9 12

9_11

9_13

Visit 9 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 12's page at Knotilus! Visit 9 12's page at the original Knot Atlas! 9 12 Quick Notes

9 12 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 10 16 14 2 18 8 6 12
Conway Notation	[4212]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 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 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 8.83664 A-Polynomial See Data:9 12/A-polynomial

[edit Notes for 9 12'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 2} Rasmussen s-Invariant -2

[edit Notes for 9 12'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 -2 t^2+9 t-13+9 t^{-1} -2 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 -2 z^4+z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 35, -2 }
Jones polynomial	$q-2+4q^{-1}-5q^{-2}+6q^{-3}-6q^{-4}+5q^{-5}-3q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+2z^{2}a^{6}+2a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+3z^{3}a^{7}-za^{7}+z^{8}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+4z^{7}a^{5}-11z^{5}a^{5}+13z^{3}a^{5}-4za^{5}+z^{8}a^{4}-z^{4}a^{4}+3z^{2}a^{4}-a^{4}+2z^{7}a^{3}-3z^{5}a^{3}+4z^{3}a^{3}-2za^{3}+2z^{6}a^{2}-z^{4}a^{2}-2z^{2}a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{26}-q^{24}+q^{22}+q^{18}+2q^{16}-q^{14}-q^{10}+q^{6}-q^{4}+2q^{2}+q^{-4}$
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}+2 q^{124}-3 q^{122}+2 q^{120}-2 q^{118}-2 q^{116}+7 q^{114}-10 q^{112}+10 q^{110}-10 q^{108}+4 q^{106}+4 q^{104}-15 q^{102}+20 q^{100}-20 q^{98}+14 q^{96}-q^{94}-12 q^{92}+19 q^{90}-18 q^{88}+15 q^{86}-3 q^{84}-10 q^{82}+14 q^{80}-11 q^{78}+2 q^{76}+10 q^{74}-16 q^{72}+18 q^{70}-8 q^{68}-4 q^{66}+17 q^{64}-26 q^{62}+29 q^{60}-21 q^{58}+6 q^{56}+11 q^{54}-23 q^{52}+30 q^{50}-25 q^{48}+13 q^{46}-13 q^{42}+16 q^{40}-14 q^{38}+3 q^{36}+8 q^{34}-13 q^{32}+10 q^{30}-2 q^{28}-9 q^{26}+17 q^{24}-19 q^{22}+15 q^{20}-6 q^{18}-5 q^{16}+14 q^{14}-16 q^{12}+17 q^{10}-10 q^8+5 q^6+q^4-7 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

Further Quantum Invariants

Further quantum knot invariants for 9_12.

