9 45

9_44

9_46

Visit 9 45's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 45's page at Knotilus!

Visit 9 45's page at the original Knot Atlas!

9 45 Quick Notes

9 45 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,6,11,5 X₈₃₉₄ X_2,9,3,10 X_7,14,8,15 X_18,15,1,16 X_16,11,17,12 X_12,17,13,18 X_13,6,14,7
Gauss code	1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6
Dowker-Thistlethwaite code	4 8 10 -14 2 16 -6 18 12
Conway Notation	[211,21,2-]

Minimum Braid Representative:

Length is 9, width is 4.

Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	2
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 \{4,5\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[-10][1]
Hyperbolic Volume	8.60203
A-Polynomial	See Data:9 45/A-polynomial

[edit Notes for 9 45'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	$1$
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 2}
Rasmussen s-Invariant	2

[edit Notes for 9 45'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 -t^2+6 t-9+6 t^{-1} - t^{-2} }
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 -z^4+2 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	{ 23, -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 2 q^{-1} -3 q^{-2} +4 q^{-3} -4 q^{-4} +4 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }
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^8+2 z^2 a^6+2 a^6-z^4 a^4-2 z^2 a^4-2 a^4+2 z^2 a^2+2 a^2}
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-3z^{3}a^{9}+2za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+z^{7}a^{7}-5z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-10z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+z^{7}a^{5}-z^{3}a^{5}+2z^{6}a^{4}-4z^{4}a^{4}+6z^{2}a^{4}-2a^{4}+z^{5}a^{3}+z^{3}a^{3}+3z^{2}a^{2}-2a^{2}$
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^{26}-q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{10}+q^8+q^6+2 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^{128}-q^{126}+3 q^{124}-4 q^{122}+2 q^{120}-5 q^{116}+9 q^{114}-10 q^{112}+8 q^{110}-3 q^{108}-7 q^{106}+10 q^{104}-12 q^{102}+8 q^{100}-3 q^{98}-6 q^{96}+9 q^{94}-7 q^{92}+2 q^{90}+6 q^{88}-9 q^{86}+10 q^{84}-4 q^{82}-2 q^{80}+9 q^{78}-12 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+7 q^{68}-12 q^{66}+14 q^{64}-12 q^{62}+5 q^{60}+2 q^{58}-9 q^{56}+9 q^{54}-8 q^{52}+q^{50}+6 q^{48}-10 q^{46}+6 q^{44}-q^{42}-7 q^{40}+11 q^{38}-11 q^{36}+8 q^{34}-5 q^{30}+9 q^{28}-8 q^{26}+8 q^{24}-q^{22}-q^{20}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8}

Further Quantum Invariants

Further quantum knot invariants for 9_45.

