9 35

9_34

9_36

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 35's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 35 at Knotilus!
	9_35 is also known as the pretzel knot P(3,3,3).

Three-fold symmetric decorative knot

Another three-fold symmetric decorative form

Knot presentations

Planar diagram presentation	X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5
Gauss code	-1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4
Dowker-Thistlethwaite code	8 12 16 14 18 4 2 6 10
Conway Notation	[3,3,3]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 14, width is 5, Braid index is 5		[{8, 4}, {3, 7}, {4, 2}, {1, 3}, {9, 12}, {11, 8}, {12, 10}, {6, 9}, {7, 5}, {2, 6}, {5, 11}, {10, 1}]

[edit Notes on presentations of 9 35]

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 35"];

In[4]:=

PD[K]

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

Out[4]=

X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

8 12 16 14 18 4 2 6 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,3,3]

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]=

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

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, 14, 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[{8, 4}, {3, 7}, {4, 2}, {1, 3}, {9, 12}, {11, 8}, {12, 10}, {6, 9}, {7, 5}, {2, 6}, {5, 11}, {10, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 1 Bridge index 3 Super bridge index 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 \{4,6\}} Nakanishi index 2 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 7.94058 A-Polynomial See Data:9 35/A-polynomial

[edit Notes for 9 35'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 35's four dimensional invariants]

Polynomial invariants

Alexander 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 7 t-13+7 t^{-1} }
Conway polynomial	$7z^{2}+1$
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 \{3,t+1\}}
Determinant and Signature	{ 27, -2 }
Jones polynomial	$q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+5q^{-5}-3q^{-6}+4q^{-7}-3q^{-8}+q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+z^{2}a^{8}-a^{8}+3z^{2}a^{6}+3a^{6}+2z^{2}a^{4}+z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-6z^{5}a^{11}+12z^{3}a^{11}-8za^{11}+z^{8}a^{10}-4z^{6}a^{10}+3z^{4}a^{10}+z^{2}a^{10}+a^{10}+4z^{7}a^{9}-18z^{5}a^{9}+23z^{3}a^{9}-9za^{9}+z^{8}a^{8}+z^{6}a^{8}-15z^{4}a^{8}+16z^{2}a^{8}-a^{8}+3z^{7}a^{7}-8z^{5}a^{7}+3z^{3}a^{7}-za^{7}+5z^{6}a^{6}-15z^{4}a^{6}+12z^{2}a^{6}-3a^{6}+4z^{5}a^{5}-6z^{3}a^{5}+3z^{4}a^{4}-2z^{2}a^{4}+2z^{3}a^{3}+z^{2}a^{2}$
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}-2 q^{26}-q^{24}+q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{14}-q^{10}+q^8-q^4+q^2}
The G2 invariant	$q^{156}+3q^{152}-3q^{150}+2q^{148}-q^{146}-2q^{144}+7q^{142}-9q^{140}+6q^{138}-2q^{136}-2q^{134}+8q^{132}-12q^{130}+5q^{128}-2q^{126}-3q^{124}+3q^{122}-10q^{120}-2q^{118}+4q^{116}-2q^{114}+q^{112}-8q^{110}-2q^{108}+6q^{106}-6q^{104}+5q^{102}-11q^{100}+6q^{98}+8q^{96}-3q^{94}+8q^{92}-10q^{90}+12q^{88}+4q^{86}-5q^{84}+7q^{82}-5q^{80}+5q^{78}+7q^{76}-3q^{74}+2q^{72}+q^{70}-2q^{68}+4q^{66}-6q^{64}+3q^{62}-2q^{60}-2q^{58}+4q^{56}-4q^{54}+3q^{52}-2q^{50}+q^{48}-q^{46}-q^{44}+2q^{42}-3q^{40}+3q^{38}+q^{36}+q^{34}-q^{30}+2q^{28}-2q^{26}+2q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+q^{10}$

Further Quantum Invariants

Further quantum knot invariants for 9_35.

A1 Invariants.