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}+2 q^{11}-q^9+q^5-q^3+2 q- q^{-1} + q^{-3} }
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}-q^{44}+3 q^{42}-2 q^{40}-4 q^{38}+5 q^{36}+q^{34}-6 q^{32}+5 q^{30}+3 q^{28}-7 q^{26}+2 q^{24}+3 q^{22}-3 q^{20}-2 q^{18}+2 q^{16}+4 q^{14}-5 q^{12}+7 q^8-5 q^6-2 q^4+7 q^2-2-2 q^{-2} +3 q^{-4} - q^{-6} - q^{-8} + q^{-10} }
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}+q^{89}-q^{87}-2 q^{85}+2 q^{83}+5 q^{81}-2 q^{79}-8 q^{77}+10 q^{73}+3 q^{71}-12 q^{69}-10 q^{67}+12 q^{65}+15 q^{63}-6 q^{61}-19 q^{59}+3 q^{57}+21 q^{55}+2 q^{53}-21 q^{51}-7 q^{49}+19 q^{47}+7 q^{45}-14 q^{43}-9 q^{41}+9 q^{39}+10 q^{37}-5 q^{35}-10 q^{33}-3 q^{31}+11 q^{29}+10 q^{27}-10 q^{25}-16 q^{23}+9 q^{21}+20 q^{19}-5 q^{17}-21 q^{15}+q^{13}+20 q^{11}+4 q^9-15 q^7-5 q^5+11 q^3+5 q-5 q^{-1} -4 q^{-3} +3 q^{-5} +2 q^{-7} -2 q^{-9} +2 q^{-13} - q^{-17} - q^{-19} + q^{-21} }
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}-q^{148}+q^{146}+2 q^{142}-4 q^{140}-3 q^{138}+4 q^{136}+3 q^{134}+8 q^{132}-8 q^{130}-13 q^{128}+2 q^{126}+8 q^{124}+23 q^{122}-4 q^{120}-25 q^{118}-17 q^{116}-4 q^{114}+40 q^{112}+23 q^{110}-12 q^{108}-36 q^{106}-41 q^{104}+28 q^{102}+50 q^{100}+33 q^{98}-24 q^{96}-77 q^{94}-14 q^{92}+44 q^{90}+69 q^{88}+12 q^{86}-80 q^{84}-49 q^{82}+16 q^{80}+74 q^{78}+38 q^{76}-56 q^{74}-51 q^{72}-5 q^{70}+52 q^{68}+39 q^{66}-25 q^{64}-39 q^{62}-15 q^{60}+29 q^{58}+32 q^{56}+3 q^{54}-30 q^{52}-29 q^{50}+31 q^{46}+43 q^{44}-17 q^{42}-50 q^{40}-36 q^{38}+24 q^{36}+79 q^{34}+13 q^{32}-50 q^{30}-68 q^{28}-5 q^{26}+84 q^{24}+42 q^{22}-18 q^{20}-67 q^{18}-38 q^{16}+48 q^{14}+42 q^{12}+18 q^{10}-33 q^8-40 q^6+10 q^4+14 q^2+24- q^{-2} -18 q^{-4} -2 q^{-6} -5 q^{-8} +10 q^{-10} +4 q^{-12} -3 q^{-14} +3 q^{-16} -6 q^{-18} +4 q^{-26} - q^{-28} - q^{-32} - q^{-34} + q^{-36} }
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}+q^{221}-q^{219}+2 q^{211}+2 q^{209}-4 q^{207}-6 q^{205}-q^{203}+4 q^{201}+9 q^{199}+8 q^{197}-4 q^{195}-19 q^{193}-17 q^{191}+3 q^{189}+23 q^{187}+33 q^{185}+13 q^{183}-24 q^{181}-50 q^{179}-36 q^{177}+12 q^{175}+56 q^{173}+66 q^{171}+23 q^{169}-47 q^{167}-91 q^{165}-71 q^{163}+6 q^{161}+91 q^{159}+119 q^{157}+58 q^{155}-56 q^{153}-149 q^{151}-138 q^{149}-4 q^{147}+147 q^{145}+202 q^{143}+99 q^{141}-106 q^{139}-247 q^{137}-186 q^{135}+40 q^{133}+245 q^{131}+256 q^{129}+44 q^{127}-220 q^{125}-299 q^{123}-113 q^{121}+170 q^{119}+300 q^{117}+164 q^{115}-109 q^{113}-278 q^{111}-188 q^{109}+65 q^{107}+231 q^{105}+181 q^{103}-19 q^{101}-182 q^{99}-164 q^{97}-3 q^{95}+138 q^{93}+134 q^{91}+17 q^{89}-96 q^{87}-114 q^{85}-35 q^{83}+68 q^{81}+106 q^{79}+51 q^{77}-34 q^{75}-103 q^{73}-93 q^{71}+q^{69}+108 q^{67}+140 q^{65}+48 q^{63}-107 q^{61}-193 q^{59}-114 q^{57}+89 q^{55}+238 q^{53}+187 q^{51}-47 q^{49}-263 q^{47}-256 q^{45}-19 q^{43}+249 q^{41}+307 q^{39}+93 q^{37}-201 q^{35}-316 q^{33}-160 q^{31}+122 q^{29}+287 q^{27}+207 q^{25}-35 q^{23}-224 q^{21}-210 q^{19}-36 q^{17}+139 q^{15}+184 q^{13}+84 q^{11}-65 q^9-135 q^7-90 q^5+7 q^3+80 q+81 q^{-1} +26 q^{-3} -39 q^{-5} -57 q^{-7} -32 q^{-9} +8 q^{-11} +33 q^{-13} +28 q^{-15} +7 q^{-17} -15 q^{-19} -20 q^{-21} -8 q^{-23} +3 q^{-25} +9 q^{-27} +9 q^{-29} +4 q^{-31} -5 q^{-33} -6 q^{-35} -2 q^{-37} -2 q^{-39} +2 q^{-41} +4 q^{-43} + q^{-45} - q^{-47} - q^{-51} - q^{-53} + q^{-55} }

