10 4

10_3

10_5

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 4 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X20,8,1,7 X18,10,19,9 X4,13,5,14 X10,18,11,17 X8,20,9,19
Gauss code	1, -4, 3, -8, 5, -1, 6, -10, 7, -9, 2, -3, 8, -5, 4, -2, 9, -7, 10, -6
Dowker-Thistlethwaite code	6 12 14 20 18 16 4 2 10 8
Conway Notation	[613]

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

[edit Notes on presentations of 10 4]

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 4"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X20,8,1,7 X18,10,19,9 X4,13,5,14 X10,18,11,17 X8,20,9,19

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 12 14 20 18 16 4 2 10 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[613]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 5.81713 A-Polynomial See Data:10 4/A-polynomial

[edit Notes for 10 4'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 10 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+7t-7+7t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}-5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, -2 }
Jones polynomial	$q^{5}-q^{4}+2q^{3}-3q^{2}+3q-4+4q^{-1}-3q^{-2}+3q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-z^{4}-2z^{2}-z^{4}a^{-2}-3z^{2}a^{-2}-2a^{-2}+z^{2}a^{-4}+2a^{-4}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+3z^{8}a^{-2}+z^{8}a^{-4}+2z^{8}+3az^{7}-3z^{7}a^{-1}-6z^{7}a^{-3}+3a^{2}z^{6}-17z^{6}a^{-2}-7z^{6}a^{-4}-7z^{6}+3a^{3}z^{5}-10az^{5}-2z^{5}a^{-1}+11z^{5}a^{-3}+3a^{4}z^{4}-6a^{2}z^{4}+29z^{4}a^{-2}+16z^{4}a^{-4}+4z^{4}+2a^{5}z^{3}-4a^{3}z^{3}+8az^{3}+7z^{3}a^{-1}-7z^{3}a^{-3}+a^{6}z^{2}-3a^{4}z^{2}-16z^{2}a^{-2}-13z^{2}a^{-4}+z^{2}-3az-za^{-1}+2za^{-3}+a^{4}+2a^{-2}+2a^{-4}$
The A2 invariant	$q^{16}+q^{10}+q^{6}-q^{-2}-q^{-4}-q^{-6}-q^{-8}+q^{-10}+q^{-12}+q^{-14}+q^{-16}$
The G2 invariant	$q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3q^{72}-2q^{70}+2q^{68}-q^{66}+q^{62}-2q^{60}+3q^{58}-2q^{56}+2q^{48}-q^{46}+2q^{44}-q^{42}+q^{40}+2q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{26}-q^{24}+q^{22}-q^{20}-q^{14}-q^{10}+2q^{4}-4q^{2}+2-3q^{-4}+4q^{-6}-5q^{-8}+2q^{-10}-q^{-12}-q^{-14}+2q^{-16}-3q^{-18}+2q^{-20}-2q^{-22}-q^{-26}-q^{-28}+2q^{-36}-2q^{-38}+q^{-40}+q^{-42}-2q^{-44}+5q^{-46}-5q^{-48}+3q^{-50}+q^{-52}-2q^{-54}+5q^{-56}-4q^{-58}+3q^{-60}+q^{-66}-2q^{-68}+2q^{-70}+q^{-74}$

Further Quantum Invariants

Further quantum knot invariants for 10_4.

A1 Invariants.