Weight	Invariant
1	$-q^{21}-2q^{17}+q^{15}+q^{13}+2q^{11}+q^{9}-q^{7}+q^{5}-q^{3}+q$
2	$q^{60}-q^{56}+2q^{54}+2q^{52}-3q^{50}+q^{46}-4q^{44}-3q^{42}+q^{40}-q^{38}-2q^{36}+3q^{34}+3q^{32}+3q^{26}-q^{24}-2q^{22}+q^{20}+q^{18}-2q^{16}+q^{14}+3q^{12}-q^{10}+q^{8}-q^{4}+q^{2}$
3	$-q^{117}+q^{113}+q^{111}-2q^{109}-3q^{107}+5q^{103}+2q^{101}-4q^{99}-4q^{97}+4q^{95}+7q^{93}+3q^{91}-4q^{89}-3q^{87}+4q^{85}+4q^{83}-q^{81}-8q^{79}-3q^{77}+2q^{75}+q^{73}-7q^{71}-4q^{69}+q^{67}+6q^{65}-2q^{63}-4q^{61}+4q^{59}+9q^{57}-q^{55}-8q^{53}+6q^{49}+2q^{47}-6q^{45}-4q^{43}-q^{41}+6q^{39}+4q^{37}-3q^{35}-6q^{33}+4q^{31}+8q^{29}-5q^{25}+3q^{21}-q^{19}-q^{17}+2q^{13}+q^{11}-q^{5}+q^{3}$
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}+3 q^{182}+3 q^{180}+q^{178}-2 q^{176}-8 q^{174}-2 q^{172}+4 q^{170}+9 q^{168}+6 q^{166}-8 q^{164}-11 q^{162}-10 q^{160}+3 q^{158}+15 q^{156}+8 q^{154}-q^{152}-17 q^{150}-15 q^{148}+4 q^{146}+15 q^{144}+22 q^{142}+2 q^{140}-17 q^{138}-15 q^{136}-2 q^{134}+23 q^{132}+23 q^{130}-q^{128}-17 q^{126}-21 q^{124}+q^{122}+19 q^{120}+14 q^{118}-7 q^{116}-25 q^{114}-15 q^{112}+9 q^{110}+17 q^{108}+3 q^{106}-15 q^{104}-12 q^{102}+8 q^{100}+17 q^{98}+6 q^{96}-14 q^{94}-12 q^{92}+10 q^{90}+18 q^{88}+2 q^{86}-18 q^{84}-21 q^{82}+6 q^{80}+20 q^{78}+13 q^{76}-6 q^{74}-25 q^{72}-15 q^{70}+6 q^{68}+21 q^{66}+23 q^{64}-4 q^{62}-27 q^{60}-20 q^{58}+6 q^{56}+35 q^{54}+21 q^{52}-14 q^{50}-31 q^{48}-15 q^{46}+20 q^{44}+22 q^{42}-15 q^{38}-11 q^{36}+8 q^{34}+10 q^{32}+2 q^{30}-3 q^{28}-5 q^{26}+5 q^{24}+q^{22}-2 q^{20}-q^{18}-q^{16}+4 q^{14}-q^6+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}-3 q^{273}-4 q^{271}-q^{269}+2 q^{267}+5 q^{265}+8 q^{263}+3 q^{261}-6 q^{259}-12 q^{257}-11 q^{255}-2 q^{253}+12 q^{251}+20 q^{249}+16 q^{247}-18 q^{243}-27 q^{241}-18 q^{239}+4 q^{237}+27 q^{235}+37 q^{233}+20 q^{231}-11 q^{229}-38 q^{227}-44 q^{225}-23 q^{223}+20 q^{221}+50 q^{219}+46 q^{217}+10 q^{215}-36 q^{213}-66 q^{211}-51 q^{209}+3 q^{207}+57 q^{205}+69 q^{203}+39 q^{201}-26 q^{199}-74 q^{197}-67 q^{195}-10 q^{193}+55 q^{191}+87 q^{189}+58 q^{187}-16 q^{185}-76 q^{183}-76 q^{181}-19 q^{179}+57 q^{177}+92 q^{175}+49 q^{173}-28 q^{171}-83 q^{169}-76 q^{167}-9 q^{165}+65 q^{163}+76 q^{161}+27 q^{159}-46 q^{157}-76 q^{155}-44 q^{153}+29 q^{151}+66 q^{149}+43 q^{147}-11 q^{145}-49 q^{143}-37 q^{141}+15 q^{139}+42 q^{137}+23 q^{135}-19 q^{133}-40 q^{131}-15 q^{129}+30 q^{127}+48 q^{125}+13 q^{123}-44 q^{121}-64 q^{119}-24 q^{117}+36 q^{115}+73 q^{113}+47 q^{111}-23 q^{109}-76 q^{107}-69 q^{105}-16 q^{103}+53 q^{101}+85 q^{99}+61 q^{97}-7 q^{95}-75 q^{93}-95 q^{91}-44 q^{89}+46 q^{87}+106 q^{85}+93 q^{83}-3 q^{81}-100 q^{79}-114 q^{77}-40 q^{75}+66 q^{73}+115 q^{71}+63 q^{69}-34 q^{67}-94 q^{65}-66 q^{63}+11 q^{61}+68 q^{59}+60 q^{57}+4 q^{55}-44 q^{53}-42 q^{51}-3 q^{49}+24 q^{47}+25 q^{45}+3 q^{43}-15 q^{41}-14 q^{39}+2 q^{37}+9 q^{35}+5 q^{33}-q^{31}-q^{29}+q^{25}+q^{23}-2 q^{21}-2 q^{19}+q^{17}+3 q^{15}-q^7+q^5}