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}+2 q^{16}-q^{14}-q^{10}+q^6-q^4+2 q^2+ q^{-4} }
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^{68}-2 q^{66}+4 q^{64}-8 q^{62}+17 q^{60}-24 q^{58}+32 q^{56}-48 q^{54}+57 q^{52}-64 q^{50}+62 q^{48}-58 q^{46}+46 q^{44}-18 q^{42}-6 q^{40}+38 q^{38}-65 q^{36}+90 q^{34}-112 q^{32}+116 q^{30}-122 q^{28}+110 q^{26}-90 q^{24}+68 q^{22}-37 q^{20}+10 q^{18}+20 q^{16}-36 q^{14}+47 q^{12}-54 q^{10}+56 q^8-48 q^6+42 q^4-36 q^2+30-20 q^{-2} +15 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }
2,0	$q^{66}+q^{64}-2q^{60}-q^{58}+q^{56}-q^{54}-3q^{52}-q^{50}+4q^{48}+3q^{46}-2q^{44}+4q^{40}-5q^{36}-3q^{34}+2q^{32}-2q^{28}+2q^{26}+q^{24}-q^{22}+3q^{20}+2q^{18}-3q^{16}-q^{14}+4q^{12}+q^{10}-4q^{8}+6q^{4}-3+q^{-2}+2q^{-4}-q^{-8}+q^{-12}$

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^{48}-3 q^{46}+q^{44}+2 q^{42}-5 q^{40}+2 q^{38}+4 q^{36}-6 q^{34}+q^{32}+4 q^{30}-3 q^{28}+3 q^{24}+q^{22}-q^{20}-q^{18}+4 q^{16}-2 q^{14}-4 q^{12}+6 q^{10}-q^8-5 q^6+5 q^4+q^2-2+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }
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}+2 q^{21}-q^{19}-q^{15}-q^{13}+q^7-q^5+2 q^3+ q^{-1} + q^{-5} }

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}-2 q^{50}+3 q^{48}-5 q^{46}+5 q^{44}-6 q^{42}+5 q^{40}-4 q^{38}+4 q^{36}-q^{32}+6 q^{30}-7 q^{28}+10 q^{26}-11 q^{24}+11 q^{22}-11 q^{20}+7 q^{18}-6 q^{16}+2 q^{14}-2 q^{10}+5 q^8-5 q^6+7 q^4-5 q^2+4-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }
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}+q^{80}+2 q^{78}-q^{76}-4 q^{74}-q^{72}+4 q^{70}+4 q^{68}-3 q^{66}-6 q^{64}+6 q^{60}+4 q^{58}-5 q^{56}-5 q^{54}+q^{52}+5 q^{50}-4 q^{46}-q^{44}+4 q^{42}+2 q^{40}-3 q^{38}-2 q^{36}+3 q^{34}+4 q^{32}-2 q^{30}-4 q^{28}+q^{26}+5 q^{24}-5 q^{20}-2 q^{18}+5 q^{16}+5 q^{14}-3 q^{12}-6 q^{10}-q^8+6 q^6+3 q^4-2 q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }

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^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-2 q^{118}-2 q^{116}+7 q^{114}-10 q^{112}+10 q^{110}-10 q^{108}+4 q^{106}+4 q^{104}-15 q^{102}+20 q^{100}-20 q^{98}+14 q^{96}-q^{94}-12 q^{92}+19 q^{90}-18 q^{88}+15 q^{86}-3 q^{84}-10 q^{82}+14 q^{80}-11 q^{78}+2 q^{76}+10 q^{74}-16 q^{72}+18 q^{70}-8 q^{68}-4 q^{66}+17 q^{64}-26 q^{62}+29 q^{60}-21 q^{58}+6 q^{56}+11 q^{54}-23 q^{52}+30 q^{50}-25 q^{48}+13 q^{46}-13 q^{42}+16 q^{40}-14 q^{38}+3 q^{36}+8 q^{34}-13 q^{32}+10 q^{30}-2 q^{28}-9 q^{26}+17 q^{24}-19 q^{22}+15 q^{20}-6 q^{18}-5 q^{16}+14 q^{14}-16 q^{12}+17 q^{10}-10 q^8+5 q^6+q^4-7 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

.

