9 33

9_32

9_34

Visit 9 33's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 33's page at Knotilus! Visit 9 33's page at the original Knot Atlas! 9 33 Quick Notes

9 33 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,8,13,7 X8394 X2,9,3,10 X18,13,1,14 X14,5,15,6 X6,17,7,18 X16,12,17,11 X10,16,11,15
Gauss code	1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5
Dowker-Thistlethwaite code	4 8 14 12 2 16 18 10 6
Conway Notation	[.21.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 13.2805 A-Polynomial See Data:9 33/A-polynomial

[edit Notes for 9 33'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 0

[edit Notes for 9 33's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+6 t^2-14 t+19-14 t^{-1} +6 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 61, 0 }
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+7 q^2-9 q+11-10 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+z^4 a^{-2} -3 z^4-a^4 z^2+4 a^2 z^2+z^2 a^{-2} -3 z^2-a^4+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^2 z^8+2 z^8+4 a^3 z^7+10 a z^7+6 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+7 z^6 a^{-2} +9 z^6+a^5 z^5-6 a^3 z^5-16 a z^5-5 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-16 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-2 a^5 z^3+a^3 z^3+5 a z^3-z^3 a^{-1} -3 z^3 a^{-3} +4 a^4 z^2+10 a^2 z^2+3 z^2 a^{-2} +9 z^2+a^5 z+a^3 z-a^4-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{12}-2 q^{10}+2 q^8+3 q^2-1+3 q^{-2} -2 q^{-4} + q^{-8} -2 q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-7 q^{70}-2 q^{68}+17 q^{66}-35 q^{64}+50 q^{62}-52 q^{60}+28 q^{58}+13 q^{56}-67 q^{54}+113 q^{52}-123 q^{50}+92 q^{48}-23 q^{46}-64 q^{44}+129 q^{42}-148 q^{40}+111 q^{38}-31 q^{36}-54 q^{34}+104 q^{32}-98 q^{30}+43 q^{28}+39 q^{26}-101 q^{24}+121 q^{22}-81 q^{20}-5 q^{18}+102 q^{16}-171 q^{14}+188 q^{12}-138 q^{10}+43 q^8+71 q^6-159 q^4+198 q^2-170+88 q^{-2} +17 q^{-4} -101 q^{-6} +132 q^{-8} -101 q^{-10} +29 q^{-12} +54 q^{-14} -99 q^{-16} +89 q^{-18} -33 q^{-20} -51 q^{-22} +119 q^{-24} -142 q^{-26} +110 q^{-28} -38 q^{-30} -44 q^{-32} +103 q^{-34} -124 q^{-36} +105 q^{-38} -58 q^{-40} +6 q^{-42} +31 q^{-44} -54 q^{-46} +53 q^{-48} -36 q^{-50} +20 q^{-52} -2 q^{-54} -6 q^{-56} +9 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_33.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-3 q^7+3 q^5-q^3+q+2 q^{-1} -2 q^{-3} +3 q^{-5} -3 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+8 q^{26}-6 q^{24}-11 q^{22}+19 q^{20}-q^{18}-24 q^{16}+19 q^{14}+10 q^{12}-23 q^{10}+7 q^8+14 q^6-9 q^4-7 q^2+10+10 q^{-2} -19 q^{-4} +24 q^{-8} -19 q^{-10} -10 q^{-12} +24 q^{-14} -8 q^{-16} -11 q^{-18} +10 q^{-20} -3 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+q^{59}-4 q^{57}-5 q^{55}+9 q^{53}+17 q^{51}-14 q^{49}-37 q^{47}+7 q^{45}+64 q^{43}+17 q^{41}-89 q^{39}-53 q^{37}+97 q^{35}+98 q^{33}-83 q^{31}-142 q^{29}+55 q^{27}+160 q^{25}-14 q^{23}-163 q^{21}-24 q^{19}+146 q^{17}+55 q^{15}-116 q^{13}-71 q^{11}+78 q^9+88 q^7-37 q^5-96 q^3-7 q+100 q^{-1} +55 q^{-3} -99 q^{-5} -100 q^{-7} +84 q^{-9} +144 q^{-11} -59 q^{-13} -163 q^{-15} +19 q^{-17} +164 q^{-19} +19 q^{-21} -141 q^{-23} -48 q^{-25} +103 q^{-27} +55 q^{-29} -58 q^{-31} -50 q^{-33} +26 q^{-35} +35 q^{-37} -11 q^{-39} -15 q^{-41} + q^{-43} +7 q^{-45} -3 