9 22

9_21

9_23

Visit 9 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 22's page at Knotilus! Visit 9 22's page at the original Knot Atlas! 9 22 Quick Notes

9 22 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17
Gauss code	1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code	4 8 10 14 2 16 18 6 12
Conway Notation	[211,3,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,7\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-8] Hyperbolic Volume 10.6207 A-Polynomial See Data:9 22/A-polynomial

[edit Notes for 9 22'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{ 3 }[/math] Rasmussen s-Invariant -2

[edit Notes for 9 22's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+10 t-11+10 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 43, 2 }
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-5 q^4+7 q^3-7 q^2+7 q-6+4 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+6 z^2 a^{-2} -2 z^2 a^{-4} -6 z^2+2 a^2+4 a^{-2} - a^{-4} -4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} +a^2 z^6+7 z^6 a^{-2} +6 z^6 a^{-4} +2 z^6-7 a z^5-16 z^5 a^{-1} -4 z^5 a^{-3} +5 z^5 a^{-5} -4 a^2 z^4-23 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} -15 z^4+7 a z^3+10 z^3 a^{-1} -2 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +5 a^2 z^2+17 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} +16 z^2-2 a z-2 z a^{-1} +z a^{-3} +z a^{-5} -2 a^2-4 a^{-2} - a^{-4} -4 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}+q^8+q^4-2 q^2-1- q^{-4} +3 q^{-6} +2 q^{-10} - q^{-14} + q^{-16} - q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-3 q^{36}-3 q^{34}+14 q^{32}-20 q^{30}+25 q^{28}-17 q^{26}+2 q^{24}+19 q^{22}-35 q^{20}+43 q^{18}-35 q^{16}+14 q^{14}+11 q^{12}-35 q^{10}+41 q^8-32 q^6+10 q^4+12 q^2-28+24 q^{-2} -14 q^{-4} -11 q^{-6} +29 q^{-8} -38 q^{-10} +31 q^{-12} -9 q^{-14} -20 q^{-16} +46 q^{-18} -56 q^{-20} +50 q^{-22} -25 q^{-24} -5 q^{-26} +37 q^{-28} -52 q^{-30} +54 q^{-32} -30 q^{-34} +5 q^{-36} +23 q^{-38} -34 q^{-40} +28 q^{-42} -9 q^{-44} -11 q^{-46} +25 q^{-48} -25 q^{-50} +12 q^{-52} +8 q^{-54} -28 q^{-56} +36 q^{-58} -33 q^{-60} +17 q^{-62} +2 q^{-64} -21 q^{-66} +28 q^{-68} -29 q^{-70} +24 q^{-72} -11 q^{-74} +8 q^{-78} -14 q^{-80} +14 q^{-82} -11 q^{-84} +8 q^{-86} -2 q^{-88} - q^{-90} +2 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_22.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^7-q^5+2 q^3-2 q+ q^{-1} +2 q^{-7} -2 q^{-9} +2 q^{-11} - q^{-13} }[/math]
2	[math]\displaystyle{ q^{22}-q^{20}-2 q^{18}+4 q^{16}-7 q^{12}+6 q^{10}+5 q^8-10 q^6+3 q^4+9 q^2-8-2 q^{-2} +8 q^{-4} -2 q^{-6} -5 q^{-8} +3 q^{-10} +5 q^{-12} -5 q^{-14} -5 q^{-16} +10 q^{-18} -2 q^{-20} -9 q^{-22} +10 q^{-24} + q^{-26} -7 q^{-28} +4 q^{-30} -2 q^{-34} + q^{-36} }[/math]