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 12"];

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 -2 t^2+9 t-13+9 t^{-1} -2 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 -2 z^4+z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+2z^{2}a^{6}+2a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+3z^{3}a^{7}-za^{7}+z^{8}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+4z^{7}a^{5}-11z^{5}a^{5}+13z^{3}a^{5}-4za^{5}+z^{8}a^{4}-z^{4}a^{4}+3z^{2}a^{4}-a^{4}+2z^{7}a^{3}-3z^{5}a^{3}+4z^{3}a^{3}-2za^{3}+2z^{6}a^{2}-z^{4}a^{2}-2z^{2}a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$

Vassiliev invariants

V2 and V3:

(1, -3)

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 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 -24} 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 \frac{302}{3}} ${\frac {58}{3}}$ 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 -400} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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 -120} 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{32}{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 288} 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{1208}{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{232}{3}} ${\frac {49471}{30}}$ 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{5942}{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{49622}{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{1793}{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{4831}{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 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 $j-2r=s+1$ 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 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 1   -1 -1               3 1   2 -3             3 2     -1 -5           3 2       1 -7         3 3         0 -9       2 3           -1 -11     1 3             2 -13   1 2               -1 -15   1                 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} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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}^{3}\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}^{3}\oplus{\mathbb Z}_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 {\mathbb Z}^{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 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}^{3}\oplus{\mathbb Z}_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 {\mathbb Z}^{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 r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_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 {\mathbb Z}^{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 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}^{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}^{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 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} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_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}}

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, 12]]
Out[2]=	9
In[3]:=	PD[Knot[9, 12]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 16, 6, 17], X[11, 1, 12, 18], X[17, 13, 18, 12], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[9, 12]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4]
In[5]:=	BR[Knot[9, 12]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, -3, -4, 3, -4}]
In[6]:=	alex = Alexander[Knot[9, 12]][t]
Out[6]=	2 9 2 -13 - -- + - + 9 t - 2 t 2 t t
In[7]:=	Conway[Knot[9, 12]][z]
Out[7]=	2 4 1 + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 12], Knot[11, NonAlternating, 84]}
In[9]:=	{KnotDet[Knot[9, 12]], KnotSignature[Knot[9, 12]]}
Out[9]=	{35, -2}
In[10]:=	J=Jones[Knot[9, 12]][q]
Out[10]=	-8 2 3 5 6 6 5 4 -2 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 12], Knot[11, NonAlternating, 15]}
In[12]:=	A2Invariant[Knot[9, 12]][q]
Out[12]=	-26 -24 -22 -18 2 -14 -10 -6 -4 2 4 -q - q + q + q + --- - q - q + q - q + -- + q 16 2 q q
In[13]:=	Kauffman[Knot[9, 12]][a, z]
Out[13]=	4 6 8 3 5 7 9 2 2 2 1 - a - 2 a - a - 2 a z - 4 a z - a z + a z - 2 z - 2 a z + 4 2 6 2 8 2 3 3 3 5 3 7 3 3 a z + 7 a z + 4 a z - 3 a z + 4 a z + 13 a z + 3 a z - 9 3 4 2 4 4 4 6 4 8 4 5 3 5 3 a z + z - a z - a z - 5 a z - 6 a z + 2 a z - 3 a z - 5 5 7 5 9 5 2 6 8 6 3 7 5 7 11 a z - 5 a z + a z + 2 a z + 2 a z + 2 a z + 4 a z + 7 7 4 8 6 8 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 12]], Vassiliev[3][Knot[9, 12]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[9, 12]][q, t]
Out[15]=	2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 3 3 3 3 2 3 t 3 2 ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t