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-q^{100}+4 q^{98}+q^{96}+2 q^{94}-15 q^{92}-11 q^{90}+22 q^{88}+29 q^{86}+29 q^{84}-62 q^{82}-100 q^{80}-q^{78}+114 q^{76}+209 q^{74}-22 q^{72}-287 q^{70}-263 q^{68}+41 q^{66}+528 q^{64}+357 q^{62}-249 q^{60}-678 q^{58}-460 q^{56}+546 q^{54}+887 q^{52}+285 q^{50}-737 q^{48}-1090 q^{46}+37 q^{44}+1015 q^{42}+925 q^{40}-283 q^{38}-1272 q^{36}-567 q^{34}+639 q^{32}+1143 q^{30}+245 q^{28}-957 q^{26}-813 q^{24}+157 q^{22}+933 q^{20}+515 q^{18}-486 q^{16}-764 q^{14}-200 q^{12}+594 q^{10}+626 q^8-30 q^6-653 q^4-533 q^2+214+742 q^{-2} +505 q^{-4} -478 q^{-6} -908 q^{-8} -311 q^{-10} +735 q^{-12} +1078 q^{-14} -47 q^{-16} -1054 q^{-18} -913 q^{-20} +347 q^{-22} +1318 q^{-24} +534 q^{-26} -683 q^{-28} -1152 q^{-30} -260 q^{-32} +938 q^{-34} +781 q^{-36} -51 q^{-38} -793 q^{-40} -535 q^{-42} +312 q^{-44} +507 q^{-46} +254 q^{-48} -261 q^{-50} -350 q^{-52} -11 q^{-54} +144 q^{-56} +170 q^{-58} -21 q^{-60} -107 q^{-62} -27 q^{-64} +7 q^{-66} +45 q^{-68} +6 q^{-70} -18 q^{-72} -3 q^{-74} -2 q^{-76} +7 q^{-78} -3 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+q^{151}-4 q^{149}-q^{147}+2 q^{145}+4 q^{143}+9 q^{141}+3 q^{139}-25 q^{137}-35 q^{135}-5 q^{133}+46 q^{131}+94 q^{129}+70 q^{127}-58 q^{125}-221 q^{123}-235 q^{121}-7 q^{119}+348 q^{117}+555 q^{115}+310 q^{113}-364 q^{111}-1004 q^{109}-932 q^{107}+45 q^{105}+1342 q^{103}+1869 q^{101}+852 q^{99}-1269 q^{97}-2883 q^{95}-2321 q^{93}+449 q^{91}+3483 q^{89}+4137 q^{87}+1278 q^{85}-3263 q^{83}-5786 q^{81}-3654 q^{79}+1968 q^{77}+6658 q^{75}+6189 q^{73}+321 q^{71}-6439 q^{69}-8261 q^{67}-3059 q^{65}+5080 q^{63}+9299 q^{61}+5719 q^{59}-2916 q^{57}-9238 q^{55}-7670 q^{53}+518 q^{51}+8149 q^{49}+8637 q^{47}+1662 q^{45}-6495 q^{43}-8639 q^{41}-3221 q^{39}+4654 q^{37}+7933 q^{35}+4089 q^{33}-2980 q^{31}-6839 q^{29}-4415 q^{27}+1618 q^{25}+5711 q^{23}+4441 q^{21}-578 q^{19}-4685 q^{17}-4449 q^{15}-358 q^{13}+3874 q^{11}+4636 q^9+1353 q^7-3141 q^5-5037 q^3-2629 q+2274 q^{-1} +5598 q^{-3} +4247 q^{-5} -1081 q^{-7} -6023 q^{-9} -6083 q^{-11} -652 q^{-13} +5991 q^{-15} +7924 q^{-17} +2837 q^{-19} -5227 q^{-21} -9225 q^{-23} -5283 q^{-25} +3573 q^{-27} +9657 q^{-29} +7474 q^{-31} -1241 q^{-33} -8894 q^{-35} -8890 q^{-37} -1354 q^{-39} +7044 q^{-41} +9129 q^{-43} +3583 q^{-45} -4489 q^{-47} -8184 q^{-49} -4898 q^{-51} +1895 q^{-53} +6290 q^{-55} +5125 q^{-57} +182 q^{-59} -4094 q^{-61} -4423 q^{-63} -1321 q^{-65} +2102 q^{-67} +3189 q^{-69} +1648 q^{-71} -711 q^{-73} -1951 q^{-75} -1400 q^{-77} +17 q^{-79} +967 q^{-81} +917 q^{-83} +237 q^{-85} -379 q^{-87} -521 q^{-89} -207 q^{-91} +124 q^{-93} +217 q^{-95} +129 q^{-97} -16 q^{-99} -86 q^{-101} -64 q^{-103} +4 q^{-105} +32 q^{-107} +16 q^{-109} - q^{-111} -6 q^{-113} -6 q^{-115} -2 q^{-117} +7 q^{-119} -3 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{12}-2 q^{10}+2 q^8+3 q^2-1+3 q^{-2} -2 q^{-4} + q^{-8} -2 q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+60 q^{36}-114 q^{34}+194 q^{32}-296 q^{30}+411 q^{28}-526 q^{26}+600 q^{24}-622 q^{22}+565 q^{20}-416 q^{18}+186 q^{16}+102 q^{14}-411 q^{12}+716 q^{10}-966 q^8+1136 q^6-1197 q^4+1156 q^2-1002+762 q^{-2} -462 q^{-4} +146 q^{-6} +148 q^{-8} -384 q^{-10} +538 q^{-12} -606 q^{-14} +596 q^{-16} -528 q^{-18} +423 q^{-20} -312 