7q^{108}+6q^{106}+q^{104}-5q^{102}-7q^{100}-7q^{98}-2q^{96}+4q^{94}+9q^{92}+9q^{90}+5q^{88}-q^{86}-8q^{84}-10q^{82}-7q^{80}-q^{78}+4q^{76}+8q^{74}+8q^{72}+2q^{70}-4q^{68}-8q^{66}-9q^{64}-7q^{62}+6q^{58}+8q^{56}+5q^{54}-6q^{50}-10q^{48}-7q^{46}-q^{44}+7q^{42}+10q^{40}+9q^{38}+2q^{36}-6q^{34}-9q^{32}-6q^{30}+q^{28}+7q^{26}+11q^{24}+6q^{22}-2q^{20}-7q^{18}-8q^{16}-2q^{14}+4q^{12}+9q^{10}+6q^{8}-q^{6}-5q^{4}-6q^{2}-1+4q^{-2}+7q^{-4}+5q^{-6}-q^{-8}-4q^{-10}-5q^{-12}-2q^{-14}+q^{-16}+4q^{-18}+2q^{-20}-q^{-22}-2q^{-24}-q^{-26}+2q^{-28}+q^{-30}-2q^{-34}-3q^{-36}-q^{-38}+q^{-40}+5q^{-42}+5q^{-44}+3q^{-46}-q^{-48}-4q^{-50}-5q^{-52}-6q^{-54}-3q^{-56}+2q^{-58}+7q^{-60}+9q^{-62}+8q^{-64}+q^{-66}-7q^{-68}-14q^{-70}-13q^{-72}-6q^{-74}+5q^{-76}+16q^{-78}+15q^{-80}+8q^{-82}-5q^{-84}-15q^{-86}-18q^{-88}-9q^{-90}+6q^{-92}+14q^{-94}+14q^{-96}+5q^{-98}-4q^{-100}-12q^{-102}-9q^{-104}+6q^{-108}+8q^{-110}+5q^{-112}+q^{-114}-4q^{-116}-5q^{-118}-2q^{-120}+q^{-122}+2q^{-124}+2q^{-126}+q^{-128}-2q^{-130}-4q^{-132}-2q^{-134}+q^{-138}+2q^{-140}+2q^{-142}+q^{-144}-q^{-146}-q^{-148}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+3+2q^{-2}+q^{-4}-q^{-8}-q^{-10}-q^{-12}$
1,1	$q^{36}+2q^{32}-2q^{30}+2q^{28}-2q^{26}+2q^{24}-2q^{22}-4q^{20}-2q^{18}-6q^{16}-5q^{12}+2q^{10}+6q^{6}+5q^{4}+8q^{2}+8+6q^{-2}+5q^{-4}-2q^{-6}-2q^{-8}-4q^{-10}-5q^{-12}-2q^{-14}-2q^{-16}+2q^{-20}-2q^{-26}+q^{-36}$
2,0	$q^{34}+q^{32}+q^{30}-q^{24}-2q^{22}-3q^{20}-3q^{18}-2q^{16}-2q^{14}-2q^{12}-q^{10}+q^{8}+3q^{6}+5q^{4}+6q^{2}+8+5q^{-2}+4q^{-4}-q^{-8}-3q^{-10}-2q^{-12}-3q^{-14}-3q^{-16}-2q^{-18}-q^{-20}-q^{-24}+q^{-26}+q^{-28}+q^{-30}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{26}+q^{22}-q^{18}-2q^{16}-3q^{14}-4q^{12}-5q^{10}-2q^{8}+5q^{4}+7q^{2}+10+8q^{-2}+5q^{-4}+q^{-6}-2q^{-8}-4q^{-10}-4q^{-12}-3q^{-14}-2q^{-16}-q^{-18}+q^{-30}$
1,0,0	$-q^{15}-q^{13}-2q^{11}-q^{9}-q^{7}+q^{5}+3q^{3}+4q+4q^{-1}+3q^{-3}+q^{-5}-q^{-7}-q^{-9}-2q^{-11}-q^{-13}-q^{-15}$
1,0,1	$q^{44}+2q^{40}+q^{36}+q^{32}+q^{30}+q^{26}-4q^{24}-2q^{22}-9q^{20}-7q^{18}-14q^{16}-11q^{14}-11q^{12}-5q^{10}+5q^{8}+11q^{6}+24q^{4}+22q^{2}+30+18q^{-2}+16q^{-4}+3q^{-6}-6q^{-8}-11q^{-10}-17q^{-12}-13q^{-14}-15q^{-16}-5q^{-18}-4q^{-20}+4q^{-22}+3q^{-24}+5q^{-26}+3q^{-28}-q^{-30}-2q^{-34}+q^{-48}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{32}+q^{30}+2q^{28}+2q^{26}+2q^{24}-3q^{20}-6q^{18}-9q^{16}-11q^{14}-12q^{12}-8q^{10}-3q^{8}+5q^{6}+11q^{4}+18q^{2}+21+19q^{-2}+14q^{-4}+8q^{-6}-6q^{-10}-9q^{-12}-10q^{-14}-10q^{-16}-7q^{-18}-4q^{-20}-2q^{-22}-q^{-24}+q^{-26}+2q^{-28}+q^{-30}+q^{-32}+q^{-34}+q^{-36}$
1,0,0,0	$-q^{18}-q^{16}-2q^{14}-2q^{12}-2q^{10}-q^{8}+q^{6}+3q^{4}+5q^{2}+5+5q^{-2}+3q^{-4}+q^{-6}-q^{-8}-2q^{-10}-2q^{-12}-2q^{-14}-q^{-16}-q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{26}-q^{22}-q^{18}-q^{14}+q^{10}+2q^{6}+q^{4}+3q^{2}+2q^{-2}+q^{-4}+q^{-6}-q^{-14}-q^{-18}-q^{-30}$
1,0	$q^{44}+q^{36}-q^{32}-q^{30}-q^{26}-2q^{24}-2q^{22}-q^{20}-q^{18}-2q^{16}-q^{14}-q^{12}+q^{10}+q^{8}+3q^{6}+2q^{4}+4q^{2}+4+4q^{-2}+2q^{-4}+3q^{-6}+2q^{-8}+q^{-10}-q^{-12}-q^{-14}-q^{-16}-q^{-18}-2q^{-20}-2q^{-22}-q^{-24}-q^{-26}-q^{-28}-q^{-30}+q^{-48}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{34}+q^{30}+q^{26}-q^{24}-q^{22}-3q^{20}-4q^{18}-5q^{16}-6q^{14}-4q^{12}-3q^{10}+q^{8}+3q^{6}+8q^{4}+9q^{2}+12+9q^{-2}+8q^{-4}+4q^{-6}+q^{-8}-2q^{-10}-4q^{-12}-5q^{-14}-5q^{-16}-3q^{-18}-3q^{-20}-q^{-22}-q^{-24}+q^{-42}$

