9 2

9_1

9_3

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 2 at Knotilus!

[edit 9 2 Quick Notes]

[edit 9 2 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_3,12,4,13 X_5,18,6,1 X_7,16,8,17 X_9,14,10,15 X_13,10,14,11 X_15,8,16,9 X_17,6,18,7 X_11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3
Dowker-Thistlethwaite code	4 12 18 16 14 2 10 8 6
Conway Notation	[72]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 12, width is 5,

Braid index is 5

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

[edit Notes on presentations of 9 2]

Knot 9_2.

A graph, knot 9_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["9 2"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_3,12,4,13 X_5,18,6,1 X_7,16,8,17 X_9,14,10,15 X_13,10,14,11 X_15,8,16,9 X_17,6,18,7 X_11,2,12,3

In[5]:= GaussCode[K]

Out[5]= -1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3

In[6]:= DTCode[K]

Out[6]= 4 12 18 16 14 2 10 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [72]

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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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[{11, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 10}, {9, 11}, {10, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	1
Bridge index	2
Super bridge index	$\{4,7\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[-12][1]
Hyperbolic Volume	3.48666
A-Polynomial	See Data:9 2/A-polynomial

[edit Notes for 9 2's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t-7+4t^{-1}$
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 15, -2 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^{10}+z^2 a^8+a^8+z^2 a^6+z^2 a^4+z^2 a^2+a^2}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-4 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+11 z^4 a^{10}-7 z^2 a^{10}+a^{10}+2 z^7 a^9-10 z^5 a^9+13 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+8 z^4 a^8-6 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z^6 a^6-2 z^4 a^6+z^5 a^5-z^3 a^5+z^4 a^4+z^3 a^3+z^2 a^2-a^2}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{32}-q^{30}+q^{24}+q^{22}+q^8+q^6+q^2}
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{156}+q^{152}-q^{150}+q^{142}-2 q^{140}+q^{138}-q^{136}-q^{134}-2 q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2 q^{102}+q^{98}+q^{94}+q^{92}-2 q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10}}

Further Quantum Invariants

Further quantum knot invariants for 9_2.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{21}+q^{15}+q^5+q}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-q^{56}-q^{50}+q^{46}-q^{30}-q^{28}+q^{18}+q^{16}+q^{14}+q^8+q^2}
3	$-q^{117}+q^{113}+q^{111}-q^{107}+q^{103}-q^{99}-q^{97}+q^{93}+q^{73}+q^{71}-q^{67}-q^{61}-q^{59}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{39}+q^{37}-q^{33}-q^{31}+q^{29}+2q^{27}-q^{23}+q^{21}+2q^{19}+q^{17}+q^{11}+q^{3}$
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{192}-q^{188}-q^{186}-q^{184}+q^{182}+q^{180}+q^{178}-2 q^{174}+q^{170}+q^{168}+q^{166}-q^{164}-q^{162}-q^{160}+q^{156}-q^{134}-q^{132}+q^{128}+2 q^{126}-q^{122}+q^{118}+2 q^{116}-2 q^{112}-q^{110}+q^{106}-2 q^{102}-q^{100}+q^{96}+q^{94}+q^{90}+q^{88}+q^{86}-q^{82}-q^{80}+q^{76}-q^{72}-q^{70}-q^{68}-q^{66}+q^{62}-q^{60}-2 q^{58}-2 q^{56}+2 q^{52}+q^{50}-2 q^{46}+q^{44}+2 q^{42}+q^{40}-q^{38}-2 q^{36}+q^{34}+2 q^{32}+q^{30}-q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{14}+q^4}
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{285}+q^{281}+q^{279}+q^{277}-q^{273}-2 q^{271}-q^{269}+q^{265}+2 q^{263}+q^{261}-q^{259}-2 q^{257}-q^{255}+q^{251}+2 q^{249}+q^{247}-q^{243}-q^{241}-q^{239}+q^{235}+q^{213}+q^{211}-q^{207}-2 q^{205}-2 q^{203}+2 q^{199}+2 q^{197}-2 q^{193}-2 q^{191}-q^{189}+2 q^{187}+4 q^{185}+2 q^{183}-q^{181}-2 q^{179}-2 q^{177}+2 q^{173}+2 q^{171}-2 q^{167}-2 q^{165}-q^{163}+q^{159}+q^{157}-q^{155}-q^{153}-q^{151}+q^{147}+2 q^{145}+q^{143}-q^{139}-q^{137}+q^{135}+2 q^{133}+q^{131}-q^{127}+q^{123}-q^{119}-2 q^{117}-q^{115}+q^{113}+2 q^{111}+q^{109}-q^{105}-q^{103}-2 q^{101}+q^{97}+2 q^{95}+2 q^{93}+q^{91}-2 q^{89}-3 q^{87}-2 q^{85}+q^{81}+q^{79}-q^{77}-2 q^{75}-3 q^{73}-q^{71}+q^{69}+q^{67}-q^{63}+q^{57}-q^{53}+2 q^{49}+3 q^{47}+q^{45}-q^{43}-q^{41}-q^{39}+q^{37}+2 q^{35}+q^{33}+q^{23}+q^{21}+q^{19}+q^{17}+q^5}