q^{-22} +216 q^{-24} -134 q^{-26} +73 q^{-28} -38 q^{-30} +18 q^{-32} -6 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{42}-2 q^{38}-q^{36}+4 q^{34}+4 q^{32}-6 q^{30}-6 q^{28}+6 q^{26}+4 q^{24}-10 q^{22}-6 q^{20}+9 q^{18}+6 q^{16}-9 q^{14}+q^{12}+11 q^{10}-3 q^8-2 q^6+6 q^4+q^2-5+4 q^{-2} +6 q^{-4} -10 q^{-6} -5 q^{-8} +11 q^{-10} +4 q^{-12} -13 q^{-14} + q^{-16} +10 q^{-18} - q^{-20} -6 q^{-22} - q^{-24} +5 q^{-26} - q^{-28} -2 q^{-30} + q^{-32} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}-q^{40}+q^{38}+2 q^{36}-3 q^{34}-5 q^{32}+3 q^{30}+4 q^{28}-11 q^{26}-5 q^{24}+14 q^{22}+q^{20}-14 q^{18}+6 q^{16}+15 q^{14}-8 q^{12}-9 q^{10}+13 q^8+7 q^6-13 q^4+8 q^2+17-10 q^{-2} -6 q^{-4} +17 q^{-6} -4 q^{-8} -18 q^{-10} +4 q^{-12} +11 q^{-14} -9 q^{-16} -9 q^{-18} +11 q^{-20} +6 q^{-22} -8 q^{-24} - q^{-26} +6 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{22}-q^{20}+q^{18}-2 q^{16}+3 q^{14}+q^{10}+2 q^8+2 q^4+2- q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} - q^{-10} - q^{-12} +2 q^{-14} -2 q^{-16} + q^{-18} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+8 q^{28}-13 q^{26}+17 q^{24}-19 q^{22}+20 q^{20}-18 q^{18}+13 q^{16}-5 q^{14}-4 q^{12}+15 q^{10}-24 q^8+33 q^6-36 q^4+39 q^2-35+29 q^{-2} -18 q^{-4} +9 q^{-6} + q^{-8} -10 q^{-10} +16 q^{-12} -20 q^{-14} +20 q^{-16} -19 q^{-18} +15 q^{-20} -10 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+6 q^{46}-q^{44}-11 q^{42}-6 q^{40}+12 q^{38}+14 q^{36}-7 q^{34}-21 q^{32}-4 q^{30}+21 q^{28}+13 q^{26}-15 q^{24}-18 q^{22}+6 q^{20}+19 q^{18}+q^{16}-15 q^{14}-4 q^{12}+14 q^{10}+7 q^8-10 q^6-9 q^4+10 q^2+14-5 q^{-2} -15 q^{-4} +4 q^{-6} +17 q^{-8} +2 q^{-10} -19 q^{-12} -10 q^{-14} +16 q^{-16} +17 q^{-18} -10 q^{-20} -22 q^{-22} - q^{-24} +18 q^{-26} +10 q^{-28} -10 q^{-30} -12 q^{-32} +2 q^{-34} +9 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{46}-2 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-11 q^{36}+11 q^{34}-14 q^{32}+15 q^{30}-18 q^{28}+13 q^{26}-13 q^{24}+12 q^{22}-5 q^{20}-q^{18}+7 q^{16}-7 q^{14}+19 q^{12}-23 q^{10}+25 q^8-25 q^6+32 q^4-28 q^2+26-23 q^{-2} +22 q^{-4} -11 q^{-6} +5 q^{-8} -5 q^{-10} -3 q^{-12} +10 q^{-14} -13 q^{-16} +12 q^{-18} -17 q^{-20} +18 q^{-22} -13 q^{-24} +12 q^{-26} -12 q^{-28} +9 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-7 q^{70}-2 q^{68}+17 q^{66}-35 q^{64}+50 q^{62}-52 q^{60}+28 q^{58}+13 q^{56}-67 q^{54}+113 q^{52}-123 q^{50}+92 q^{48}-23 q^{46}-64 q^{44}+129 q^{42}-148 q^{40}+111 q^{38}-31 q^{36}-54 q^{34}+104 q^{32}-98 q^{30}+43 q^{28}+39 q^{26}-101 q^{24}+121 q^{22}-81 q^{20}-5 q^{18}+102 q^{16}-171 q^{14}+188 q^{12}-138 q^{10}+43 q^8+71 q^6-159 q^4+198 q^2-170+88 q^{-2} +17 q^{-4} -101 q^{-6} +132 q^{-8} -101 q^{-10} +29 q^{-12} +54 q^{-14} -99 q^{-16} +89 q^{-18} -33 q^{-20} -51 q^{-22} +119 q^{-24} -142 q^{-26} +110 q^{-28} -38 q^{-30} -44 q^{-32} +103 q^{-34} -124 q^{-36} +105 q^{-38} -58 q^{-40} +6 q^{-42} +31 q^{-44} -54 q^{-46} +53 q^{-48} -36 q^{-50} +20 q^{-52} -2 q^{-54} -6 q^{-56} +9 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/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 33"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+6 t^2-14 t+19-14 t^{-1} +6 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6+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]=