Weight	Invariant
1	$q^{11}-q^{9}+q^{7}+q^{3}-q^{-1}-q^{-5}+q^{-7}+q^{-11}$
2	$q^{30}-q^{28}+2q^{24}-q^{22}-q^{16}+q^{6}-q^{4}+1+q^{-6}+2q^{-8}+q^{-14}-q^{-16}-2q^{-18}+q^{-20}-2q^{-24}+q^{-26}+q^{-28}-q^{-30}+q^{-34}$
3	$q^{57}-q^{55}+q^{51}+q^{49}-q^{47}-2q^{45}+q^{43}+q^{41}-q^{39}-2q^{37}+q^{35}+3q^{33}-q^{31}-3q^{29}-q^{27}+4q^{25}+q^{23}-3q^{21}-q^{19}+2q^{17}+3q^{15}-q^{11}-2q^{9}+q^{7}+3q^{5}+2q^{3}-3q-q^{-1}+2q^{-3}+q^{-5}-2q^{-7}-q^{-9}+q^{-11}-q^{-15}-q^{-17}+q^{-19}-q^{-25}+q^{-29}+2q^{-31}+q^{-33}-q^{-37}+q^{-39}+2q^{-41}-3q^{-45}-2q^{-47}+2q^{-49}+2q^{-51}-q^{-53}-3q^{-55}+2q^{-59}+q^{-61}-q^{-63}-q^{-65}+q^{-69}$
4	$q^{92}-q^{90}+q^{86}+q^{82}-3q^{80}+q^{76}+q^{72}-4q^{70}+2q^{68}+3q^{66}-3q^{62}-6q^{60}+4q^{58}+6q^{56}+2q^{54}-5q^{52}-7q^{50}+4q^{48}+8q^{46}+3q^{44}-5q^{42}-8q^{40}+2q^{38}+7q^{36}+5q^{34}-2q^{32}-9q^{30}-3q^{28}+2q^{26}+6q^{24}+5q^{22}-2q^{20}-6q^{18}-5q^{16}+q^{14}+7q^{12}+4q^{10}-3q^{8}-6q^{6}-4q^{4}+4q^{2}+6-3q^{-4}-2q^{-6}+3q^{-8}+4q^{-10}-2q^{-12}-3q^{-14}-q^{-16}+5q^{-18}+6q^{-20}-2q^{-22}-5q^{-24}-3q^{-26}+4q^{-28}+6q^{-30}-4q^{-34}-5q^{-36}-q^{-38}+5q^{-40}+2q^{-42}-3q^{-46}-4q^{-48}+q^{-50}+q^{-52}+2q^{-54}+q^{-56}-3q^{-58}-q^{-60}-2q^{-62}+4q^{-66}+2q^{-68}+3q^{-70}-2q^{-72}-4q^{-74}-q^{-76}+q^{-78}+6q^{-80}+2q^{-82}-3q^{-84}-4q^{-86}-4q^{-88}+3q^{-90}+4q^{-92}+2q^{-94}-q^{-96}-5q^{-98}-q^{-100}+q^{-102}+2q^{-104}+2q^{-106}-q^{-108}-q^{-110}-q^{-112}+q^{-116}$
5	$q^{135}-q^{133}+q^{129}-q^{123}-q^{121}+q^{117}-q^{113}+q^{109}+2q^{107}-q^{105}-4q^{103}-4q^{101}+2q^{99}+7q^{97}+5q^{95}-q^{93}-8q^{91}-6q^{89}+2q^{87}+10q^{85}+6q^{83}-3q^{81}-8q^{79}-4q^{77}+3q^{75}+7q^{73}+2q^{71}-6q^{69}-8q^{67}+q^{65}+10q^{63}+8q^{61}-11q^{57}-13q^{55}-2q^{53}+11q^{51}+13q^{49}+4q^{47}-8q^{45}-12q^{43}-9q^{41}+q^{39}+10q^{37}+11q^{35}+6q^{33}-q^{31}-10q^{29}-13q^{27}-5q^{25}+7q^{23}+14q^{21}+11q^{19}-q^{17}-12q^{15}-12q^{13}-2q^{11}+8q^{9}+11q^{7}+3q^{5}-3q^{3}-4q-q^{-1}+4q^{-3}+4q^{-5}-2q^{-7}-5q^{-9}-4q^{-11}+3q^{-13}+7q^{-15}+6q^{-17}-2q^{-19}-11q^{-21}-10q^{-23}-q^{-25}+9q^{-27}+12q^{-29}+5q^{-31}-9q^{-33}-15q^{-35}-9q^{-37}+5q^{-39}+15q^{-41}+14q^{-43}-14q^{-47}-15q^{-49}-2q^{-51}+11q^{-53}+17q^{-55}+8q^{-57}-6q^{-59}-15q^{-61}-11q^{-63}+q^{-65}+11q^{-67}+12q^{-69}+5q^{-71}-5q^{-73}-10q^{-75}-8q^{-77}+6q^{-81}+6q^{-83}+4q^{-85}-5q^{-89}-5q^{-91}-2q^{-93}-q^{-95}+q^{-97}+3q^{-99}+3q^{-101}+q^{-103}-3q^{-107}-5q^{-109}-4q^{-111}+q^{-113}+5q^{-115}+7q^{-117}+5q^{-119}-q^{-121}-7q^{-123}-8q^{-125}-3q^{-127}+4q^{-129}+8q^{-131}+7q^{-133}+q^{-135}-5q^{-137}-8q^{-139}-5q^{-141}+q^{-143}+5q^{-145}+6q^{-147}+3q^{-149}-2q^{-151}-5q^{-153}-3q^{-155}-q^{-157}+q^{-159}+3q^{-161}+2q^{-163}-q^{-167}-q^{-169}-q^{-171}+q^{-175}$

A2 Invariants.