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^{17}+q^{15}-q^{13}+q^{11}+q^5-q^3+2 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^{48}-q^{46}-2 q^{44}+3 q^{42}+q^{40}-4 q^{38}+2 q^{36}+3 q^{34}-4 q^{32}-q^{30}+3 q^{28}-2 q^{26}-2 q^{24}+2 q^{22}+q^{20}-2 q^{18}-q^{16}+5 q^{14}-q^{12}-3 q^{10}+5 q^8-3 q^4+2 q^2+1}
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^{93}+q^{91}+2 q^{89}-4 q^{85}-3 q^{83}+5 q^{81}+6 q^{79}-3 q^{77}-9 q^{75}-q^{73}+11 q^{71}+6 q^{69}-9 q^{67}-10 q^{65}+5 q^{63}+13 q^{61}-3 q^{59}-13 q^{57}-q^{55}+13 q^{53}+4 q^{51}-11 q^{49}-4 q^{47}+9 q^{45}+4 q^{43}-7 q^{41}-7 q^{39}+2 q^{37}+7 q^{35}+q^{33}-9 q^{31}-6 q^{29}+9 q^{27}+12 q^{25}-6 q^{23}-14 q^{21}+4 q^{19}+15 q^{17}-10 q^{13}-3 q^{11}+8 q^9+4 q^7-4 q^5-q^3+2 q^{-1} }
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^{152}-q^{150}-2 q^{148}+q^{144}+6 q^{142}+q^{140}-5 q^{138}-7 q^{136}-7 q^{134}+11 q^{132}+13 q^{130}+5 q^{128}-9 q^{126}-25 q^{124}-5 q^{122}+14 q^{120}+28 q^{118}+16 q^{116}-26 q^{114}-32 q^{112}-15 q^{110}+27 q^{108}+45 q^{106}+4 q^{104}-33 q^{102}-46 q^{100}+49 q^{96}+34 q^{94}-12 q^{92}-51 q^{90}-23 q^{88}+33 q^{86}+42 q^{84}+4 q^{82}-40 q^{80}-26 q^{78}+18 q^{76}+34 q^{74}+7 q^{72}-27 q^{70}-23 q^{68}+6 q^{66}+27 q^{64}+11 q^{62}-13 q^{60}-23 q^{58}-12 q^{56}+17 q^{54}+25 q^{52}+15 q^{50}-23 q^{48}-42 q^{46}-5 q^{44}+33 q^{42}+48 q^{40}-4 q^{38}-57 q^{36}-35 q^{34}+17 q^{32}+61 q^{30}+25 q^{28}-37 q^{26}-40 q^{24}-11 q^{22}+36 q^{20}+31 q^{18}-6 q^{16}-19 q^{14}-16 q^{12}+7 q^{10}+12 q^8+3 q^6-q^4-5 q^2-1+2 q^{-2} + 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^{225}+q^{223}+2 q^{221}-q^{217}-3 q^{215}-4 q^{213}-q^{211}+7 q^{209}+9 q^{207}+5 q^{205}-3 q^{203}-14 q^{201}-18 q^{199}-7 q^{197}+16 q^{195}+28 q^{193}+24 q^{191}+q^{189}-32 q^{187}-48 q^{185}-30 q^{183}+17 q^{181}+59 q^{179}+67 q^{177}+25 q^{175}-47 q^{173}-97 q^{171}-77 q^{169}+8 q^{167}+96 q^{165}+125 q^{163}+56 q^{161}-67 q^{159}-151 q^{157}-122 q^{155}+12 q^{153}+144 q^{151}+170 q^{149}+58 q^{147}-109 q^{145}-196 q^{143}-116 q^{141}+61 q^{139}+187 q^{137}+159 q^{135}-9 q^{133}-165 q^{131}-175 q^{129}-31 q^{127}+133 q^{125}+168 q^{123}+56 q^{121}-102 q^{119}-153 q^{117}-61 q^{115}+76 q^{113}+131 q^{111}+57 q^{109}-59 q^{107}-108 q^{105}-49 q^{103}+50 q^{101}+92 q^{99}+44 q^{97}-39 q^{95}-78 q^{93}-47 q^{91}+22 q^{89}+72 q^{87}+63 q^{85}+4 q^{83}-63 q^{81}-87 q^{79}-47 q^{77}+41 q^{75}+112 q^{73}+98 q^{71}-9 q^{69}-128 q^{67}-155 q^{65}-44 q^{63}+121 q^{61}+202 q^{59}+108 q^{57}-94 q^{55}-222 q^{53}-164 q^{51}+41 q^{49}+209 q^{47}+200 q^{45}+22 q^{43}-170 q^{41}-199 q^{39}-67 q^{37}+101 q^{35}+170 q^{33}+94 q^{31}-42 q^{29}-119 q^{27}-85 q^{25}+q^{23}+64 q^{21}+64 q^{19}+21 q^{17}-25 q^{15}-38 q^{13}-13 q^{11}+4 q^9+13 q^7+13 q^5-6 q-3 q^{-1} +2 q^{-7} }

