9 35

9_34

9_36

Visit 9 35's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 35's page at Knotilus! Visit 9 35's page at the original Knot Atlas!

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]

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 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} Topological 4 genus 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} Concordance genus 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} Rasmussen s-Invariant -2

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

Polynomial invariants

Alexander polynomial	$7t-13+7t^{-1}$
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 7 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 \{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)	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}+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}
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	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}}

Further Quantum Invariants

Further quantum knot invariants for 9_35.

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}-2 q^{17}+q^{15}+q^{13}+2 q^{11}+q^9-q^7+q^5-q^3+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}+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}
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^{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}
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	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}+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}
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}+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}

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}+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}
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}-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}

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	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}+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}

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]=

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

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 7 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 \{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]=

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}+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}

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 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 28} $-144$ 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 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}} 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 -4032} 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 -7584} 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 -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}} 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{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}} 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{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}} 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{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 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 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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 35]]
Out[2]=	9
In[3]:=	PD[Knot[9, 35]]
Out[3]=	PD[X[1, 8, 2, 9], X[7, 14, 8, 15], X[5, 16, 6, 17], X[9, 18, 10, 1], X[15, 6, 16, 7], X[17, 10, 18, 11], X[13, 2, 14, 3], X[3, 12, 4, 13], X[11, 4, 12, 5]]
In[4]:=	GaussCode[Knot[9, 35]]
Out[4]=	GaussCode[-1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4]
In[5]:=	BR[Knot[9, 35]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -2, -3, 2, 2, -4, 3, -2, -4, -3}]
In[6]:=	alex = Alexander[Knot[9, 35]][t]
Out[6]=	7 -13 + - + 7 t t
In[7]:=	Conway[Knot[9, 35]][z]
Out[7]=	2 1 + 7 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 35]}
In[9]:=	{KnotDet[Knot[9, 35]], KnotSignature[Knot[9, 35]]}
Out[9]=	{27, -2}
In[10]:=	J=Jones[Knot[9, 35]][q]
Out[10]=	-10 -9 3 4 3 5 4 3 2 1 -q + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 35]}
In[12]:=	A2Invariant[Knot[9, 35]][q]
Out[12]=	-32 -30 2 -24 -22 -20 3 2 -14 -10 -q - q - --- - q + q + q + --- + --- + q - q + 26 18 16 q q q -8 -4 -2 q - q + q
In[13]:=	Kauffman[Knot[9, 35]][a, z]
Out[13]=	6 8 10 7 9 11 2 2 4 2 -3 a - a + a - a z - 9 a z - 8 a z + a z - 2 a z + 6 2 8 2 10 2 3 3 5 3 7 3 12 a z + 16 a z + a z + 2 a z - 6 a z + 3 a z + 9 3 11 3 4 4 6 4 8 4 10 4 23 a z + 12 a z + 3 a z - 15 a z - 15 a z + 3 a z + 5 5 7 5 9 5 11 5 6 6 8 6 4 a z - 8 a z - 18 a z - 6 a z + 5 a z + a z - 10 6 7 7 9 7 11 7 8 8 10 8 4 a z + 3 a z + 4 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 35]], Vassiliev[3][Knot[9, 35]]}
Out[14]=	{0, -18}
In[15]:=	Kh[Knot[9, 35]][q, t]
Out[15]=	-3 1 1 1 3 1 3 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 17 8 17 7 15 6 13 6 13 5 q t q t q t q t q t q t 1 3 2 1 3 2 1 2 ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t