7 2

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7 1.gif

7_1

7 3.gif

7_3

7 2.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 7 2 at Knotilus!

[edit 7 2 Quick Notes]
[edit 7 2 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X5,14,6,1 X7,12,8,13 X11,8,12,9 X13,6,14,7 X9,2,10,3
Gauss code -1, 7, -2, 1, -3, 6, -4, 5, -7, 2, -5, 4, -6, 3
Dowker-Thistlethwaite code 4 10 14 12 2 8 6
Conway Notation [52]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif

Length is 9, width is 4,

Braid index is 4

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

[edit Notes on presentations of 7 2]

Knot 7_2.
A graph, knot 7_2.
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 2"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3,10,4,11 X5,14,6,1 X7,12,8,13 X11,8,12,9 X13,6,14,7 X9,2,10,3
In[5]:= GaussCode[K]
Out[5]= -1, 7, -2, 1, -3, 6, -4, 5, -7, 2, -5, 4, -6, 3
In[6]:= DTCode[K]
Out[6]= 4 10 14 12 2 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [52]
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,-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, 9, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.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 2 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{9, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 8}, {7, 9}, {8, 1}]
In[14]:= Draw[ap]
7 2 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index [math]\displaystyle{ \{3,4\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [math]\displaystyle{ \text{$\$$Failed} }[/math]
Hyperbolic Volume 3.33174
A-Polynomial See Data:7 2/A-polynomial

[edit Notes for 7 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ \textrm{ConcordanceGenus}(\textrm{Knot}(7,2)) }[/math]
Rasmussen s-Invariant -2