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^{26}-q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{10}+q^8+q^6+2 q^2}
1,1	$q^{68}-2q^{66}+6q^{64}-12q^{62}+17q^{60}-24q^{58}+30q^{56}-30q^{54}+25q^{52}-18q^{50}+6q^{48}+12q^{46}-27q^{44}+38q^{42}-48q^{40}+52q^{38}-54q^{36}+44q^{34}-36q^{32}+22q^{30}-6q^{28}-6q^{26}+20q^{24}-26q^{22}+31q^{20}-28q^{18}+22q^{16}-16q^{14}+13q^{12}-4q^{10}+2q^{8}+2q^{4}+2q^{2}$
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^{66}+q^{64}-3 q^{60}-2 q^{58}+q^{56}+2 q^{54}+3 q^{48}+2 q^{46}-2 q^{44}-4 q^{42}-q^{40}-2 q^{38}-2 q^{36}-q^{34}+2 q^{32}+2 q^{30}+q^{28}+2 q^{26}-q^{24}-q^{22}+2 q^{20}+2 q^{18}-2 q^{16}-q^{14}+3 q^{12}+2 q^{10}-q^8-q^6+3 q^4+q^2}

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^{54}-q^{52}+q^{50}+q^{48}-3 q^{46}+q^{44}-q^{42}-4 q^{40}+2 q^{38}+q^{36}-2 q^{34}+2 q^{32}+2 q^{30}+q^{22}-2 q^{20}-2 q^{18}+3 q^{16}-2 q^{14}+5 q^{10}+3 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^{35}-q^{33}-q^{31}+q^{29}+2 q^{25}+q^{23}+q^{21}-q^{19}-q^{17}-q^{15}-q^{13}+q^{11}+q^9+2 q^7+2 q^3}

A4 Invariants.

Weight

Invariant

0,1,0,0

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^{72}+q^{70}+q^{64}-q^{62}-4 q^{60}-2 q^{58}+q^{56}-2 q^{54}-3 q^{52}+3 q^{50}+4 q^{48}-q^{46}-q^{44}+2 q^{42}-q^{40}-3 q^{38}+q^{36}+3 q^{34}-q^{32}+q^{30}+4 q^{28}-2 q^{26}-4 q^{24}+q^{20}-2 q^{18}+5 q^{14}+4 q^{12}+q^{10}+q^8+3 q^6}

1,0,0,0

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^{44}-q^{42}-q^{40}-q^{38}+q^{36}+2 q^{32}+2 q^{30}+q^{28}+q^{26}-q^{24}-q^{22}-2 q^{20}-q^{18}-q^{16}+q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^4}

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^{54}+q^{52}-3 q^{50}+3 q^{48}-3 q^{46}+3 q^{44}-3 q^{42}+2 q^{40}-q^{36}+4 q^{34}-4 q^{32}+6 q^{30}-6 q^{28}+6 q^{26}-6 q^{24}+3 q^{22}-2 q^{20}+q^{16}-2 q^{14}+4 q^{12}-3 q^{10}+4 q^8-2 q^6+3 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^{88}-q^{84}-q^{82}+2 q^{80}+2 q^{78}-2 q^{76}-3 q^{74}+3 q^{70}+q^{68}-4 q^{66}-3 q^{64}+2 q^{62}+3 q^{60}-3 q^{56}+2 q^{52}+q^{50}-2 q^{48}-q^{46}+2 q^{44}+2 q^{42}-q^{40}-3 q^{38}+q^{36}+3 q^{34}-3 q^{30}-q^{28}+3 q^{26}+2 q^{24}-2 q^{22}-3 q^{20}+2 q^{18}+4 q^{16}+q^{14}-2 q^{12}-q^{10}+q^8+3 q^6}

D4 Invariants.

Weight

Invariant

1,0,0,0

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^{74}-q^{72}+2 q^{70}-2 q^{68}+3 q^{66}-3 q^{64}+q^{62}-4 q^{60}+q^{58}-3 q^{56}-q^{54}-q^{50}+3 q^{48}-2 q^{46}+6 q^{44}-2 q^{42}+6 q^{40}-4 q^{38}+5 q^{36}-4 q^{34}+3 q^{32}-4 q^{30}-q^{28}-2 q^{26}-q^{24}+q^{22}-2 q^{20}+3 q^{18}-q^{16}+6 q^{14}+3 q^{10}-q^8+3 q^6}

G2 Invariants.