3	[math]\displaystyle{ q^{45}-q^{43}-2 q^{41}+5 q^{37}+2 q^{35}-8 q^{33}-7 q^{31}+10 q^{29}+15 q^{27}-7 q^{25}-24 q^{23}+29 q^{19}+12 q^{17}-29 q^{15}-25 q^{13}+25 q^{11}+33 q^9-14 q^7-39 q^5+4 q^3+41 q+6 q^{-1} -37 q^{-3} -14 q^{-5} +33 q^{-7} +21 q^{-9} -26 q^{-11} -25 q^{-13} +19 q^{-15} +27 q^{-17} -6 q^{-19} -29 q^{-21} -7 q^{-23} +26 q^{-25} +21 q^{-27} -21 q^{-29} -34 q^{-31} +12 q^{-33} +41 q^{-35} - q^{-37} -42 q^{-39} -5 q^{-41} +35 q^{-43} +10 q^{-45} -25 q^{-47} -10 q^{-49} +16 q^{-51} +7 q^{-53} -10 q^{-55} -2 q^{-57} +3 q^{-59} + q^{-61} -2 q^{-63} +2 q^{-67} - q^{-69} }[/math]
4	[math]\displaystyle{ q^{76}-q^{74}-2 q^{72}+q^{68}+7 q^{66}-8 q^{62}-8 q^{60}-5 q^{58}+22 q^{56}+18 q^{54}-5 q^{52}-28 q^{50}-41 q^{48}+19 q^{46}+51 q^{44}+44 q^{42}-13 q^{40}-95 q^{38}-46 q^{36}+32 q^{34}+108 q^{32}+78 q^{30}-79 q^{28}-121 q^{26}-74 q^{24}+89 q^{22}+172 q^{20}+30 q^{18}-106 q^{16}-177 q^{14}-20 q^{12}+172 q^{10}+138 q^8-13 q^6-194 q^4-119 q^2+99+172 q^{-2} +70 q^{-4} -149 q^{-6} -156 q^{-8} +29 q^{-10} +160 q^{-12} +102 q^{-14} -102 q^{-16} -152 q^{-18} -16 q^{-20} +133 q^{-22} +114 q^{-24} -44 q^{-26} -137 q^{-28} -75 q^{-30} +81 q^{-32} +126 q^{-34} +53 q^{-36} -87 q^{-38} -148 q^{-40} -30 q^{-42} +100 q^{-44} +171 q^{-46} +25 q^{-48} -171 q^{-50} -152 q^{-52} +4 q^{-54} +212 q^{-56} +142 q^{-58} -100 q^{-60} -184 q^{-62} -95 q^{-64} +142 q^{-66} +159 q^{-68} -8 q^{-70} -106 q^{-72} -107 q^{-74} +48 q^{-76} +89 q^{-78} +18 q^{-80} -28 q^{-82} -56 q^{-84} +10 q^{-86} +26 q^{-88} +4 q^{-90} +2 q^{-92} -17 q^{-94} +3 q^{-96} +5 q^{-98} -2 q^{-100} +3 q^{-102} -3 q^{-104} +2 q^{-106} -2 q^{-110} + q^{-112} }[/math]
5	[math]\displaystyle{ q^{115}-q^{113}-2 q^{111}+q^{107}+3 q^{105}+5 q^{103}-10 q^{99}-10 q^{97}-3 q^{95}+9 q^{93}+24 q^{91}+21 q^{89}-8 q^{87}-41 q^{85}-47 q^{83}-16 q^{81}+44 q^{79}+90 q^{77}+69 q^{75}-20 q^{73}-120 q^{71}-146 q^{69}-57 q^{67}+105 q^{65}+222 q^{63}+186 q^{61}-16 q^{59}-251 q^{57}-324 q^{55}-155 q^{53}+173 q^{51}+423 q^{49}+375 q^{47}+12 q^{45}-413 q^{43}-563 q^{41}-286 q^{39}+257 q^{37}+662 q^{35}+579 q^{33}+12 q^{31}-616 q^{29}-795 q^{27}-352 q^{25}+419 q^{23}+903 q^{21}+668 q^{19}-136 q^{17}-856 q^{15}-894 q^{13}-189 q^{11}+700 q^9+1010 q^7+464 q^5-482 q^3-1003 q-655 q^{-1} +255 q^{-3} +925 q^{-5} +753 q^{-7} -78 q^{-9} -803 q^{-11} -765 q^{-13} -37 q^{-15} +679 q^{-17} +733 q^{-19} +97 q^{-21} -587 q^{-23} -679 q^{-25} -126 q^{-27} +520 q^{-29} +642 q^{-31} +153 q^{-33} -461 q^{-35} -629 q^{-37} -222 q^{-39} +383 q^{-41} +638 q^{-43} +332 q^{-45} -242 q^{-47} -624 q^{-49} -507 q^{-51} +28 q^{-53} +568 q^{-55} +687 q^{-57} +264 q^{-59} -413 q^{-61} -829 q^{-63} -605 q^{-65} +162 q^{-67} +876 