[edit Notes for 7 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t+3 t^{-1} -5 }[/math]
Conway polynomial [math]\displaystyle{ 3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 11, -2 }
Jones polynomial [math]\displaystyle{ - q^{-8} + q^{-7} - q^{-6} +2 q^{-5} -2 q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^8+a^6 z^2+a^6+a^4 z^2+a^2 z^2+a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^9 z^5-4 a^9 z^3+3 a^9 z+a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^5-6 a^7 z^3+3 a^7 z+a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+a^5 z^5-a^5 z^3+a^4 z^4+a^3 z^3+a^2 z^2-a^2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}-q^{24}+q^{18}+q^{16}+q^8+q^6+q^2 }[/math]
The G2 invariant [math]\displaystyle{ q^{128}+q^{124}-q^{122}-q^{116}+2 q^{114}-2 q^{112}-q^{110}-q^{106}-2 q^{102}-2 q^{100}-q^{92}-q^{90}+2 q^{88}+q^{78}+3 q^{74}+q^{70}+q^{68}-q^{66}+2 q^{64}-q^{60}+q^{54}-q^{50}-q^{40}+q^{38}+q^{36}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 7_2.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+q^{11}+q^5+q }[/math]
2 [math]\displaystyle{ q^{48}-q^{44}-q^{38}+q^{34}-q^{26}-q^{24}+2 q^{14}+q^{12}+q^8+q^2 }[/math]
3 [math]\displaystyle{ -q^{93}+q^{89}+q^{87}-q^{83}+q^{79}-q^{75}-q^{73}+q^{69}+q^{61}+q^{59}-q^{55}-q^{49}-q^{47}-q^{41}-q^{39}+q^{33}-q^{29}+q^{25}+q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^3 }[/math]
4 [math]\displaystyle{ q^{152}-q^{148}-q^{146}-q^{144}+q^{142}+q^{140}+q^{138}-2 q^{134}+q^{130}+q^{128}+q^{126}-q^{124}-q^{122}-q^{120}+q^{116}-q^{110}-q^{108}+q^{104}+2 q^{102}-q^{98}+q^{94}+2 q^{92}-2 q^{88}-q^{86}+q^{82}-q^{78}+q^{74}+q^{72}-q^{68}+q^{64}-q^{60}-q^{58}-2 q^{56}-2 q^{54}-q^{52}+q^{50}+q^{48}-q^{46}-q^{44}-2 q^{42}+q^{40}+3 q^{38}+2 q^{36}-2 q^{32}+2 q^{28}+q^{26}+q^{24}-q^{22}+q^{18}+q^{16}+2 q^{14}+q^4 }[/math]
5 [math]\displaystyle{ -q^{225}+q^{221}+q^{219}+q^{217}-q^{213}-2 q^{211}-q^{209}+q^{205}+2 q^{203}+q^{201}-q^{199}-2 q^{197}-q^{195}+q^{191}+2 q^{189}+q^{187}-q^{183}-q^{181}-q^{179}+q^{175}+q^{173}+q^{171}-q^{167}-2 q^{165}-2 q^{163}+2 q^{159}+2 q^{157}-2 q^{153}-2 q^{151}-q^{149}+2 q^{147}+4 q^{145}+2 q^{143}-q^{141}-2 q^{139}-2 q^{137}+2 q^{133}+q^{131}-q^{129}-3 q^{127}-2 q^{125}+2 q^{121}+2 q^{119}+q^{117}-q^{115}-q^{113}+q^{111}+2 q^{109}+q^{107}-q^{103}+2 q^{99}+q^{97}-q^{93}-q^{91}-q^{89}-3 q^{77}-3 q^{75}-2 q^{73}+q^{71}+2 q^{69}+3 q^{67}+q^{65}-3 q^{63}-5 q^{61}-3 q^{59}+2 q^{57}+4 q^{55}+3 q^{53}-q^{51}-4 q^{49}-3 q^{47}+q^{45}+4 q^{43}+3 q^{41}-q^{37}-q^{35}+q^{33}+2 q^{31}+2 q^{29}-q^{25}+q^{19}+2 q^{17}+q^{15}+q^5 }[/math]
6 [math]\displaystyle{ q^{312}-q^{308}-q^{306}-q^{304}+2 q^{298}+2 q^{296}+q^{294}-q^{290}-2 q^{288}-3 q^{286}+q^{282}+2 q^{280}+2 q^{278}+q^{276}-3 q^{272}-2 q^{270}-q^{268}+q^{264}+2 q^{262}+2 q^{260}-q^{254}-q^{252}-2 q^{250}-q^{248}+q^{246}+q^{244}+2 q^{242}+2 q^{240}+2 q^{238}-q^{236}-3 q^{234}-3 q^{232}-2 q^{230}+q^{228}+3 q^{226}+4 q^{224}+q^{222}-2 q^{220}-4 q^{218}-5 q^{216}-2 q^{214}+2 q^{212}+5 q^{210}+4 q^{208}+2 q^{206}-q^{204}-4 q^{202}-3 q^{200}+3 q^{196}+4 q^{194}+3 q^{192}-4 q^{188}-5 q^{186}-3 q^{184}+2 q^{180}+3 q^{178}+2 q^{176}-2 q^{174}-4 q^{172}-2 q^{170}+q^{168}+3 q^{166}+3 q^{164}+2 q^{162}-q^{160}-4 q^{158}-2 q^{156}+q^{154}+2 q^{152}+2 q^{150}+q^{148}-q^{146}-2 q^{144}+2 q^{140}+q^{138}-q^{134}-q^{132}-q^{130}+q^{128}+2 q^{126}+2 q^{124}+q^{122}-q^{114}-q^{112}+q^{110}+2 q^{108}+2 q^{106}+q^{104}-q^{102}-4 q^{100}-6 q^{98}-4 q^{96}-q^{94}+2 q^{92}+6 q^{90}+4 q^{88}-q^{86}-6 q^{84}-7 q^{82}-5 q^{80}+6 q^{76}+7 q^{74}+3 q^{72}-3 q^{70}-5 q^{68}-4 q^{66}-q^{64}+4 q^{62}+4 q^{60}+q^{58}-2 q^{56}-2 q^{54}-q^{52}+3 q^{48}+2 q^{46}-q^{42}+q^{38}+q^{36}+3 q^{34}+q^{32}-q^{28}+2 q^{20}+q^{18}+q^{16}+q^6 }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{26}-q^{24}+q^{18}+q^{16}+q^8+q^6+q^2 }[/math]