Weight

Invariant

1,0

q^{128}-q^{126}+3q^{124}-4q^{122}+2q^{120}-5q^{116}+9q^{114}-10q^{112}+8q^{110}-3q^{108}-7q^{106}+10q^{104}-12q^{102}+8q^{100}-3q^{98}-6q^{96}+9q^{94}-7q^{92}+2q^{90}+6q^{88}-9q^{86}+10q^{84}-4q^{82}-2q^{80}+9q^{78}-12q^{76}+15q^{74}-9q^{72}+3q^{70}+7q^{68}-12q^{66}+14q^{64}-12q^{62}+5q^{60}+2q^{58}-9q^{56}+9q^{54}-8q^{52}+q^{50}+6q^{48}-10q^{46}+6q^{44}-q^{42}-7q^{40}+11q^{38}-11q^{36}+8q^{34}-5q^{30}+9q^{28}-8q^{26}+8q^{24}-q^{22}-q^{20}+2q^{18}-2q^{16}+3q^{14}-q^{12}+2q^{10}+q^{8}

.

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 45"];

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 -t^2+6 t-9+6 t^{-1} - t^{-2} }

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 -z^4+2 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]= { 23, -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 2 q^{-1} -3 q^{-2} +4 q^{-3} -4 q^{-4} +4 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }

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^8+2 z^2 a^6+2 a^6-z^4 a^4-2 z^2 a^4-2 a^4+2 z^2 a^2+2 a^2}

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

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

Out[10]= $z^{5}a^{9}-3z^{3}a^{9}+2za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+z^{7}a^{7}-5z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-10z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+z^{7}a^{5}-z^{3}a^{5}+2z^{6}a^{4}-4z^{4}a^{4}+6z^{2}a^{4}-2a^{4}+z^{5}a^{3}+z^{3}a^{3}+3z^{2}a^{2}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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}} ): {...}

Vassiliev invariants

V₂ and V₃:

(2, -4)

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 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 -32}

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 32}

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{412}{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{68}{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 -256}

-{\frac {1856}{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{320}{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 -96}

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{256}{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 512}

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{3296}{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{544}{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{42751}{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{284}{15}}

{\frac {56284}{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{257}{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{2431}{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 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} , 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 45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

χ

-1

2

-3

2

1

-1

-5

2

1

-7

2

0

-9

2

0

-11

1

2

1

-13

1

2

-1

-15

1

-17

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=-7}	${\mathbb {Z} }$
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}^{2}\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}^{2}\oplus{\mathbb Z}_2^{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}^{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 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}^{2}\oplus{\mathbb Z}_2^{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}^{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 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}^{2}\oplus{\mathbb Z}_2^{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}^{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 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}\oplus{\mathbb Z}_2^{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}^{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 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}\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}^{2}}

