7 1

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6 3.gif

6_3

7 2.gif

7_2

7 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 7 1 at Knotilus!

[edit 7 1 Quick Notes]

7_1 should perhaps be called "The Septafoil Knot", following the trefoil knot and the cinquefoil knot. See also T(7,2).

[edit 7 1 Further Notes and Views]


Interlaced form of 7/2 star polygon or "septagram"
Decorative interlaced form of 7/2 star polygon or "septagram"
3D depiction
Heptagram of intersecting circles.

Knot presentations

Planar diagram presentation X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7
Gauss code -1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4
Dowker-Thistlethwaite code 8 10 12 14 2 4 6
Conway Notation [7]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif

Length is 7, width is 2,

Braid index is 2

7 1 ML.gif 7 1 AP.gif
[{9, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]

[edit Notes on presentations of 7 1]

Knot 7_1.
A graph, knot 7_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.org/wiki/KnotTheory.

In[3]:= K = Knot["7 1"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7
In[5]:= GaussCode[K]
Out[5]= -1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4
In[6]:= DTCode[K]
Out[6]= 8 10 12 14 2 4 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [7]
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}(2,\{-1,-1,-1,-1,-1,-1,-1\}) }[/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]= { 2, 7, 2 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
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."
7 1 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{9, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]
In[14]:= Draw[ap]
7 1 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [math]\displaystyle{ \text{$\$$Failed} }[/math]
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:7 1/A-polynomial

[edit Notes for 7 1'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{ \textrm{ConcordanceGenus}(\textrm{Knot}(7,1)) }[/math]
Rasmussen s-Invariant -6