1,1 [math]\displaystyle{ q^{68}+2 q^{64}-2 q^{62}+2 q^{60}-4 q^{58}-2 q^{54}+2 q^{50}+4 q^{46}-3 q^{44}+2 q^{42}-4 q^{40}+2 q^{38}-4 q^{36}-2 q^{32}-2 q^{30}+2 q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+2 q^{12}+2 q^8+q^4 }[/math]
2,0 [math]\displaystyle{ 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^{38}-2 q^{36}-2 q^{34}-q^{32}+q^{26}+q^{24}+q^{22}+3 q^{20}+2 q^{18}+q^{16}+q^{12}+q^{10}+q^4 }[/math]
3,0 [math]\displaystyle{ -q^{120}-q^{118}-q^{116}+2 q^{112}+2 q^{110}+2 q^{108}-2 q^{96}-2 q^{94}-2 q^{92}+q^{88}+q^{86}+q^{84}+q^{82}+2 q^{80}+3 q^{78}+2 q^{76}-q^{72}-2 q^{70}-q^{68}-3 q^{66}-3 q^{64}-3 q^{62}-q^{60}-q^{56}-q^{54}-q^{52}-q^{48}-q^{40}-q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+2 q^{30}+2 q^{26}+3 q^{24}+3 q^{22}+q^{20}+q^{16}+q^{14}+q^6 }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{54}+q^{50}-q^{46}-q^{44}-2 q^{42}-2 q^{40}-q^{38}-q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{16}+q^{12}+2 q^{10}+q^8+q^4 }[/math]
1,0,0 [math]\displaystyle{ -q^{35}-q^{33}-q^{31}+q^{25}+q^{23}+q^{21}+q^{11}+q^9+q^7+q^3 }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{72}+q^{70}+q^{68}+q^{66}+q^{64}-q^{62}-2 q^{60}-2 q^{58}-2 q^{56}-3 q^{54}-3 q^{52}-q^{50}-q^{48}-q^{46}+q^{42}+q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+q^{32}+q^{30}+q^{28}+q^{22}+q^{20}+q^{18}+q^{16}+2 q^{14}+2 q^{12}+q^{10}+q^6 }[/math]
1,0,0,0 [math]\displaystyle{ -q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{14}+q^{12}+q^{10}+q^8+q^4 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{54}-q^{50}-q^{46}+q^{44}+q^{38}+q^{34}-q^{32}+q^{30}-q^{28}+q^{26}-q^{24}+q^{22}+q^{16}+q^{12}+q^8+q^4 }[/math]
1,0 [math]\displaystyle{ q^{88}+q^{80}-q^{76}-q^{74}-q^{68}-q^{66}-q^{64}-q^{56}+q^{44}+q^{42}+q^{36}+q^{34}+q^{26}+q^{18}+q^{16}+q^{14}+q^6 }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{74}+q^{70}+q^{66}-q^{64}-q^{62}-2 q^{60}-2 q^{58}-2 q^{56}-2 q^{54}-q^{52}-q^{50}+q^{48}+2 q^{44}+q^{42}+2 q^{40}+2 q^{36}+q^{32}+q^{22}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^6 }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}+q^{124}-q^{122}-q^{116}+2 q^{114}-2 q^{112}-q^{110}-q^{106}-2 q^{102}-2 q^{100}-q^{92}-q^{90}+2 q^{88}+q^{78}+3 q^{74}+q^{70}+q^{68}-q^{66}+2 q^{64}-q^{60}+q^{54}-q^{50}-q^{40}+q^{38}+q^{36}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/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 2"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t+3 t^{-1} -5 }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 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]= { 11, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ - q^{-8} + q^{-7} - q^{-6} +2 q^{-5} -2 q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^8+a^6 z^2+a^6+a^4 z^2+a^2 z^2+a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^9 z^5-4 a^9 z^3+3 a^9 z+a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^5-6 a^7 z^3+3 a^7 z+a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+a^5 z^5-a^5 z^3+a^4 z^4+a^3 z^3+a^2 z^2-a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n88,}