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{32}-q^{30}+q^{24}+q^{22}+q^8+q^6+q^2}
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}+2 q^{80}-2 q^{78}+2 q^{76}-4 q^{74}+2 q^{72}-4 q^{70}+4 q^{62}-q^{60}+4 q^{58}-4 q^{56}+2 q^{54}-4 q^{52}+2 q^{50}-2 q^{48}+2 q^{46}-2 q^{42}-2 q^{40}-2 q^{38}-2 q^{36}+2 q^{30}+2 q^{28}+2 q^{26}+2 q^{24}+q^{20}+2 q^{16}+2 q^{12}+2 q^8+q^4}
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{82}+q^{80}+q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}+q^{64}+q^{62}+q^{60}-q^{44}-2 q^{42}-2 q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+2 q^{26}+q^{24}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^4}

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{62}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{46}-q^{42}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{16}+q^{12}+2 q^{10}+q^8+q^4}
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{43}-q^{41}-q^{39}+q^{33}+q^{31}+q^{29}+q^{11}+q^9+q^7+q^3}

B2 Invariants.

Weight	Invariant
0,1	$-q^{66}-q^{62}-q^{58}+q^{56}-q^{54}+q^{52}+q^{50}+q^{46}+q^{42}-2q^{40}+q^{38}-q^{36}+q^{34}-q^{32}+q^{30}+q^{16}+q^{12}+q^{8}+q^{4}$
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{108}+q^{100}-q^{96}-q^{94}-q^{88}-q^{86}+q^{82}-q^{76}-q^{68}-q^{66}+q^{56}+q^{54}+q^{48}+q^{46}+q^{26}+q^{18}+q^{16}+q^{14}+q^6}

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{156}+q^{152}-q^{150}+q^{142}-2 q^{140}+q^{138}-q^{136}-q^{134}-2 q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2 q^{102}+q^{98}+q^{94}+q^{92}-2 q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10}}

.

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.

In[3]:= K = Knot["9 2"];

In[4]:= Alexander[K][t]

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

Out[4]= $4t-7+4t^{-1}$

In[5]:= Conway[K][z]

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 z^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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 15, -2 }

In[8]:= Jones[K][q]

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

Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^{10}+z^2 a^8+a^8+z^2 a^6+z^2 a^4+z^2 a^2+a^2}

In[10]:= Kauffman[K][a, z]

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

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-4 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+11 z^4 a^{10}-7 z^2 a^{10}+a^{10}+2 z^7 a^9-10 z^5 a^9+13 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+8 z^4 a^8-6 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z^6 a^6-2 z^4 a^6+z^5 a^5-z^3 a^5+z^4 a^4+z^3 a^3+z^2 a^2-a^2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {7_4,}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {K11n13,}

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.