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^{-1} + q^{-2} -5 q^{-3} +4 q^{-4} +6 q^{-5} -13 q^{-6} +6 q^{-7} +12 q^{-8} -19 q^{-9} +5 q^{-10} +15 q^{-11} -18 q^{-12} + q^{-13} +15 q^{-14} -13 q^{-15} -3 q^{-16} +12 q^{-17} -6 q^{-18} -4 q^{-19} +6 q^{-20} - q^{-21} -2 q^{-22} + q^{-23} }
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 2 q^{-1} -2 q^{-2} - q^{-3} -3 q^{-4} +10 q^{-5} +2 q^{-6} -12 q^{-7} -10 q^{-8} +20 q^{-9} +17 q^{-10} -23 q^{-11} -28 q^{-12} +28 q^{-13} +35 q^{-14} -26 q^{-15} -43 q^{-16} +25 q^{-17} +45 q^{-18} -20 q^{-19} -48 q^{-20} +16 q^{-21} +45 q^{-22} -9 q^{-23} -43 q^{-24} +3 q^{-25} +38 q^{-26} +6 q^{-27} -34 q^{-28} -11 q^{-29} +26 q^{-30} +16 q^{-31} -18 q^{-32} -19 q^{-33} +11 q^{-34} +17 q^{-35} -3 q^{-36} -14 q^{-37} - q^{-38} +9 q^{-39} +3 q^{-40} -5 q^{-41} -2 q^{-42} + q^{-43} +2 q^{-44} - q^{-45} }
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 1+ q^{-1} -3 q^{-2} -4 q^{-3} +4 q^{-4} +5 q^{-5} +10 q^{-6} -8 q^{-7} -27 q^{-8} + q^{-9} +18 q^{-10} +47 q^{-11} -3 q^{-12} -74 q^{-13} -28 q^{-14} +21 q^{-15} +109 q^{-16} +33 q^{-17} -118 q^{-18} -80 q^{-19} - q^{-20} +162 q^{-21} +85 q^{-22} -133 q^{-23} -118 q^{-24} -38 q^{-25} +181 q^{-26} +123 q^{-27} -123 q^{-28} -126 q^{-29} -67 q^{-30} +170 q^{-31} +133 q^{-32} -99 q^{-33} -110 q^{-34} -88 q^{-35} +141 q^{-36} +129 q^{-37} -65 q^{-38} -83 q^{-39} -104 q^{-40} +97 q^{-41} +115 q^{-42} -21 q^{-43} -45 q^{-44} -113 q^{-45} +41 q^{-46} +87 q^{-47} +18 q^{-48} + q^{-49} -98 q^{-50} -8 q^{-51} +41 q^{-52} +31 q^{-53} +38 q^{-54} -57 q^{-55} -26 q^{-56} - q^{-57} +14 q^{-58} +44 q^{-59} -15 q^{-60} -14 q^{-61} -15 q^{-62} -5 q^{-63} +24 q^{-64} + q^{-65} -7 q^{-67} -7 q^{-68} +6 q^{-69} + q^{-70} +2 q^{-71} - q^{-72} -2 q^{-73} + q^{-74} }
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 2 q-2-3 q^{-2} -3 q^{-3} +6 q^{-4} +15 q^{-5} -2 q^{-6} -9 q^{-7} -20 q^{-8} -28 q^{-9} +19 q^{-10} +61 q^{-11} +41 q^{-12} -9 q^{-13} -83 q^{-14} -114 q^{-15} -15 q^{-16} +138 q^{-17} +177 q^{-18} +67 q^{-19} -152 q^{-20} -282 q^{-21} -147 q^{-22} +167 q^{-23} +369 q^{-24} +245 q^{-25} -143 q^{-26} -450 q^{-27} -352 q^{-28} +109 q^{-29} +497 q^{-30} +447 q^{-31} -49 q^{-32} -531 q^{-33} -517 q^{-34} -2 q^{-35} +524 q^{-36} +566 q^{-37} +58 q^{-38} -517 q^{-39} -588 q^{-40} -90 q^{-41} +484 q^{-42} +590 q^{-43} +125 q^{-44} -458 q^{-45} -579 q^{-46} -140 q^{-47} +415 q^{-48} +559 q^{-49} +164 q^{-50} -375 q^{-51} -531 q^{-52} -182 q^{-53} +316 q^{-54} +500 q^{-55} +213 q^{-56} -259 q^{-57} -457 q^{-58} -237 q^{-59} +179 q^{-60} +408 q^{-61} +264 q^{-62} -101 q^{-63} -345 q^{-64} -272 q^{-65} +15 q^{-66} +264 q^{-67} +274 q^{-68} +55 