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 2"];
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{ 3 t+3 t^{-1} -5 }[/math], [math]\displaystyle{ - q^{-8} + q^{-7} - q^{-6} +2 q^{-5} -2 q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/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]= {K11n88,}

Vassiliev invariants

V2 and V3: (3, -6)
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{ 12 }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 222 }[/math] [math]\displaystyle{ 34 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -1152 }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -176 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2664 }[/math] [math]\displaystyle{ 408 }[/math] [math]\displaystyle{ \frac{60431}{10} }[/math] [math]\displaystyle{ -\frac{826}{15} }[/math] [math]\displaystyle{ \frac{38942}{15} }[/math] [math]\displaystyle{ \frac{497}{6} }[/math] [math]\displaystyle{ \frac{3471}{10} }[/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]-2 is the signature of 7 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-10χ
-1       11
-3      110
-5     1  1
-7    11  0
-9   11   0
-11   1    1
-13 11     0
-15        0
-171       -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/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}\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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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^{-2} - q^{-3} +2 q^{-5} -2 q^{-6} + q^{-7} +3 q^{-8} -4 q^{-9} + q^{-10} +3 q^{-11} -4 q^{-12} +3 q^{-14} -3 q^{-15} +3 q^{-17} -2 q^{-18} - q^{-19} +2 q^{-20} - q^{-21} - q^{-22} + q^{-23} }[/math]
3 [math]\displaystyle{ q^{-3} - q^{-4} +2 q^{-7} - q^{-8} + q^{-11} - q^{-12} + q^{-13} -2 q^{-16} +2 q^{-17} + q^{-18} - q^{-19} -2 q^{-20} + q^{-21} + q^{-22} -2 q^{-24} + q^{-26} + q^{-27} -2 q^{-28} - q^{-29} +2 q^{-30} +2 q^{-31} -2 q^{-32} -2 q^{-33} +2 q^{-34} +2 q^{-35} - q^{-36} -3 q^{-37} + q^{-38} +2 q^{-39} -2 q^{-41} + q^{-43} + q^{-44} - q^{-45} }[/math]
4 [math]\displaystyle{ q^{-4} - q^{-5} +3 q^{-9} -2 q^{-10} - q^{-12} - q^{-13} +5 q^{-14} -2 q^{-15} + q^{-16} -3 q^{-17} -3 q^{-18} +7 q^{-19} +2 q^{-21} -5 q^{-22} -6 q^{-23} +8 q^{-24} +4 q^{-26} -5 q^{-27} -8 q^{-28} +7 q^{-29} +5 q^{-31} -5 q^{-32} -7 q^{-33} +8 q^{-34} - q^{-35} +4 q^{-36} -4 q^{-37} -6 q^{-38} +8 q^{-39} -2 q^{-40} +3 q^{-41} -3 q^{-42} -5 q^{-43} +7 q^{-44} -3 q^{-45} +2 q^{-46} - q^{-47} -3 q^{-48} +6 q^{-49} -4 q^{-50} + q^{-51} - q^{-53} +5 q^{-54} -5 q^{-55} +5 q^{-59} -4 q^{-60} - q^{-61} - q^{-62} +5 q^{-64} -2 q^{-65} - q^{-66} - q^{-67} - q^{-68} +3 q^{-69} - q^{-72} - q^{-73} + q^{-74} }[/math]