q^{-69} +912 q^{-71} +168 q^{-73} -782 q^{-75} -1123 q^{-77} -518 q^{-79} +571 q^{-81} +1183 q^{-83} +780 q^{-85} -275 q^{-87} -1073 q^{-89} -926 q^{-91} -8 q^{-93} +845 q^{-95} +911 q^{-97} +213 q^{-99} -565 q^{-101} -768 q^{-103} -311 q^{-105} +314 q^{-107} +568 q^{-109} +309 q^{-111} -146 q^{-113} -362 q^{-115} -237 q^{-117} +40 q^{-119} +203 q^{-121} +162 q^{-123} -2 q^{-125} -110 q^{-127} -82 q^{-129} -9 q^{-131} +43 q^{-133} +45 q^{-135} +8 q^{-137} -21 q^{-139} -19 q^{-141} +8 q^{-145} +4 q^{-147} +2 q^{-149} - q^{-151} -4 q^{-153} - q^{-155} +3 q^{-157} -2 q^{-159} +2 q^{-163} - q^{-165} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{10}+q^8+q^4-2 q^2-1- q^{-4} +3 q^{-6} +2 q^{-10} - q^{-14} + q^{-16} - q^{-18} }[/math]
1,1	[math]\displaystyle{ q^{28}-2 q^{26}+8 q^{24}-16 q^{22}+31 q^{20}-54 q^{18}+78 q^{16}-106 q^{14}+126 q^{12}-140 q^{10}+138 q^8-110 q^6+76 q^4-16 q^2-44+112 q^{-2} -171 q^{-4} +214 q^{-6} -248 q^{-8} +250 q^{-10} -236 q^{-12} +200 q^{-14} -148 q^{-16} +90 q^{-18} -24 q^{-20} -26 q^{-22} +70 q^{-24} -96 q^{-26} +108 q^{-28} -108 q^{-30} +98 q^{-32} -86 q^{-34} +70 q^{-36} -54 q^{-38} +42 q^{-40} -32 q^{-42} +20 q^{-44} -12 q^{-46} +8 q^{-48} -4 q^{-50} + q^{-52} }[/math]
2,0	[math]\displaystyle{ q^{28}+q^{26}-2 q^{22}+2 q^{18}-q^{16}-5 q^{14}-q^{12}+4 q^{10}+2 q^8-q^6+4 q^4+7 q^2-1-3 q^{-2} + q^{-4} -2 q^{-6} -5 q^{-8} + q^{-12} -2 q^{-14} +6 q^{-18} + q^{-20} -5 q^{-22} +3 q^{-24} +5 q^{-26} -2 q^{-28} -3 q^{-30} +3 q^{-32} + q^{-34} -2 q^{-36} -2 q^{-38} + q^{-40} - q^{-44} + q^{-46} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}-q^{18}+2 q^{16}+2 q^{14}-3 q^{12}+4 q^{10}+q^8-7 q^6+5 q^4-8+6 q^{-2} +2 q^{-4} -6 q^{-6} +3 q^{-8} +4 q^{-10} - q^{-14} +2 q^{-16} +5 q^{-18} -5 q^{-20} -2 q^{-22} +8 q^{-24} -7 q^{-26} -3 q^{-28} +8 q^{-30} -4 q^{-32} -3 q^{-34} +5 q^{-36} - q^{-38} -2 q^{-40} + q^{-42} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^{11}+2 q^9+q^5-3 q^3-q-3 q^{-1} +2 q^{-7} +3 q^{-9} + q^{-11} +2 q^{-13} - q^{-15} + q^{-17} -2 q^{-19} + q^{-21} - q^{-23} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{26}+q^{24}+q^{22}+2 q^{20}+2 q^{18}-q^{14}-2 q^{10}-6 q^8-q^6+4 q^4-2 q^2-1+9 q^{-2} +6 q^{-4} -5 q^{-6} -2 q^{-8} +4 q^{-10} -4 q^{-12} -8 q^{-14} +3 q^{-16} +6 q^{-18} -4 q^{-20} +4 q^{-22} +9 q^{-24} -2 q^{-26} -4 q^{-28} +5 q^{-30} + q^{-32} -7 q^{-34} - q^{-36} +5 q^{-38} - q^{-40} -5 q^{-42} +2 q^{-44} +3 q^{-46} -2 q^{-48} - q^{-50} + q^{-52} }[/math]