G2 Invariants.

Weight	Invariant
1,0	$q^{60}+q^{56}-q^{54}-q^{48}-2q^{44}-q^{40}-2q^{38}-q^{36}-q^{34}-2q^{32}-2q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2q^{14}+q^{12}+2q^{10}+2q^{8}+2q^{6}+2q^{4}+3q^{2}+1+2q^{-2}+2q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-14}+2q^{-16}-q^{-18}+q^{-20}-q^{-24}+q^{-26}-q^{-28}-q^{-30}-q^{-34}-q^{-36}-q^{-38}-2q^{-40}-q^{-44}-q^{-46}-q^{-50}-q^{-56}+q^{-72}$

.

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["10 125"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+4z^{4}+3z^{2}+1$

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]=

{ 11, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+q^{3}-q^{2}+2q-1+2q^{-1}-q^{-2}+q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-4a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}-3a^{2}-3a^{-2}+7$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+a^{3}z^{7}+2az^{7}+z^{7}a^{-1}-6a^{2}z^{6}-6z^{6}-6a^{3}z^{5}-11az^{5}-5z^{5}a^{-1}+11a^{2}z^{4}+2z^{4}a^{-2}+13z^{4}+10a^{3}z^{3}+17az^{3}+8z^{3}a^{-1}+z^{3}a^{-3}-8a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-15z^{2}-4a^{3}z-8az-6za^{-1}-za^{-3}+za^{-5}+3a^{2}+3a^{-2}+7$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 125"];