5 [math]\displaystyle{ q^{-5} - q^{-6} + q^{-10} +2 q^{-11} -2 q^{-12} - q^{-13} - q^{-15} +2 q^{-16} +4 q^{-17} -2 q^{-18} -2 q^{-19} -2 q^{-20} - q^{-21} +3 q^{-22} +7 q^{-23} - q^{-24} -5 q^{-25} -6 q^{-26} -2 q^{-27} +6 q^{-28} +11 q^{-29} -7 q^{-31} -11 q^{-32} -4 q^{-33} +8 q^{-34} +15 q^{-35} +2 q^{-36} -8 q^{-37} -12 q^{-38} -7 q^{-39} +7 q^{-40} +15 q^{-41} +5 q^{-42} -8 q^{-43} -12 q^{-44} -7 q^{-45} +7 q^{-46} +14 q^{-47} +5 q^{-48} -8 q^{-49} -11 q^{-50} -6 q^{-51} +8 q^{-52} +12 q^{-53} +4 q^{-54} -7 q^{-55} -10 q^{-56} -5 q^{-57} +7 q^{-58} +10 q^{-59} +4 q^{-60} -5 q^{-61} -9 q^{-62} -5 q^{-63} +5 q^{-64} +8 q^{-65} +3 q^{-66} -3 q^{-67} -7 q^{-68} -4 q^{-69} +3 q^{-70} +6 q^{-71} +3 q^{-72} -2 q^{-73} -4 q^{-74} -2 q^{-75} + q^{-76} +3 q^{-77} +2 q^{-78} -2 q^{-79} -2 q^{-80} + q^{-82} + q^{-83} -2 q^{-85} - q^{-86} + q^{-87} +2 q^{-88} + q^{-89} -3 q^{-91} -2 q^{-92} + q^{-93} +2 q^{-94} +2 q^{-95} + q^{-96} -2 q^{-97} -3 q^{-98} + q^{-100} + q^{-101} +2 q^{-102} -2 q^{-104} - q^{-105} + q^{-108} + q^{-109} - q^{-110} }[/math]
6 [math]\displaystyle{ q^{-6} - q^{-7} + q^{-11} +2 q^{-13} -3 q^{-14} - q^{-17} +2 q^{-18} + q^{-19} +4 q^{-20} -5 q^{-21} - q^{-23} -2 q^{-24} +3 q^{-25} +3 q^{-26} +5 q^{-27} -8 q^{-28} - q^{-29} -2 q^{-30} -2 q^{-31} +6 q^{-32} +6 q^{-33} +5 q^{-34} -13 q^{-35} -4 q^{-36} -3 q^{-37} +12 q^{-39} +10 q^{-40} +4 q^{-41} -19 q^{-42} -9 q^{-43} -5 q^{-44} + q^{-45} +17 q^{-46} +15 q^{-47} +6 q^{-48} -23 q^{-49} -12 q^{-50} -8 q^{-51} - q^{-52} +19 q^{-53} +18 q^{-54} +8 q^{-55} -22 q^{-56} -12 q^{-57} -9 q^{-58} -3 q^{-59} +19 q^{-60} +19 q^{-61} +8 q^{-62} -22 q^{-63} -11 q^{-64} -8 q^{-65} -3 q^{-66} +18 q^{-67} +17 q^{-68} +8 q^{-69} -22 q^{-70} -10 q^{-71} -7 q^{-72} -2 q^{-73} +16 q^{-74} +15 q^{-75} +9 q^{-76} -20 q^{-77} -9 q^{-78} -7 q^{-79} -3 q^{-80} +13 q^{-81} +13 q^{-82} +12 q^{-83} -17 q^{-84} -8 q^{-85} -7 q^{-86} -5 q^{-87} +10 q^{-88} +11 q^{-89} +14 q^{-90} -13 q^{-91} -7 q^{-92} -8 q^{-93} -7 q^{-94} +7 q^{-95} +9 q^{-96} +15 q^{-97} -9 q^{-98} -4 q^{-99} -7 q^{-100} -8 q^{-101} +4 q^{-102} +6 q^{-103} +14 q^{-104} -6 q^{-105} - q^{-106} -5 q^{-107} -7 q^{-108} + q^{-109} +2 q^{-110} +11 q^{-111} -5 q^{-112} + q^{-113} -2 q^{-114} -4 q^{-115} +8 q^{-118} -6 q^{-119} + q^{-120} - q^{-122} +7 q^{-125} -6 q^{-126} - q^{-127} - q^{-128} + q^{-131} +7 q^{-132} -4 q^{-133} - q^{-134} -2 q^{-135} - q^{-136} - q^{-137} +6 q^{-139} - q^{-140} - q^{-142} - q^{-143} -2 q^{-144} - q^{-145} +3 q^{-146} + q^{-148} - q^{-151} - q^{-152} + q^{-153} }[/math]