1,0,0,0	[math]\displaystyle{ q^{16}+q^{14}+2 q^{12}+2 q^{10}+q^6-3 q^4-2 q^2-3-3 q^{-2} +3 q^{-8} +2 q^{-10} +4 q^{-12} + q^{-14} +2 q^{-16} - q^{-18} -2 q^{-24} + q^{-26} - q^{-28} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{20}-q^{18}+4 q^{16}-4 q^{14}+7 q^{12}-8 q^{10}+9 q^8-9 q^6+7 q^4-6 q^2+2 q^{-2} -8 q^{-4} +12 q^{-6} -15 q^{-8} +18 q^{-10} -16 q^{-12} +17 q^{-14} -12 q^{-16} +9 q^{-18} -3 q^{-20} +4 q^{-24} -7 q^{-26} +9 q^{-28} -10 q^{-30} +8 q^{-32} -7 q^{-34} +5 q^{-36} -3 q^{-38} +2 q^{-40} - q^{-42} }[/math]
1,0	[math]\displaystyle{ q^{34}-q^{30}-q^{28}+3 q^{26}+3 q^{24}-2 q^{22}-5 q^{20}+7 q^{16}+4 q^{14}-7 q^{12}-8 q^{10}+3 q^8+10 q^6+2 q^4-10 q^2-5+6 q^{-2} +8 q^{-4} -2 q^{-6} -7 q^{-8} +6 q^{-12} + q^{-14} -6 q^{-16} - q^{-18} +6 q^{-20} +4 q^{-22} -5 q^{-24} -5 q^{-26} +4 q^{-28} +8 q^{-30} -2 q^{-32} -8 q^{-34} - q^{-36} +9 q^{-38} +4 q^{-40} -7 q^{-42} -8 q^{-44} +3 q^{-46} +9 q^{-48} + q^{-50} -6 q^{-52} -5 q^{-54} +2 q^{-56} +5 q^{-58} + q^{-60} -2 q^{-62} -2 q^{-64} + q^{-68} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{26}-q^{24}+3 q^{22}-2 q^{20}+6 q^{18}-4 q^{16}+7 q^{14}-6 q^{12}+8 q^{10}-8 q^8+4 q^6-7 q^4+2 q^2-3-4 q^{-2} +3 q^{-4} -6 q^{-6} +9 q^{-8} -10 q^{-10} +15 q^{-12} -10 q^{-14} +16 q^{-16} -11 q^{-18} +13 q^{-20} -8 q^{-22} +9 q^{-24} -6 q^{-26} + q^{-28} - q^{-30} -2 q^{-32} +4 q^{-34} -7 q^{-36} +5 q^{-38} -7 q^{-40} +9 q^{-42} -6 q^{-44} +4 q^{-46} -5 q^{-48} +5 q^{-50} -2 q^{-52} + q^{-54} -2 q^{-56} + q^{-58} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-3 q^{36}-3 q^{34}+14 q^{32}-20 q^{30}+25 q^{28}-17 q^{26}+2 q^{24}+19 q^{22}-35 q^{20}+43 q^{18}-35 q^{16}+14 q^{14}+11 q^{12}-35 q^{10}+41 q^8-32 q^6+10 q^4+12 q^2-28+24 q^{-2} -14 q^{-4} -11 q^{-6} +29 q^{-8} -38 q^{-10} +31 q^{-12} -9 q^{-14} -20 q^{-16} +46 q^{-18} -56 q^{-20} +50 q^{-22} -25 q^{-24} -5 q^{-26} +37 q^{-28} -52 q^{-30} +54 q^{-32} -30 q^{-34} +5 q^{-36} +23 q^{-38} -34 q^{-40} +28 q^{-42} -9 q^{-44} -11 q^{-46} +25 q^{-48} -25 q^{-50} +12 q^{-52} +8 q^{-54} -28 q^{-56} +36 q^{-58} -33 q^{-60} +17 q^{-62} +2 q^{-64} -21 q^{-66} +28 q^{-68} -29 q^{-70} +24 q^{-72} -11 q^{-74} +8 q^{-78} -14 q^{-80} +14 q^{-82} -11 q^{-84} +8 q^{-86} -2 q^{-88} - q^{-90} +2 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/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 22"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-5 t^2+10 t-11+10 t^{-1} -5 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+z^4-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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} +a^2 z^6+7 z^6 a^{-2} +6 z^6 a^{-4} +2 z^6-7 a z^5-16 z^5 a^{-1} -4 z^5 a^{-3} +5 z^5 a^{-5} -4 a^2 z^4-23 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} -15 z^4+7 a z^3+10 z^3 a^{-1} -2 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +5 a^2 z^2+17 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} +16 z^2-2 a z-2 z a^{-1} +z a^{-3} +z a^{-5} -2 a^2-4 a^{-2} - a^{-4} -4 }[/math]