[edit Notes for 7 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1 }[/math]
Conway polynomial [math]\displaystyle{ z^6+5 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 7, -6 }
Jones polynomial [math]\displaystyle{ - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^8 \left(-z^4\right)-4 a^8 z^2-3 a^8+a^6 z^6+6 a^6 z^4+10 a^6 z^2+4 a^6 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^{13} z+a^{12} z^2+a^{11} z^3-a^{11} z+a^{10} z^4-2 a^{10} z^2+a^9 z^5-3 a^9 z^3+a^9 z+a^8 z^6-5 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^5-4 a^7 z^3+3 a^7 z+a^6 z^6-6 a^6 z^4+10 a^6 z^2-4 a^6 }[/math]
The A2 invariant [math]\displaystyle{ -q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10} }[/math]
The G2 invariant [math]\displaystyle{ q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2 q^{84}+2 q^{80}+q^{78}+q^{72}+3 q^{70}+2 q^{68}+q^{64}+2 q^{62}+2 q^{60}+q^{58}+q^{54}+q^{52}+q^{50} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 7_1.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{21}+q^9+q^7+q^5 }[/math]
2 [math]\displaystyle{ q^{56}-q^{44}-q^{42}-q^{40}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10} }[/math]
3 [math]\displaystyle{ -q^{105}+q^{93}+q^{91}+q^{89}-q^{67}-q^{65}-q^{63}-q^{61}-q^{59}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15} }[/math]
4 [math]\displaystyle{ q^{168}-q^{156}-q^{154}-q^{152}+q^{130}+q^{128}+q^{126}+q^{124}+q^{122}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20} }[/math]
5 [math]\displaystyle{ -q^{245}+q^{233}+q^{231}+q^{229}-q^{207}-q^{205}-q^{203}-q^{201}-q^{199}+q^{167}+q^{165}+q^{163}+q^{161}+q^{159}+q^{157}+q^{155}-q^{113}-q^{111}-q^{109}-q^{107}-q^{105}-q^{103}-q^{101}-q^{99}-q^{97}+q^{45}+q^{43}+q^{41}+q^{39}+q^{37}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25} }[/math]
6 [math]\displaystyle{ q^{336}-q^{324}-q^{322}-q^{320}+q^{298}+q^{296}+q^{294}+q^{292}+q^{290}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}+q^{204}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}-q^{136}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30} }[/math]
8 [math]\displaystyle{ q^{560}-q^{548}-q^{546}-q^{544}+q^{522}+q^{520}+q^{518}+q^{516}+q^{514}-q^{482}-q^{480}-q^{478}-q^{476}-q^{474}-q^{472}-q^{470}+q^{428}+q^{426}+q^{424}+q^{422}+q^{420}+q^{418}+q^{416}+q^{414}+q^{412}-q^{360}-q^{358}-q^{356}-q^{354}-q^{352}-q^{350}-q^{348}-q^{346}-q^{344}-q^{342}-q^{340}+q^{278}+q^{276}+q^{274}+q^{272}+q^{270}+q^{268}+q^{266}+q^{264}+q^{262}+q^{260}+q^{258}+q^{256}+q^{254}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}-q^{172}-q^{170}-q^{168}-q^{166}-q^{164}-q^{162}-q^{160}-q^{158}-q^{156}-q^{154}+q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10} }[/math]
1,1 [math]\displaystyle{ q^{84}-2 q^{48}-2 q^{46}-4 q^{44}-4 q^{42}-4 q^{40}-2 q^{38}-q^{36}+2 q^{34}+4 q^{32}+4 q^{30}+5 q^{28}+4 q^{26}+4 q^{24}+2 q^{22}+q^{20} }[/math]
2,0 [math]\displaystyle{ q^{74}+q^{72}+2 q^{70}+q^{68}+q^{66}-q^{62}-2 q^{60}-3 q^{58}-3 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{36}+q^{34}+2 q^{32}+2 q^{30}+3 q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{20} }[/math]
3,0 [math]\displaystyle{ -q^{132}-q^{130}-2 q^{128}-2 q^{126}-2 q^{124}-q^{122}+2 q^{118}+4 q^{116}+4 q^{114}+5 q^{112}+4 q^{110}+4 q^{108}+2 q^{106}+q^{104}-q^{94}-2 q^{92}-3 q^{90}-4 q^{88}-5 q^{86}-5 q^{84}-5 q^{82}-4 q^{80}-3 q^{78}-2 q^{76}-q^{74}+q^{54}+q^{52}+2 q^{50}+2 q^{48}+3 q^{46}+3 q^{44}+4 q^{42}+3 q^{40}+3 q^{38}+2 q^{36}+2 q^{34}+q^{32}+q^{30} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{70}-q^{48}-2 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+3 q^{30}+4 q^{28}+3 q^{26}+3 q^{24}+q^{22}+q^{20} }[/math]
1,0,0 [math]\displaystyle{ -q^{39}-q^{37}-2 q^{35}-q^{33}-q^{31}+q^{27}+q^{25}+2 q^{23}+2 q^{21}+2 q^{19}+q^{17}+q^{15} }[/math]
1,0,1 [math]\displaystyle{ q^{112}+q^{78}+q^{76}+3 q^{74}+3 q^{72}+4 q^{70}+3 q^{68}+q^{66}-3 q^{64}-7 q^{62}-10 q^{60}-14 q^{58}-14 q^{56}-13 q^{54}-8 q^{52}-3 q^{50}+3 q^{48}+8 q^{46}+11 q^{44}+12 q^{42}+11 q^{40}+10 q^{38}+7 q^{36}+5 q^{34}+2 q^{32}+q^{30} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{88}+q^{86}+q^{84}+q^{82}+q^{80}-q^{66}-2 q^{64}-4 q^{62}-5 q^{60}-7 q^{58}-7 q^{56}-6 q^{54}-4 q^{52}-q^{50}+2 q^{48}+5 q^{46}+6 q^{44}+8 q^{42}+6 q^{40}+6 q^{38}+4 q^{36}+3 q^{34}+q^{32}+q^{30} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{48}-q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-q^{38}+q^{34}+2 q^{32}+2 q^{30}+3 q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{70}-q^{48}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+q^{30}+2 q^{28}+q^{26}+q^{24}+q^{22}+q^{20} }[/math]
1,0 [math]\displaystyle{ q^{112}-q^{78}-q^{76}-q^{74}-q^{72}-2 q^{70}-q^{68}-q^{66}-q^{64}-q^{62}+q^{54}+q^{50}+q^{48}+2 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{36}+q^{34}+q^{30} }[/math]