In[3]:= K = Knot["9 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]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 t-7+4 t^{-1} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} } }

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]= {7_4,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

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

Out[6]= {K11n13,}

Vassiliev invariants

V₂ and V₃:

(4, -10)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -80}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128}

{\frac {1304}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{184}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1280}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{8000}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1280}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -400}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2048}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3200}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{20864}{3}}

{\frac {2944}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{249422}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{856}{5}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{315368}{45}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2482}{9}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{13742}{15}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed

j

, alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -2 is the signature of 9 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-1

1

-3

1

0

-5

1

-7

1

0

-9

1

0

-11

1

0

-13

1

0

-15

1

-17

1

0

-19

0

-21

1

-1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-9}	${\mathbb {Z} }$
$r=-8$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-7}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} - q^{-3} +2 q^{-5} -2 q^{-6} +3 q^{-8} -2 q^{-9} +2 q^{-11} -2 q^{-12} +2 q^{-14} -3 q^{-15} +3 q^{-17} -3 q^{-18} +3 q^{-20} -3 q^{-21} +3 q^{-23} -2 q^{-24} - q^{-25} +2 q^{-26} - q^{-27} - q^{-28} + q^{-29} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} - q^{-4} +2 q^{-7} -2 q^{-8} + q^{-10} +3 q^{-11} -3 q^{-12} -2 q^{-13} +2 q^{-14} +5 q^{-15} -4 q^{-16} -4 q^{-17} +2 q^{-18} +6 q^{-19} -3 q^{-20} -6 q^{-21} +2 q^{-22} +6 q^{-23} -2 q^{-24} -5 q^{-25} +2 q^{-26} +5 q^{-27} -3 q^{-28} -4 q^{-29} +2 q^{-30} +4 q^{-31} -3 q^{-32} -3 q^{-33} +2 q^{-34} +3 q^{-35} -2 q^{-36} -2 q^{-37} +2 q^{-38} +2 q^{-39} -2 q^{-40} -2 q^{-41} +2 q^{-42} +2 q^{-43} -2 q^{-44} -2 q^{-45} +2 q^{-46} +2 q^{-47} - q^{-48} -3 q^{-49} + q^{-50} +2 q^{-51} -2 q^{-53} + q^{-55} + q^{-56} - q^{-57} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-4} - q^{-5} +2 q^{-9} -2 q^{-10} + q^{-11} +2 q^{-14} -4 q^{-15} +2 q^{-16} + q^{-17} + q^{-18} + q^{-19} -7 q^{-20} +3 q^{-21} +3 q^{-22} +2 q^{-23} -10 q^{-25} +5 q^{-26} +4 q^{-27} +3 q^{-28} -2 q^{-29} -12 q^{-30} +5 q^{-31} +5 q^{-32} +5 q^{-33} -3 q^{-34} -13 q^{-35} +5 q^{-36} +5 q^{-37} +5 q^{-38} -2 q^{-39} -12 q^{-40} +4 q^{-41} +4 q^{-42} +5 q^{-43} - q^{-44} -11 q^{-45} +4 q^{-46} +4 q^{-47} +4 q^{-48} -11 q^{-50} +3 q^{-51} +3 q^{-52} +3 q^{-53} +2 q^{-54} -10 q^{-55} +2 q^{-56} +2 q^{-57} +2 q^{-58} +4 q^{-59} -8 q^{-60} + q^{-61} + q^{-62} + q^{-63} +5 q^{-64} -6 q^{-65} +5 q^{-69} -5 q^{-70} +5 q^{-74} -5 q^{-75} +5 q^{-79} -4 q^{-80} - q^{-81} - q^{-82} +5 q^{-84} -2 q^{-85} - q^{-86} - q^{-87} - q^{-88} +3 q^{-89} - q^{-92} - q^{-93} + q^{-94} }

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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9 2

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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