Vassiliev invariants

V2 and V3:

(-1, 1)

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{ -4 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ -\frac{10}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -\frac{208}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ -\frac{751}{30} }[/math] [math]\displaystyle{ -\frac{766}{5} }[/math] [math]\displaystyle{ \frac{7018}{45} }[/math] [math]\displaystyle{ -\frac{305}{18} }[/math] [math]\displaystyle{ \frac{1169}{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 22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               3 1   -2 7             4 2     2 5           3 3       0 3         4 4         0 1       3 4           1 -1     1 3             -2 -3   1 3               2 -5   1                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 22]]
Out[2]=	9
In[3]:=	PD[Knot[9, 22]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[16, 12, 17, 11], X[14, 7, 15, 8], X[6, 15, 7, 16], X[18, 14, 1, 13], X[12, 18, 13, 17]]
In[4]:=	GaussCode[Knot[9, 22]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -9, 8, -6, 7, -5, 9, -8]
In[5]:=	BR[Knot[9, 22]]
Out[5]=	BR[4, {-1, 2, -1, 2, -3, 2, 2, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 22]][t]
Out[6]=	-3 5 10 2 3 -11 + t - -- + -- + 10 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[9, 22]][z]
Out[7]=	2 4 6 1 - z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 22], Knot[11, NonAlternating, 128]}
In[9]:=	{KnotDet[Knot[9, 22]], KnotSignature[Knot[9, 22]]}
Out[9]=	{43, 2}
In[10]:=	J=Jones[Knot[9, 22]][q]
Out[10]=	-3 2 4 2 3 4 5 6 -6 + q - -- + - + 7 q - 7 q + 7 q - 5 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 22], Knot[11, NonAlternating, 3]}
In[12]:=	A2Invariant[Knot[9, 22]][q]
Out[12]=	-10 -8 -4 2 4 6 10 14 16 18 -1 + q + q + q - -- - q + 3 q + 2 q - q + q - q 2 q
In[13]:=	Kauffman[Knot[9, 22]][a, z]
Out[13]=	2 2 -4 4 2 z z 2 z 2 z 5 z -4 - a - -- - 2 a + -- + -- - --- - 2 a z + 16 z - -- + ---- + 2 5 3 a 6 4 a a a a a 2 3 3 3 3 4 17 z 2 2 z 4 z 2 z 10 z 3 4 3 z ----- + 5 a z + -- - ---- - ---- + ----- + 7 a z - 15 z + ---- - 2 7 5 3 a 6 a a a a a 4 4 5 5 5 6 9 z 23 z 2 4 5 z 4 z 16 z 5 6 6 z ---- - ----- - 4 a z + ---- - ---- - ----- - 7 a z + 2 z + ---- + 4 2 5 3 a 4 a a a a a 6 7 7 8 7 z 2 6 4 z 6 z 7 8 z ---- + a z + ---- + ---- + 2 a z + z + -- 2 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[9, 22]], Vassiliev[3][Knot[9, 22]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[9, 22]][q, t]
Out[15]=	3 1 1 1 3 1 3 3 q 4 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 4 q t + 3 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 11 4 13 5 2 q t + q t