{ 61, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-4 q^3+7 q^2-9 q+11-10 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^2 z^8+2 z^8+4 a^3 z^7+10 a z^7+6 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+7 z^6 a^{-2} +9 z^6+a^5 z^5-6 a^3 z^5-16 a z^5-5 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-16 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-2 a^5 z^3+a^3 z^3+5 a z^3-z^3 a^{-1} -3 z^3 a^{-3} +4 a^4 z^2+10 a^2 z^2+3 z^2 a^{-2} +9 z^2+a^5 z+a^3 z-a^4-2 a^2 }[/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{62}{3} }[/math] [math]\displaystyle{ \frac{10}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -\frac{176}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ \frac{5071}{30} }[/math] [math]\displaystyle{ -\frac{422}{15} }[/math] [math]\displaystyle{ \frac{3062}{45} }[/math] [math]\displaystyle{ \frac{593}{18} }[/math] [math]\displaystyle{ \frac{271}{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]0 is the signature of 9 33. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               4 1   3 3             5 3     -2 1           6 4       2 -1         5 6         1 -3       4 5           -1 -5     2 5             3 -7   1 4               -3 -9   2                 2 -11 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=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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, 33]]
Out[2]=	9
In[3]:=	PD[Knot[9, 33]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 8, 13, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], X[18, 13, 1, 14], X[14, 5, 15, 6], X[6, 17, 7, 18], X[16, 12, 17, 11], X[10, 16, 11, 15]]
In[4]:=	GaussCode[Knot[9, 33]]
Out[4]=	GaussCode[1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5]
In[5]:=	BR[Knot[9, 33]]
Out[5]=	BR[4, {-1, 2, -1, 2, 2, -1, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 33]][t]
Out[6]=	-3 6 14 2 3 19 - t + -- - -- - 14 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[9, 33]][z]
Out[7]=	2 6 1 + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 33], Knot[11, NonAlternating, 55]}
In[9]:=	{KnotDet[Knot[9, 33]], KnotSignature[Knot[9, 33]]}
Out[9]=	{61, 0}
In[10]:=	J=Jones[Knot[9, 33]][q]
Out[10]=	-5 3 6 9 10 2 3 4 11 - q + -- - -- + -- - -- - 9 q + 7 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 33]}
In[12]:=	A2Invariant[Knot[9, 33]][q]
Out[12]=	-16 -12 2 2 3 2 4 8 10 12 -1 - q + q - --- + -- + -- + 3 q - 2 q + q - 2 q + q 10 8 2 q q q
In[13]:=	Kauffman[Knot[9, 33]][a, z]
Out[13]=	2 3 2 4 3 5 2 3 z 2 2 4 2 3 z -2 a - a + a z + a z + 9 z + ---- + 10 a z + 4 a z - ---- - 2 3 a a 3 4 4 z 3 3 3 5 3 4 z 9 z 2 4 -- + 5 a z + a z - 2 a z - 20 z + -- - ---- - 16 a z - a 4 2 a a 5 5 6 4 4 4 z 5 z 5 3 5 5 5 6 7 z 6 a z + ---- - ---- - 16 a z - 6 a z + a z + 9 z + ---- + 3 a 2 a a 7 2 6 4 6 6 z 7 3 7 8 2 8 5 a z + 3 a z + ---- + 10 a z + 4 a z + 2 z + 2 a z a
In[14]:=	{Vassiliev[2][Knot[9, 33]], Vassiliev[3][Knot[9, 33]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 33]][q, t]
Out[15]=	6 1 2 1 4 2 5 4 - + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 5 5 3 3 2 5 2 5 3 7 3 ---- + --- + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 3 q t q t 9 4 q t