q^{-69} -177 q^{-70} -244 q^{-71} -111 q^{-72} +87 q^{-73} +194 q^{-74} +142 q^{-75} -10 q^{-76} -132 q^{-77} -137 q^{-78} -45 q^{-79} +60 q^{-80} +113 q^{-81} +74 q^{-82} -9 q^{-83} -68 q^{-84} -74 q^{-85} -28 q^{-86} +28 q^{-87} +54 q^{-88} +41 q^{-89} +4 q^{-90} -32 q^{-91} -36 q^{-92} -14 q^{-93} +7 q^{-94} +23 q^{-95} +20 q^{-96} + q^{-97} -13 q^{-98} -10 q^{-99} -5 q^{-100} +9 q^{-102} +5 q^{-103} -2 q^{-104} -2 q^{-105} - q^{-106} -2 q^{-107} + q^{-108} +2 q^{-109} - q^{-110} }
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 q^3+q^2-3 q-2+ q^{-2} +2 q^{-3} +9 q^{-4} +12 q^{-5} -9 q^{-6} -29 q^{-7} -21 q^{-8} -3 q^{-9} +12 q^{-10} +64 q^{-11} +80 q^{-12} +12 q^{-13} -96 q^{-14} -144 q^{-15} -105 q^{-16} -30 q^{-17} +189 q^{-18} +329 q^{-19} +209 q^{-20} -109 q^{-21} -385 q^{-22} -456 q^{-23} -320 q^{-24} +241 q^{-25} +743 q^{-26} +736 q^{-27} +164 q^{-28} -552 q^{-29} -989 q^{-30} -969 q^{-31} -14 q^{-32} +1061 q^{-33} +1437 q^{-34} +780 q^{-35} -406 q^{-36} -1393 q^{-37} -1736 q^{-38} -569 q^{-39} +1049 q^{-40} +1957 q^{-41} +1449 q^{-42} +7 q^{-43} -1459 q^{-44} -2262 q^{-45} -1126 q^{-46} +778 q^{-47} +2124 q^{-48} +1861 q^{-49} +417 q^{-50} -1283 q^{-51} -2432 q^{-52} -1447 q^{-53} +486 q^{-54} +2046 q^{-55} +1969 q^{-56} +654 q^{-57} -1066 q^{-58} -2371 q^{-59} -1536 q^{-60} +291 q^{-61} +1886 q^{-62} +1902 q^{-63} +755 q^{-64} -874 q^{-65} -2210 q^{-66} -1524 q^{-67} +129 q^{-68} +1679 q^{-69} +1773 q^{-70} +846 q^{-71} -640 q^{-72} -1974 q^{-73} -1505 q^{-74} -108 q^{-75} +1368 q^{-76} +1599 q^{-77} +993 q^{-78} -282 q^{-79} -1617 q^{-80} -1467 q^{-81} -445 q^{-82} +899 q^{-83} +1317 q^{-84} +1134 q^{-85} +183 q^{-86} -1086 q^{-87} -1302 q^{-88} -766 q^{-89} +312 q^{-90} +848 q^{-91} +1107 q^{-92} +600 q^{-93} -430 q^{-94} -901 q^{-95} -866 q^{-96} -208 q^{-97} +247 q^{-98} +788 q^{-99} +741 q^{-100} +132 q^{-101} -340 q^{-102} -623 q^{-103} -415 q^{-104} -240 q^{-105} +291 q^{-106} +515 q^{-107} +348 q^{-108} +98 q^{-109} -203 q^{-110} -256 q^{-111} -373 q^{-112} -74 q^{-113} +144 q^{-114} +215 q^{-115} +199 q^{-116} +72 q^{-117} +7 q^{-118} -209 q^{-119} -135 q^{-120} -63 q^{-121} +17 q^{-122} +73 q^{-123} +90 q^{-124} +110 q^{-125} -36 q^{-126} -38 q^{-127} -60 q^{-128} -42 q^{-129} -24 q^{-130} +14 q^{-131} +68 q^{-132} +11 q^{-133} +15 q^{-134} -9 q^{-135} -15 q^{-136} -27 q^{-137} -13 q^{-138} +18 q^{-139} +2 q^{-140} +11 q^{-141} +4 q^{-142} +3 q^{-143} -9 q^{-144} -7 q^{-145} +4 q^{-146} -2 q^{-147} +2 q^{-148} + q^{-149} +2 q^{-150} - q^{-151} -2 q^{-152} + q^{-153} }
