9 30

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9 29.gif

9_29

9 31.gif

9_31

9 30.gif Visit 9 30's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 30's page at Knotilus!

Visit 9 30's page at the original Knot Atlas!

9 30 Quick Notes


9 30 Further Notes and Views

Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
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]

Three dimensional invariants

Symmetry type Reversible
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 11.9545
A-Polynomial See Data:9 30/A-polynomial

[edit Notes for 9 30'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 30's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+5 t^2-12 t+17-12 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 { 53, 0 }
Jones polynomial [math]\displaystyle{ q^4-3 q^3+6 q^2-8 q+9-9 q^{-1} +8 q^{-2} -5 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} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^4+4 a^2+2 a^{-2} -4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+7 a z^7+4 z^7 a^{-1} +3 a^4 z^6+8 a^2 z^6+5 z^6 a^{-2} +10 z^6+a^5 z^5-3 a^3 z^5-9 a z^5-2 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4-22 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -23 z^4-2 a^5 z^3-3 a^3 z^3-2 z^3 a^{-1} -3 z^3 a^{-3} +5 a^4 z^2+16 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +17 z^2+a^5 z+2 a^3 z+a z+z a^{-1} +z a^{-3} -a^4-4 a^2-2 a^{-2} -4 }[/math]
The A2 invariant [math]\displaystyle{ -q^{16}+q^{12}-q^{10}+3 q^8+q^6+q^2-3+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-5 q^{70}-5 q^{68}+19 q^{66}-31 q^{64}+39 q^{62}-34 q^{60}+11 q^{58}+19 q^{56}-55 q^{54}+79 q^{52}-79 q^{50}+49 q^{48}-2 q^{46}-48 q^{44}+83 q^{42}-84 q^{40}+60 q^{38}-9 q^{36}-36 q^{34}+61 q^{32}-52 q^{30}+14 q^{28}+39 q^{26}-69 q^{24}+75 q^{22}-40 q^{20}-15 q^{18}+77 q^{16}-118 q^{14}+120 q^{12}-84 q^{10}+16 q^8+55 q^6-109 q^4+123 q^2-97+43 q^{-2} +15 q^{-4} -64 q^{-6} +72 q^{-8} -52 q^{-10} +6 q^{-12} +39 q^{-14} -60 q^{-16} +48 q^{-18} -8 q^{-20} -41 q^{-22} +79 q^{-24} -87 q^{-26} +65 q^{-28} -21 q^{-30} -29 q^{-32} +67 q^{-34} -77 q^{-36} +67 q^{-38} -34 q^{-40} +5 q^{-42} +19 q^{-44} -33 q^{-46} +32 q^{-48} -23 q^{-50} +13 q^{-52} -2 q^{-54} -4 q^{-56} +5 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_30.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+2 q^9-2 q^7+3 q^5-q^3+ q^{-1} -2 q^{-3} +3 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2 [math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+7 q^{26}-3 q^{24}-9 q^{22}+14 q^{20}+q^{18}-18 q^{16}+13 q^{14}+8 q^{12}-17 q^{10}+4 q^8+10 q^6-6 q^4-6 q^2+6+9 q^{-2} -13 q^{-4} - q^{-6} +19 q^{-8} -13 q^{-10} -8 q^{-12} +17 q^{-14} -6 q^{-16} -8 q^{-18} +7 q^{-20} -2 q^{-24} + q^{-26} }[/math]
3 [math]\displaystyle{ -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-7 q^{55}+3 q^{53}+16 q^{51}-2 q^{49}-26 q^{47}-7 q^{45}+39 q^{43}+23 q^{41}-47 q^{39}-45 q^{37}+48 q^{35}+68 q^{33}-36 q^{31}-91 q^{29}+19 q^{27}+98 q^{25}+4 q^{23}-96 q^{21}-26 q^{19}+87 q^{17}+40 q^{15}-65 q^{13}-50 q^{11}+41 q^9+55 q^7-12 q^5-57 q^3-15 q+55 q^{-1} +45 q^{-3} -48 q^{-5} -72 q^{-7} +37 q^{-9} +92 q^{-11} -20 q^{-13} -101 q^{-15} + q^{-17} +96 q^{-19} +20 q^{-21} -82 q^{-23} -32 q^{-25} +60 q^{-27} +33 q^{-29} -34 q^{-31} -30 q^{-33} +17 q^{-35} +21 q^{-37} -7 q^{-39} -10 q^{-41} + q^{-43} +4 q^{-45} -2 q^{-49} + q^{-51} }[/math]