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]=

{ $t^{3}-2t^{2}+2t-1+2t^{-1}-2t^{-2}+t^{-3}$, $-q^{4}+q^{3}-q^{2}+2q-1+2q^{-1}-q^{-2}+q^{-3}-q^{-4}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(3, 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 $12$ $0$ $72$ $78$ $2$ $0$ $-32$ $32$ $-64$ $288$ $0$ $936$ $24$ ${\frac {9151}{10}}$ ${\frac {2374}{15}}$ ${\frac {474}{5}}$ ${\frac {75}{2}}$ $-{\frac {289}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 10 125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{11}-q^{10}-q^{9}+q^{8}-q^{6}+q^{4}-q^{3}+q^{2}+2q^{-1}-q^{-2}+2q^{-4}-2q^{-5}+2q^{-7}-2q^{-8}-q^{-9}+2q^{-10}-q^{-11}-q^{-12}+q^{-13}$
3	$-q^{21}+2q^{20}-q^{18}-2q^{17}+3q^{16}+2q^{15}-4q^{14}-3q^{13}+4q^{12}+5q^{11}-5q^{10}-5q^{9}+3q^{8}+6q^{7}-4q^{6}-4q^{5}+2q^{4}+6q^{3}-4q^{2}-3q+2+5q^{-1}-3q^{-2}-2q^{-3}+q^{-4}+4q^{-5}-q^{-6}-2q^{-7}+2q^{-9}-q^{-10}-q^{-11}+q^{-13}-q^{-14}-q^{-15}+q^{-16}+q^{-17}-q^{-18}-2q^{-19}+q^{-20}+2q^{-21}-2q^{-23}+q^{-25}+q^{-26}-q^{-27}$
4	$-q^{33}+q^{32}+q^{31}-q^{29}-2q^{28}+3q^{27}+q^{26}-q^{25}-4q^{24}-q^{23}+7q^{22}+2q^{21}-3q^{20}-8q^{19}-q^{18}+10q^{17}+4q^{16}-2q^{15}-11q^{14}-3q^{13}+11q^{12}+4q^{11}-q^{10}-10q^{9}-3q^{8}+9q^{7}+3q^{6}-8q^{4}-3q^{3}+8q^{2}+3q+1-7q^{-1}-4q^{-2}+6q^{-3}+4q^{-4}+2q^{-5}-5q^{-6}-5q^{-7}+4q^{-8}+3q^{-9}+3q^{-10}-2q^{-11}-5q^{-12}+q^{-13}+q^{-14}+3q^{-15}+q^{-16}-4q^{-17}-q^{-18}-q^{-19}+q^{-20}+3q^{-21}-2q^{-22}-q^{-24}-q^{-25}+3q^{-26}-2q^{-27}+q^{-28}-q^{-30}+3q^{-31}-3q^{-32}+4q^{-36}-2q^{-37}-q^{-38}-q^{-39}-q^{-40}+3q^{-41}-q^{-44}-q^{-45}+q^{-46}$
5	$q^{51}-q^{50}-q^{48}-q^{47}+q^{46}+2q^{45}+q^{44}-q^{43}-q^{42}-2q^{41}+q^{40}+q^{39}+q^{38}+q^{37}+q^{36}-q^{35}-2q^{34}-5q^{33}-q^{32}+3q^{31}+8q^{30}+5q^{29}-4q^{28}-10q^{27}-7q^{26}+q^{25}+10q^{24}+11q^{23}-10q^{21}-10q^{20}-2q^{19}+8q^{18}+11q^{17}+2q^{16}-8q^{15}-9q^{14}-q^{13}+6q^{12}+9q^{11}-7q^{9}-7q^{8}+5q^{6}+8q^{5}-4q^{3}-6q^{