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 2 q^5-2 q^4-2 q^2-3 q+1+6 q^{-1} +7 q^{-2} +6 q^{-3} -2 q^{-4} -8 q^{-5} -18 q^{-6} -34 q^{-7} -16 q^{-8} +29 q^{-9} +65 q^{-10} +73 q^{-11} +46 q^{-12} -11 q^{-13} -101 q^{-14} -202 q^{-15} -199 q^{-16} -19 q^{-17} +190 q^{-18} +369 q^{-19} +405 q^{-20} +224 q^{-21} -147 q^{-22} -632 q^{-23} -869 q^{-24} -589 q^{-25} +54 q^{-26} +846 q^{-27} +1385 q^{-28} +1241 q^{-29} +400 q^{-30} -963 q^{-31} -2082 q^{-32} -2135 q^{-33} -1084 q^{-34} +838 q^{-35} +2644 q^{-36} +3207 q^{-37} +2149 q^{-38} -366 q^{-39} -3085 q^{-40} -4318 q^{-41} -3411 q^{-42} -412 q^{-43} +3201 q^{-44} +5271 q^{-45} +4772 q^{-46} +1462 q^{-47} -2997 q^{-48} -5978 q^{-49} -6027 q^{-50} -2620 q^{-51} +2505 q^{-52} +6357 q^{-53} +7045 q^{-54} +3739 q^{-55} -1857 q^{-56} -6403 q^{-57} -7743 q^{-58} -4699 q^{-59} +1135 q^{-60} +6235 q^{-61} +8154 q^{-62} +5394 q^{-63} -510 q^{-64} -5911 q^{-65} -8262 q^{-66} -5857 q^{-67} -27 q^{-68} +5562 q^{-69} +8234 q^{-70} +6087 q^{-71} +373 q^{-72} -5224 q^{-73} -8059 q^{-74} -6166 q^{-75} -628 q^{-76} +4927 q^{-77} +7873 q^{-78} +6152 q^{-79} +773 q^{-80} -4673 q^{-81} -7635 q^{-82} -6090 q^{-83} -914 q^{-84} +4409 q^{-85} +7408 q^{-86} +6027 q^{-87} +1058 q^{-88} -4113 q^{-89} -7122 q^{-90} -5982 q^{-91} -1282 q^{-92} +3734 q^{-93} +6811 q^{-94} +5939 q^{-95} +1572 q^{-96} -3231 q^{-97} -6397 q^{-98} -5908 q^{-99} -1973 q^{-100} +2617 q^{-101} +5884 q^{-102} +5825 q^{-103} +2418 q^{-104} -1845 q^{-105} -5209 q^{-106} -5683 q^{-107} -2900 q^{-108} +974 q^{-109} +4399 q^{-110} +5383 q^{-111} +3312 q^{-112} -29 q^{-113} -3407 q^{-114} -4897 q^{-115} -3620 q^{-116} -892 q^{-117} +2314 q^{-118} +4207 q^{-119} +3675 q^{-120} +1685 q^{-121} -1153 q^{-122} -3295 q^{-123} -3471 q^{-124} -2273 q^{-125} +84 q^{-126} +2259 q^{-127} +2971 q^{-128} +2515 q^{-129} +799 q^{-130} -1181 q^{-131} -2230 q^{-132} -2421 q^{-133} -1390 q^{-134} +237 q^{-135} +1378 q^{-136} +2015 q^{-137} +1592 q^{-138} +460 q^{-139} -531 q^{-140} -1398 q^{-141} -1480 q^{-142} -847 q^{-143} -114 q^{-144} +745 q^{-145} +1108 q^{-146} +892 q^{-147} +516 q^{-148} -180 q^{-149} -650 q^{-150} -709 q^{-151} -637 q^{-152} -181 q^{-153} +233 q^{-154} +421 q^{-155} +540 q^{-156} +318 q^{-157} +48 q^{-158} -124 q^{-159} -348 q^{-160} -310 q^{-161} -168 q^{-162} -49 q^{-163} +158 q^{-164} +181 q^{-165} +159 q^{-166} +145 q^{-167} -10 q^{-168} -86 q^{-169} -111 q^{-170} -125 q^{-171} -33 q^{-172} -2 q^{-173} +26 q^{-174} +92 q^{-175} +56 q^{-176} +30 q^{-177} -3 q^{-178} -47 q^{-179} -24 q^{-180} -26 q^{-181} -26 q^{-182} +11 q^{-183} +18 q^{-184} +23 q^{-185} +17 q^{-186} -9 q^{-187} -4 q^{-189} -13 q^{-190} -4 q^{-191} -2 q^{-192} +6 q^{-193} +7 q^{-194} -2 q^{-195} +2 q^{-197} -2 q^{-198} - q^{-199} -2 q^{-200} + q^{-201} +2 q^{-202} - q^{-203} }