D4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{168}-q^{136}-q^{134}-3 q^{132}-3 q^{130}-4 q^{128}-4 q^{126}-q^{124}+3 q^{120}+8 q^{118}+11 q^{116}+12 q^{114}+15 q^{112}+12 q^{110}+12 q^{108}+9 q^{106}+6 q^{104}+2 q^{102}-6 q^{98}-10 q^{96}-16 q^{94}-22 q^{92}-27 q^{90}-32 q^{88}-33 q^{86}-32 q^{84}-27 q^{82}-19 q^{80}-8 q^{78}+12 q^{74}+19 q^{72}+24 q^{70}+26 q^{68}+27 q^{66}+22 q^{64}+20 q^{62}+14 q^{60}+10 q^{58}+6 q^{56}+4 q^{54}+q^{52}+q^{50} }[/math]
1,0,0,0 [math]\displaystyle{ q^{98}-q^{66}-q^{64}-3 q^{62}-3 q^{60}-4 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+2 q^{48}+3 q^{46}+4 q^{44}+5 q^{42}+4 q^{40}+4 q^{38}+3 q^{36}+2 q^{34}+q^{32}+q^{30} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2 q^{84}+2 q^{80}+q^{78}+q^{72}+3 q^{70}+2 q^{68}+q^{64}+2 q^{62}+2 q^{60}+q^{58}+q^{54}+q^{52}+q^{50} }[/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["7 1"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1 }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+5 z^4+6 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{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 7, -6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + q^{-3} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^8 \left(-z^4\right)-4 a^8 z^2-3 a^8+a^6 z^6+6 a^6 z^4+10 a^6 z^2+4 a^6 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^{13} z+a^{12} z^2+a^{11} z^3-a^{11} z+a^{10} z^4-2 a^{10} z^2+a^9 z^5-3 a^9 z^3+a^9 z+a^8 z^6-5 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^5-4 a^7 z^3+3 a^7 z+a^6 z^6-6 a^6 z^4+10 a^6 z^2-4 a^6 }[/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["7 1"];
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^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1 }[/math], [math]\displaystyle{ - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + 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: (6, -14)
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{ 24 }[/math] [math]\displaystyle{ -112 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 684 }[/math] [math]\displaystyle{ 100 }[/math] [math]\displaystyle{ -2688 }[/math] [math]\displaystyle{ -\frac{13888}{3} }[/math] [math]\displaystyle{ -\frac{2464}{3} }[/math] [math]\displaystyle{ -560 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 6272 }[/math] [math]\displaystyle{ 16416 }[/math] [math]\displaystyle{ 2400 }[/math] [math]\displaystyle{ \frac{160231}{5} }[/math] [math]\displaystyle{ \frac{21548}{15} }[/math] [math]\displaystyle{ \frac{58148}{5} }[/math] [math]\displaystyle{ 163 }[/math] [math]\displaystyle{ \frac{7351}{5} }[/math]