A2 Invariants.

Weight	Invariant
1,0	$-q^{32}-q^{30}-2q^{26}-q^{24}+q^{22}+q^{20}+3q^{18}+2q^{16}+q^{14}-q^{10}+q^{8}-q^{4}+q^{2}$
1,1	$q^{84}+6q^{80}-6q^{78}+12q^{76}-18q^{74}+18q^{72}-22q^{70}+14q^{68}-16q^{66}+4q^{64}+8q^{62}-7q^{60}+20q^{58}-24q^{56}+30q^{54}-35q^{52}+22q^{50}-32q^{48}+16q^{46}-14q^{44}+2q^{42}+8q^{40}-2q^{38}+15q^{36}-4q^{34}+12q^{32}-10q^{30}+7q^{28}-4q^{26}+4q^{24}-6q^{22}+5q^{20}+2q^{18}+4q^{16}-4q^{14}+3q^{12}-2q^{10}+2q^{8}-2q^{6}+q^{4}$
2,0	$q^{82}+q^{80}+q^{78}-q^{76}+q^{74}+3q^{72}+3q^{70}-q^{68}-3q^{66}-q^{64}-q^{62}-5q^{60}-7q^{58}-4q^{56}-2q^{54}-2q^{52}-3q^{50}+2q^{48}+5q^{46}+6q^{44}+4q^{42}+3q^{40}+2q^{38}+q^{36}-q^{34}-q^{32}-q^{30}+q^{28}+3q^{26}-q^{24}-3q^{22}+2q^{20}+4q^{18}+q^{16}-2q^{14}+2q^{10}-q^{8}-q^{6}+q^{4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{66}+3q^{62}+2q^{60}+q^{58}+q^{56}-3q^{54}-7q^{52}-6q^{50}-7q^{48}-5q^{46}+2q^{44}+2q^{42}+5q^{40}+6q^{38}+4q^{36}+q^{28}+3q^{24}+3q^{22}-q^{20}+q^{18}+q^{16}-2q^{14}+2q^{10}-q^{8}-q^{6}+q^{4}$
1,0,0	$-q^{43}-q^{41}-q^{39}-2q^{35}-q^{33}-q^{31}+q^{29}+q^{27}+3q^{25}+3q^{23}+2q^{21}+q^{19}-q^{13}+q^{11}-q^{5}+q^{3}$

B2 Invariants.

Weight	Invariant
0,1	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^{66}-3 q^{62}+2 q^{60}-3 q^{58}+3 q^{56}-3 q^{54}+3 q^{52}-2 q^{50}+q^{48}+q^{46}-2 q^{44}+4 q^{42}-5 q^{40}+6 q^{38}-6 q^{36}+6 q^{34}-4 q^{32}+4 q^{30}-q^{28}+2 q^{26}+q^{24}-q^{22}+3 q^{20}-3 q^{18}+3 q^{16}-2 q^{14}+2 q^{12}-2 q^{10}+q^8-q^6+q^4}
1,0	$q^{108}+3q^{100}+2q^{98}-q^{96}-q^{94}+2q^{92}+2q^{90}-q^{88}-5q^{86}-4q^{84}-q^{82}-q^{80}-5q^{78}-7q^{76}-2q^{74}+2q^{72}+2q^{70}-q^{68}+q^{66}+4q^{64}+5q^{62}+2q^{60}+q^{58}+2q^{56}+3q^{54}-q^{52}-3q^{50}-q^{48}+2q^{46}+q^{44}-q^{42}-2q^{40}+2q^{38}+4q^{36}+q^{34}-2q^{32}+2q^{28}+2q^{26}-q^{24}-2q^{22}-q^{20}+q^{18}+2q^{16}-q^{12}-q^{10}+q^{6}$

