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]=

[math]\displaystyle{ \textrm{BR}(5,\{-1,-1,-2,1,-2,-2,-3,2,2,-4,3,-2,-4,-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]=

{ 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 [math]\displaystyle{ \{4,6\} }[/math] 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 [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant -2

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

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 7 t-13+7 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature	{ 27, -2 }
Jones polynomial	[math]\displaystyle{ q^{-1} -2 q^{-2} +3 q^{-3} -4 q^{-4} +5 q^{-5} -3 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} - q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^{10}+z^2 a^8-a^8+3 z^2 a^6+3 a^6+2 z^2 a^4+z^2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+12 z^3 a^{11}-8 z a^{11}+z^8 a^{10}-4 z^6 a^{10}+3 z^4 a^{10}+z^2 a^{10}+a^{10}+4 z^7 a^9-18 z^5 a^9+23 z^3 a^9-9 z a^9+z^8 a^8+z^6 a^8-15 z^4 a^8+16 z^2 a^8-a^8+3 z^7 a^7-8 z^5 a^7+3 z^3 a^7-z a^7+5 z^6 a^6-15 z^4 a^6+12 z^2 a^6-3 a^6+4 z^5 a^5-6 z^3 a^5+3 z^4 a^4-2 z^2 a^4+2 z^3 a^3+z^2 a^2 }[/math]
The A2 invariant	[math]\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 }[/math]
The G2 invariant	[math]\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} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_35.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{21}-2 q^{17}+q^{15}+q^{13}+2 q^{11}+q^9-q^7+q^5-q^3+q }[/math]
2	[math]\displaystyle{ q^{60}-q^{56}+2 q^{54}+2 q^{52}-3 q^{50}+q^{46}-4 q^{44}-3 q^{42}+q^{40}-q^{38}-2 q^{36}+3 q^{34}+3 q^{32}+3 q^{26}-q^{24}-2 q^{22}+q^{20}+q^{18}-2 q^{16}+q^{14}+3 q^{12}-q^{10}+q^8-q^4+q^2 }[/math]
3	[math]\displaystyle{ -q^{117}+q^{113}+q^{111}-2 q^{109}-3 q^{107}+5 q^{103}+2 q^{101}-4 q^{99}-4 q^{97}+4 q^{95}+7 q^{93}+3 q^{91}-4 q^{89}-3 q^{87}+4 q^{85}+4 q^{83}-q^{81}-8 q^{79}-3 q^{77}+2 q^{75}+q^{73}-7 q^{71}-4 q^{69}+q^{67}+6 q^{65}-2 q^{63}-4 q^{61}+4 q^{59}+9 q^{57}-q^{55}-8 q^{53}+6 q^{49}+2 q^{47}-6 q^{45}-4 q^{43}-q^{41}+6 q^{39}+4 q^{37}-3 q^{35}-6 q^{33}+4 q^{31}+8 q^{29}-5 q^{25}+3 q^{21}-q^{19}-q^{17}+2 q^{13}+q^{11}-q^5+q^3 }[/math]
4	[math]\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 }[/math]
5	[math]\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 }[/math]

A2 Invariants.

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

A3 Invariants.

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

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\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} }[/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["9 35"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 7 t-13+7 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 7 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{ \{3,t+1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-1} -2 q^{-2} +3 q^{-3} -4 q^{-4} +5 q^{-5} -3 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} - q^{-10} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -a^{10}+z^2 a^8-a^8+3 z^2 a^6+3 a^6+2 z^2 a^4+z^2 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+12 z^3 a^{11}-8 z a^{11}+z^8 a^{10}-4 z^6 a^{10}+3 z^4 a^{10}+z^2 a^{10}+a^{10}+4 z^7 a^9-18 z^5 a^9+23 z^3 a^9-9 z a^9+z^8 a^8+z^6 a^8-15 z^4 a^8+16 z^2 a^8-a^8+3 z^7 a^7-8 z^5 a^7+3 z^3 a^7-z a^7+5 z^6 a^6-15 z^4 a^6+12 z^2 a^6-3 a^6+4 z^5 a^5-6 z^3 a^5+3 z^4 a^4-2 z^2 a^4+2 z^3 a^3+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]): {}

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]=

{ [math]\displaystyle{ 7 t-13+7 t^{-1} }[/math], [math]\displaystyle{ q^{-1} -2 q^{-2} +3 q^{-3} -4 q^{-4} +5 q^{-5} -3 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} - q^{-10} }[/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:

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

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 [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 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)

[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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} -2 q^{-3} + q^{-4} +2 q^{-5} -4 q^{-6} +5 q^{-7} -7 q^{-9} +8 q^{-10} -10 q^{-12} +9 q^{-13} +4 q^{-14} -13 q^{-15} +9 q^{-16} +7 q^{-17} -13 q^{-18} +4 q^{-19} +8 q^{-20} -11 q^{-21} +7 q^{-23} -6 q^{-24} - q^{-25} +4 q^{-26} - q^{-27} - q^{-28} + q^{-29} }[/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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