2}-2q+2+7q^{-1}+3q^{-2}-q^{-3}-5q^{-4}-4q^{-5}-2q^{-6}+6q^{-7}+5q^{-8}+3q^{-9}-3q^{-10}-6q^{-11}-5q^{-12}+3q^{-13}+6q^{-14}+6q^{-15}-6q^{-17}-7q^{-18}-q^{-19}+4q^{-20}+6q^{-21}+4q^{-22}-3q^{-23}-6q^{-24}-3q^{-25}-q^{-26}+3q^{-27}+5q^{-28}-q^{-30}-2q^{-31}-3q^{-32}-q^{-33}+2q^{-34}+q^{-35}+2q^{-36}+q^{-37}-q^{-38}-q^{-39}-q^{-40}-q^{-41}+q^{-42}+q^{-43}+q^{-44}-q^{-47}-q^{-51}+q^{-53}+q^{-54}+q^{-55}-2q^{-57}-2q^{-58}+q^{-60}+q^{-61}+2q^{-62}-2q^{-64}-q^{-65}+q^{-68}+q^{-69}-q^{-70}$
6	$-q^{71}+2q^{69}+q^{68}-q^{66}-q^{65}-3q^{64}-2q^{63}+4q^{62}+4q^{61}+q^{60}-q^{59}-2q^{58}-6q^{57}-5q^{56}+5q^{55}+9q^{54}+5q^{53}+2q^{52}-4q^{51}-12q^{50}-14q^{49}+2q^{48}+17q^{47}+14q^{46}+11q^{45}-4q^{44}-20q^{43}-29q^{42}-7q^{41}+20q^{40}+24q^{39}+24q^{38}+3q^{37}-19q^{36}-40q^{35}-18q^{34}+13q^{33}+23q^{32}+31q^{31}+11q^{30}-12q^{29}-39q^{28}-20q^{27}+8q^{26}+18q^{25}+28q^{24}+12q^{23}-11q^{22}-36q^{21}-16q^{20}+10q^{19}+18q^{18}+24q^{17}+10q^{16}-13q^{15}-35q^{14}-14q^{13}+11q^{12}+19q^{11}+21q^{10}+9q^{9}-12q^{8}-32q^{7}-12q^{6}+8q^{5}+16q^{4}+18q^{3}+10q^{2}-9q-26-10q^{-1}+5q^{-2}+11q^{-3}+13q^{-4}+11q^{-5}-5q^{-6}-19q^{-7}-7q^{-8}+5q^{-10}+7q^{-11}+12q^{-12}-11q^{-14}-2q^{-15}-4q^{-16}-q^{-17}+9q^{-19}+3q^{-20}-3q^{-21}+5q^{-22}-3q^{-23}-4q^{-24}-8q^{-25}+3q^{-26}+q^{-28}+11q^{-29}+2q^{-30}-q^{-31}-10q^{-32}-3q^{-33}-7q^{-34}-q^{-35}+12q^{-36}+6q^{-37}+5q^{-38}-4q^{-39}-3q^{-40}-11q^{-41}-6q^{-42}+6q^{-43}+3q^{-44}+7q^{-45}+3q^{-46}+3q^{-47}-7q^{-48}-6q^{-49}+q^{-50}-3q^{-51}+2q^{-52}+3q^{-53}+5q^{-54}-q^{-55}-q^{-56}+2q^{-57}-4q^{-58}-q^{-59}-q^{-60}+q^{-61}-q^{-62}+5q^{-64}-2q^{-65}+q^{-66}-q^{-68}-2q^{-69}-2q^{-70}+4q^{-71}-3q^{-72}+q^{-73}+q^{-74}+q^{-75}-q^{-77}+4q^{-78}-4q^{-79}-q^{-80}-q^{-81}+5q^{-85}-q^{-86}-q^{-88}-q^{-89}-2q^{-90}-q^{-91}+3q^{-92}+q^{-94}-q^{-97}-q^{-98}+q^{-99}$

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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