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}+3 q^{152}-3 q^{150}+2 q^{148}-q^{146}-2 q^{144}+7 q^{142}-9 q^{140}+6 q^{138}-2 q^{136}-2 q^{134}+8 q^{132}-12 q^{130}+5 q^{128}-2 q^{126}-3 q^{124}+3 q^{122}-10 q^{120}-2 q^{118}+4 q^{116}-2 q^{114}+q^{112}-8 q^{110}-2 q^{108}+6 q^{106}-6 q^{104}+5 q^{102}-11 q^{100}+6 q^{98}+8 q^{96}-3 q^{94}+8 q^{92}-10 q^{90}+12 q^{88}+4 q^{86}-5 q^{84}+7 q^{82}-5 q^{80}+5 q^{78}+7 q^{76}-3 q^{74}+2 q^{72}+q^{70}-2 q^{68}+4 q^{66}-6 q^{64}+3 q^{62}-2 q^{60}-2 q^{58}+4 q^{56}-4 q^{54}+3 q^{52}-2 q^{50}+q^{48}-q^{46}-q^{44}+2 q^{42}-3 q^{40}+3 q^{38}+q^{36}+q^{34}-q^{30}+2 q^{28}-2 q^{26}+2 q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 35"];

In[4]:=

Alexander[K][t]

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

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 7 t-13+7 t^{-1} }

In[5]:=

Conway[K][z]

Out[5]=

$7z^{2}+1$

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 \{3,t+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]=

$q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+5q^{-5}-3q^{-6}+4q^{-7}-3q^{-8}+q^{-9}-q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{10}+z^{2}a^{8}-a^{8}+3z^{2}a^{6}+3a^{6}+2z^{2}a^{4}+z^{2}a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{11}-6z^{5}a^{11}+12z^{3}a^{11}-8za^{11}+z^{8}a^{10}-4z^{6}a^{10}+3z^{4}a^{10}+z^{2}a^{10}+a^{10}+4z^{7}a^{9}-18z^{5}a^{9}+23z^{3}a^{9}-9za^{9}+z^{8}a^{8}+z^{6}a^{8}-15z^{4}a^{8}+16z^{2}a^{8}-a^{8}+3z^{7}a^{7}-8z^{5}a^{7}+3z^{3}a^{7}-za^{7}+5z^{6}a^{6}-15z^{4}a^{6}+12z^{2}a^{6}-3a^{6}+4z^{5}a^{5}-6z^{3}a^{5}+3z^{4}a^{4}-2z^{2}a^{4}+2z^{3}a^{3}+z^{2}a^{2}$

"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["9 35"];

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]=

{ $7t-13+7t^{-1}$, $q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+5q^{-5}-3q^{-6}+4q^{-7}-3q^{-8}+q^{-9}-q^{-10}$ }

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]=

{}

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:

(7, -18)

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 $28$ $-144$ $392$ 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{3026}{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{574}{3}} $-4032$ $-7584$ $-1344$ 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 -1296} 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{10976}{3}} $10368$ 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{84728}{3}} ${\frac {16072}{3}}$ 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{1720297}{30}} 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{25154}{15}} ${\frac {1232834}{45}}$ 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{8471}{18}} ${\frac {119977}{30}}$

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 $r$). 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 9 35. 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 1 -3                 2 1 -1 -5               1     1 -7             3 2     -1 -9           2 1       1 -11         1 3         2 -13       3 2           1 -15       1             1 -17   1 3               -2 -19                     0 -21 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-2}-2q^{-3}+q^{-4}+2q^{-5}-4q^{-6}+5q^{-7}-7q^{-9}+8q^{-10}-10q^{-12}+9q^{-13}+4q^{-14}-13q^{-15}+9q^{-16}+7q^{-17}-13q^{-18}+4q^{-19}+8q^{-20}-11q^{-21}+7q^{-23}-6q^{-24}-q^{-25}+4q^{-26}-q^{-27}-q^{-28}+q^{-29}$

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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