7 [math]\displaystyle{ q^{-7} - q^{-8} + q^{-12} + q^{-15} -2 q^{-16} - q^{-19} +2 q^{-20} + q^{-21} + q^{-22} +2 q^{-23} -4 q^{-24} - q^{-26} -2 q^{-27} +2 q^{-28} +2 q^{-29} +2 q^{-30} +3 q^{-31} -5 q^{-32} - q^{-33} - q^{-34} -2 q^{-35} +3 q^{-36} + q^{-37} + q^{-38} +4 q^{-39} -6 q^{-40} - q^{-41} + q^{-42} + q^{-43} +4 q^{-44} -2 q^{-45} -3 q^{-46} - q^{-47} -7 q^{-48} +2 q^{-49} +7 q^{-50} +6 q^{-51} +7 q^{-52} -4 q^{-53} -10 q^{-54} -9 q^{-55} -11 q^{-56} +5 q^{-57} +12 q^{-58} +13 q^{-59} +13 q^{-60} -4 q^{-61} -13 q^{-62} -15 q^{-63} -17 q^{-64} +2 q^{-65} +14 q^{-66} +17 q^{-67} +17 q^{-68} -2 q^{-69} -11 q^{-70} -16 q^{-71} -20 q^{-72} - q^{-73} +12 q^{-74} +18 q^{-75} +19 q^{-76} - q^{-77} -10 q^{-78} -15 q^{-79} -19 q^{-80} -2 q^{-81} +12 q^{-82} +17 q^{-83} +18 q^{-84} -2 q^{-85} -10 q^{-86} -14 q^{-87} -18 q^{-88} +12 q^{-90} +14 q^{-91} +17 q^{-92} -2 q^{-93} -10 q^{-94} -13 q^{-95} -17 q^{-96} + q^{-97} +10 q^{-98} +10 q^{-99} +17 q^{-100} -8 q^{-102} -10 q^{-103} -16 q^{-104} +6 q^{-106} +6 q^{-107} +17 q^{-108} +3 q^{-109} -4 q^{-110} -7 q^{-111} -16 q^{-112} -3 q^{-113} +2 q^{-114} +3 q^{-115} +16 q^{-116} +7 q^{-117} + q^{-118} -3 q^{-119} -17 q^{-120} -6 q^{-121} -3 q^{-122} - q^{-123} +15 q^{-124} +9 q^{-125} +6 q^{-126} +2 q^{-127} -15 q^{-128} -9 q^{-129} -8 q^{-130} -4 q^{-131} +13 q^{-132} +9 q^{-133} +9 q^{-134} +8 q^{-135} -11 q^{-136} -9 q^{-137} -10 q^{-138} -8 q^{-139} +9 q^{-140} +6 q^{-141} +9 q^{-142} +11 q^{-143} -6 q^{-144} -6 q^{-145} -8 q^{-146} -10 q^{-147} +4 q^{-148} +2 q^{-149} +6 q^{-150} +11 q^{-151} -3 q^{-152} -2 q^{-153} -3 q^{-154} -8 q^{-155} +2 q^{-156} - q^{-157} +2 q^{-158} +8 q^{-159} -3 q^{-160} -5 q^{-163} +3 q^{-164} - q^{-165} +5 q^{-167} -3 q^{-168} - q^{-169} - q^{-170} -4 q^{-171} +4 q^{-172} + q^{-173} +5 q^{-175} -2 q^{-176} - q^{-177} -2 q^{-178} -5 q^{-179} +2 q^{-180} + q^{-181} +4 q^{-183} + q^{-184} + q^{-185} - q^{-186} -5 q^{-187} - q^{-190} +2 q^{-191} + q^{-192} +2 q^{-193} + q^{-194} -2 q^{-195} - q^{-196} - q^{-198} + q^{-201} + q^{-202} - q^{-203} }[/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).

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