V2,1 through V6,9 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 7 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-10χ
-5       11
-7       11
-9     1  1
-11        0
-13   11   0
-15        0
-17 11     0
-19        0
-211       -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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^{-6} + q^{-9} - q^{-11} + q^{-12} - q^{-14} + q^{-15} - q^{-17} + q^{-18} - q^{-20} - q^{-23} + q^{-24} - q^{-26} + q^{-27} }[/math]
3 [math]\displaystyle{ q^{-9} + q^{-13} - q^{-16} + q^{-17} - q^{-20} + q^{-21} - q^{-24} + q^{-25} - q^{-28} + q^{-29} - q^{-31} - q^{-32} + q^{-33} - q^{-35} + q^{-37} - q^{-39} + q^{-41} - q^{-43} + q^{-45} + q^{-46} - q^{-47} + q^{-50} - q^{-51} }[/math]
4 [math]\displaystyle{ q^{-12} + q^{-17} - q^{-21} + q^{-22} - q^{-26} + q^{-27} - q^{-31} + q^{-32} - q^{-36} + q^{-37} -2 q^{-41} + q^{-42} -2 q^{-46} + q^{-47} + q^{-48} -2 q^{-51} + q^{-52} + q^{-53} -2 q^{-56} + q^{-57} + q^{-58} -2 q^{-61} + q^{-62} +2 q^{-63} -2 q^{-66} + q^{-67} + q^{-68} -2 q^{-71} + q^{-72} + q^{-73} -2 q^{-76} + q^{-77} - q^{-81} + q^{-82} }[/math]
5 [math]\displaystyle{ q^{-15} + q^{-21} - q^{-26} + q^{-27} - q^{-32} + q^{-33} - q^{-38} + q^{-39} - q^{-44} + q^{-45} - q^{-50} - q^{-56} + q^{-60} - q^{-62} + q^{-66} - q^{-68} + q^{-72} - q^{-74} + q^{-78} + q^{-84} - q^{-87} + q^{-90} - q^{-93} + q^{-96} - q^{-99} - q^{-105} + q^{-107} - q^{-111} + q^{-113} + q^{-119} - q^{-120} }[/math]
6 [math]\displaystyle{ q^{-18} + q^{-25} - q^{-31} + q^{-32} - q^{-38} + q^{-39} - q^{-45} + q^{-46} - q^{-52} + q^{-53} - q^{-59} + q^{-60} - q^{-61} - q^{-66} + q^{-67} - q^{-68} + q^{-72} - q^{-73} + q^{-74} - q^{-75} + q^{-79} - q^{-80} + q^{-81} - q^{-82} + q^{-86} - q^{-87} + q^{-88} - q^{-89} + q^{-93} - q^{-94} + q^{-95} - q^{-96} + q^{-97} + q^{-100} - q^{-101} + q^{-102} - q^{-103} + q^{-104} - q^{-106} + q^{-107} - q^{-108} + q^{-109} - q^{-110} + q^{-111} - q^{-113} + q^{-114} - q^{-115} + q^{-116} - q^{-117} + q^{-118} - q^{-120} + q^{-121} - q^{-122} + q^{-123} - q^{-124} + q^{-125} - q^{-126} - q^{-127} + q^{-128} - q^{-129} + q^{-130} - q^{-131} + q^{-132} - q^{-134} + q^{-135} - q^{-136} + q^{-137} - q^{-138} + q^{-139} - q^{-141} + q^{-142} - q^{-143} + q^{-144} - q^{-145} + q^{-146} + q^{-149} - q^{-150} + q^{-151} - q^{-152} + q^{-156} - q^{-157} + q^{-158} - q^{-159} - q^{-164} + q^{-165} }[/math]
7 [math]\displaystyle{ q^{-21} + q^{-29} - q^{-36} + q^{-37} - q^{-44} + q^{-45} - q^{-52} + q^{-53} - q^{-60} + q^{-61} - q^{-68} + q^{-69} - q^{-71} - q^{-76} + q^{-77} - q^{-79} + q^{-85} - q^{-87} + q^{-93} - q^{-95} + q^{-101} - q^{-103} + q^{-109} - q^{-111} + q^{-114} + q^{-117} - q^{-119} + q^{-122} - q^{-127} + q^{-130} - q^{-135} + q^{-138} - q^{-143} + q^{-146} - q^{-150} - q^{-151} + q^{-154} - q^{-158} + q^{-162} - q^{-166} + q^{-170} - q^{-174} + q^{-178} + q^{-179} - q^{-182} + q^{-187} - q^{-190} + q^{-195} - q^{-198} - q^{-201} + q^{-203} - q^{-209} + q^{-211} + q^{-216} - q^{-217} }[/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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