Weight	Invariant
1,0	$q^{16}+q^{10}+q^{6}-q^{-2}-q^{-4}-q^{-6}-q^{-8}+q^{-10}+q^{-12}+q^{-14}+q^{-16}$
1,1	$q^{44}-2q^{42}+2q^{40}-2q^{38}+5q^{36}-4q^{34}+4q^{32}-4q^{30}+5q^{28}-4q^{26}+4q^{24}-2q^{22}+4q^{20}-4q^{18}-q^{12}-4q^{8}-4q^{4}+2q^{2}-4+10q^{-2}-7q^{-4}+20q^{-6}-10q^{-8}+18q^{-10}-14q^{-12}+12q^{-14}-12q^{-16}+2q^{-18}-5q^{-20}-2q^{-22}+6q^{-24}-10q^{-26}+11q^{-28}-12q^{-30}+12q^{-32}-10q^{-34}+6q^{-36}-4q^{-38}+4q^{-40}+q^{-44}$
2,0	$q^{40}+q^{34}+q^{32}+q^{26}-2q^{22}-2q^{16}-q^{14}+q^{12}-q^{8}-q^{6}+1+2q^{-2}+q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-12}+2q^{-14}+q^{-16}+q^{-18}-q^{-20}-2q^{-22}-2q^{-24}-2q^{-26}-2q^{-28}-2q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-42}+q^{-44}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{36}-q^{34}-q^{32}+2q^{30}-q^{26}+2q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{6}+q^{2}-1-q^{-2}+2q^{-10}+q^{-14}-2q^{-18}-q^{-20}+q^{-26}+2q^{-28}+q^{-32}$
1,0,0	$q^{21}+q^{17}+q^{13}+q^{9}-q^{-3}-q^{-5}-2q^{-7}-q^{-9}-q^{-11}+q^{-13}+q^{-15}+2q^{-17}+q^{-19}+q^{-21}$

B2 Invariants.

Weight	Invariant
0,1	$q^{36}-q^{34}+q^{32}-2q^{30}+2q^{28}-q^{26}+2q^{24}-q^{22}+2q^{20}-q^{18}+q^{16}+q^{14}+2q^{10}-2q^{8}+3q^{6}-4q^{4}+3q^{2}-5+3q^{-2}-4q^{-4}+2q^{-6}-2q^{-8}-q^{-14}+2q^{-16}-2q^{-18}+3q^{-20}-2q^{-22}+2q^{-24}-q^{-26}+2q^{-28}+q^{-32}$
1,0	$q^{58}-q^{54}-q^{52}+2q^{48}+q^{46}-q^{44}-q^{42}+2q^{38}+q^{36}-q^{32}+q^{28}+q^{26}-q^{24}-q^{22}+2q^{18}-q^{14}-q^{12}+q^{8}-q^{4}-q^{2}+1-q^{-2}-q^{-4}+2q^{-8}+q^{-10}-q^{-12}-q^{-14}+2q^{-16}+3q^{-18}+q^{-20}-2q^{-22}-q^{-24}+q^{-26}+q^{-28}-q^{-30}-3q^{-32}-q^{-34}+q^{-36}+q^{-38}-q^{-40}-q^{-42}+q^{-44}+2q^{-46}+q^{-54}$

G2 Invariants.

Weight

Invariant

1,0

$q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3q^{72}-2q^{70}+2q^{68}-q^{66}+q^{62}-2q^{60}+3q^{58}-2q^{56}+2q^{48}-q^{46}+2q^{44}-q^{42}+q^{40}+2q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{26}-q^{24}+q^{22}-q^{20}-q^{14}-q^{10}+2q^{4}-4q^{2}+2-3q^{-4}+4q^{-6}-5q^{-8}+2q^{-10}-q^{-12}-q^{-14}+2q^{-16}-3q^{-18}+2q^{-20}-2q^{-22}-q^{-26}-q^{-28}+2q^{-36}-2q^{-38}+q^{-40}+q^{-42}-2q^{-44}+5q^{-46}-5q^{-48}+3q^{-50}+q^{-52}-2q^{-54}+5q^{-56}-4q^{-58}+3q^{-60}+q^{-66}-2q^{-68}+2q^{-70}+q^{-74}$

.

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 4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+7t-7+7t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}-5z^{2}+1$

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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-z^{4}-2z^{2}-z^{4}a^{-2}-3z^{2}a^{-2}-2a^{-2}+z^{2}a^{-4}+2a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+3z^{8}a^{-2}+z^{8}a^{-4}+2z^{8}+3az^{7}-3z^{7}a^{-1}-6z^{7}a^{-3}+3a^{2}z^{6}-17z^{6}a^{-2}-7z^{6}a^{-4}-7z^{6}+3a^{3}z^{5}-10az^{5}-2z^{5}a^{-1}+11z^{5}a^{-3}+3a^{4}z^{4}-6a^{2}z^{4}+29z^{4}a^{-2}+16z^{4}a^{-4}+4z^{4}+2a^{5}z^{3}-4a^{3}z^{3}+8az^{3}+7z^{3}a^{-1}-7z^{3}a^{-3}+a^{6}z^{2}-3a^{4}z^{2}-16z^{2}a^{-2}-13z^{2}a^{-4}+z^{2}-3az-za^{-1}+2za^{-3}+a^{4}+2a^{-2}+2a^{-4}$