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 45]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[7, 14, 8, 15], X[18, 15, 1, 16], X[16, 11, 17, 12], X[12, 17, 13, 18], X[13, 6, 14, 7]]
In[3]:=	GaussCode[Knot[9, 45]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6]
In[4]:=	DTCode[Knot[9, 45]]
Out[4]=	DTCode[4, 8, 10, -14, 2, 16, -6, 18, 12]
In[5]:=	br = BR[Knot[9, 45]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 45]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 45]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[9, 45]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[9, 45]][t]
Out[10]=	-2 6 2 -9 - t + - + 6 t - t t
In[11]:=	Conway[Knot[9, 45]][z]
Out[11]=	2 4 1 + 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 45]}
In[13]:=	{KnotDet[Knot[9, 45]], KnotSignature[Knot[9, 45]]}
Out[13]=	{23, -2}
In[14]:=	Jones[Knot[9, 45]][q]
Out[14]=	-8 2 3 4 4 4 3 2 -q + -- - -- + -- - -- + -- - -- + - 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 45]}
In[16]:=	A2Invariant[Knot[9, 45]][q]
Out[16]=	-26 -24 -22 -18 -16 -14 -10 -8 -6 2 -q - q + q + q + q - q - q + q + q + -- 2 q
In[17]:=	HOMFLYPT[Knot[9, 45]][a, z]
Out[17]=	2 4 6 8 2 2 4 2 6 2 4 4 2 a - 2 a + 2 a - a + 2 a z - 2 a z + 2 a z - a z
In[18]:=	Kauffman[Knot[9, 45]][a, z]
Out[18]=	2 4 6 8 7 9 2 2 4 2 -2 a - 2 a - 2 a - a + 2 a z + 2 a z + 3 a z + 6 a z + 6 2 8 2 3 3 5 3 7 3 9 3 4 4 7 a z + 4 a z + a z - a z - 5 a z - 3 a z - 4 a z - 6 4 8 4 3 5 9 5 4 6 6 6 8 6 10 a z - 6 a z + a z + a z + 2 a z + 4 a z + 2 a z + 5 7 7 7 a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 45]], Vassiliev[3][Knot[9, 45]]}
Out[19]=	{2, -4}
In[20]:=	Kh[Knot[9, 45]][q, t]
Out[20]=	-3 2 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q t q t q t q t q t q t q t 2 2 2 2 1 2 ----- + ----- + ----- + ----- + ---- + ---- 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 45], 2][q]
Out[21]=	-23 2 -21 6 4 6 12 3 13 15 -13 q - --- - q + --- - --- - --- + --- - --- - --- + --- + q - 22 20 19 18 17 16 15 14 q q q q q q q q 18 15 5 19 12 6 13 6 4 5 -2 1 --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + - 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q

See/edit the Rolfsen_Splice_Template.

9 45

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

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