"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 4"];

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]=

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

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:

(-5, -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 $-20$ $-8$ $200$ ${\frac {890}{3}}$ ${\frac {382}{3}}$ $160$ ${\frac {880}{3}}$ ${\frac {64}{3}}$ $120$ $-{\frac {4000}{3}}$ $32$ $-{\frac {17800}{3}}$ $-{\frac {7640}{3}}$ $-{\frac {32095}{6}}$ $1330$ $-{\frac {47702}{9}}$ ${\frac {11867}{18}}$ $-{\frac {7423}{6}}$

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

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 11                     1 1 9                       0 7                 2 1   1 5               1       -1 3             2 2       0 1           2 1         -1 -1         2 2           0 -3       2 3             1 -5     1 1               0 -7   1 2                 1 -9   1                   -1 -11 1                     1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{16}-q^{15}-q^{14}+3q^{13}-q^{12}-4q^{11}+5q^{10}-7q^{8}+6q^{7}+2q^{6}-8q^{5}+6q^{4}+4q^{3}-9q^{2}+5q+4-8q^{-1}+4q^{-2}+3q^{-3}-6q^{-4}+3q^{-5}+3q^{-6}-6q^{-7}+3q^{-8}+2q^{-9}-5q^{-10}+3q^{-11}+q^{-12}-2q^{-13}+q^{-14}$
3	$q^{33}-q^{32}-q^{31}+3q^{29}-3q^{27}-3q^{26}+5q^{25}+3q^{24}-3q^{23}-7q^{22}+4q^{21}+6q^{20}-q^{19}-8q^{18}+2q^{17}+7q^{16}-7q^{14}+q^{13}+6q^{12}-q^{11}-6q^{10}+q^{9}+7q^{8}-3q^{7}-6q^{6}+2q^{5}+8q^{4}-5q^{3}-7q^{2}+5q+9-8q^{-1}-9q^{-2}+10q^{-3}+10q^{-4}-10q^{-5}-12q^{-6}+11q^{-7}+11q^{-8}-7q^{-9}-13q^{-10}+8q^{-11}+9q^{-12}-3q^{-13}-10q^{-14}+3q^{-15}+7q^{-16}-q^{-17}-6q^{-18}+q^{-19}+4q^{-20}-4q^{-22}+q^{-23}+q^{-24}+q^{-25}-2q^{-26}+q^{-27}$
4	$q^{56}-q^{55}-q^{54}+4q^{51}-q^{50}-2q^{49}-2q^{48}-4q^{47}+8q^{46}+2q^{45}-3q^{43}-11q^{42}+8q^{41}+3q^{40}+5q^{39}+q^{38}-16q^{37}+6q^{36}+7q^{34}+6q^{33}-17q^{32}+8q^{31}-4q^{30}+5q^{29}+7q^{28}-19q^{27}+12q^{26}-3q^{25}+4q^{24}+7q^{23}-24q^{22}+13q^{21}+6q^{19}+10q^{18}-30q^{17}+9q^{16}+q^{15}+10q^{14}+16q^{13}-32q^{12}+2q^{11}-q^{10}+13q^{9}+24q^{8}-33q^{7}-4q^{6}-3q^{5}+14q^{4}+30q^{3}-34q^{2}-9q-4+17q^{-1}+36q^{-2}-36q^{-3}-17q^{-4}-6q^{-5}+20q^{-6}+43q^{-7}-33q^{-8}-23q^{-9}-12q^{-10}+19q^{-11}+47q^{-12}-26q^{-13}-22q^{-14}-16q^{-15}+14q^{-16}+41q^{-17}-19q^{-18}-15q^{-19}-14q^{-20}+9q^{-21}+31q^{-22}-16q^{-23}-7q^{-24}-9q^{-25}+5q^{-26}+20q^{-27}-14q^{-28}-5q^{-30}+3q^{-31}+10q^{-32}-11q^{-33}+3q^{-34}-2q^{-35}+2q^{-36}+4q^{-37}-6q^{-38}+2q^{-39}-q^{-40}+q^{-41}+q^{-42}-2q^{-43}+q^{-44}$
5	${\textrm {NotAvailable}}(q)$
6	${\textrm {NotAvailable}}(q)$
7	${\textrm {NotAvailable}}(q)$

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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