4 [math]\displaystyle{ q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+7 q^{94}-10 q^{92}-16 q^{90}+2 q^{88}+14 q^{86}+41 q^{84}-12 q^{82}-59 q^{80}-37 q^{78}+16 q^{76}+129 q^{74}+49 q^{72}-98 q^{70}-157 q^{68}-79 q^{66}+221 q^{64}+227 q^{62}-12 q^{60}-286 q^{58}-324 q^{56}+159 q^{54}+412 q^{52}+252 q^{50}-243 q^{48}-568 q^{46}-95 q^{44}+410 q^{42}+513 q^{40}-12 q^{38}-602 q^{36}-351 q^{34}+218 q^{32}+576 q^{30}+213 q^{28}-431 q^{26}-436 q^{24}+446 q^{20}+311 q^{18}-184 q^{16}-383 q^{14}-165 q^{12}+245 q^{10}+330 q^8+66 q^6-278 q^4-306 q^2+7+324 q^{-2} +339 q^{-4} -128 q^{-6} -432 q^{-8} -272 q^{-10} +246 q^{-12} +574 q^{-14} +101 q^{-16} -431 q^{-18} -517 q^{-20} +37 q^{-22} +623 q^{-24} +326 q^{-26} -233 q^{-28} -559 q^{-30} -203 q^{-32} +420 q^{-34} +381 q^{-36} +25 q^{-38} -367 q^{-40} -279 q^{-42} +144 q^{-44} +234 q^{-46} +133 q^{-48} -125 q^{-50} -176 q^{-52} + q^{-54} +70 q^{-56} +88 q^{-58} -14 q^{-60} -59 q^{-62} -11 q^{-64} +4 q^{-66} +26 q^{-68} +3 q^{-70} -11 q^{-72} - q^{-74} -2 q^{-76} +4 q^{-78} -2 q^{-82} + q^{-84} }[/math]
5 [math]\displaystyle{ -q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+10 q^{141}+16 q^{139}-2 q^{137}-23 q^{135}-29 q^{133}-13 q^{131}+32 q^{129}+71 q^{127}+52 q^{125}-43 q^{123}-132 q^{121}-124 q^{119}+8 q^{117}+199 q^{115}+275 q^{113}+97 q^{111}-254 q^{109}-473 q^{107}-316 q^{105}+194 q^{103}+700 q^{101}+693 q^{99}+26 q^{97}-853 q^{95}-1164 q^{93}-488 q^{91}+802 q^{89}+1646 q^{87}+1166 q^{85}-454 q^{83}-1975 q^{81}-1968 q^{79}-198 q^{77}+1994 q^{75}+2703 q^{73}+1111 q^{71}-1652 q^{69}-3227 q^{67}-2060 q^{65}+989 q^{63}+3361 q^{61}+2906 q^{59}-129 q^{57}-3148 q^{55}-3462 q^{53}-718 q^{51}+2634 q^{49}+3644 q^{47}+1441 q^{45}-1965 q^{43}-3518 q^{41}-1914 q^{39}+1292 q^{37}+3139 q^{35}+2125 q^{33}-659 q^{31}-2658 q^{29}-2166 q^{27}+164 q^{25}+2139 q^{23}+2091 q^{21}+265 q^{19}-1642 q^{17}-2014 q^{15}-655 q^{13}+1175 q^{11}+1966 q^9+1091 q^7-705 q^5-1948 q^3-1594 q+160 q^{-1} +1931 q^{-3} +2165 q^{-5} +487 q^{-7} -1821 q^{-9} -2732 q^{-11} -1254 q^{-13} +1532 q^{-15} +3201 q^{-17} +2084 q^{-19} -1027 q^{-21} -3429 q^{-23} -2860 q^{-25} +307 q^{-27} +3311 q^{-29} +3458 q^{-31} +531 q^{-33} -2851 q^{-35} -3692 q^{-37} -1334 q^{-39} +2073 q^{-41} +3538 q^{-43} +1939 q^{-45} -1174 q^{-47} -3018 q^{-49} -2189 q^{-51} +321 q^{-53} +2244 q^{-55} +2098 q^{-57} +317 q^{-59} -1438 q^{-61} -1736 q^{-63} -624 q^{-65} +727 q^{-67} +1234 q^{-69} +688 q^{-71} -244 q^{-73} -766 q^{-75} -564 q^{-77} +2 q^{-79} +393 q^{-81} +378 q^{-83} +89 q^{-85} -167 q^{-87} -223 q^{-89} -84 q^{-91} +63 q^{-93} +102 q^{-95} +54 q^{-97} -13 q^{-99} -41 q^{-101} -32 q^{-103} +3 q^{-105} +19 q^{-107} +8 q^{-109} - q^{-111} -2 q^{-113} -4 q^{-115} -2 q^{-117} +4 q^{-119} -2 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{16}+q^{12}-q^{10}+3 q^8+q^6+q^2-3+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} }[/math]
1,1 [math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+54 q^{36}-96 q^{34}+150 q^{32}-214 q^{30}+280 q^{28}-336 q^{26}+366 q^{24}-352 q^{22}+299 q^{20}-192 q^{18}+42 q^{16}+138 q^{14}-321 q^{12}+486 q^{10}-622 q^8+698 q^6-714 q^4+662 q^2-550+398 q^{-2} -213 q^{-4} +36 q^{-6} +132 q^{-8} -256 q^{-10} +334 q^{-12} -360 q^{-14} +344 q^{-16} -304 q^{-18} +239 q^{-20} -174 q^{-22} +118 q^{-24} -74 q^{-26} +42 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0 [math]\displaystyle{ q^{42}-2 q^{38}-2 q^{36}+2 q^{34}+3 q^{32}-4 q^{30}-3 q^{28}+6 q^{26}+5 q^{24}-6 q^{22}-3 q^{20}+8 q^{18}+5 q^{16}-8 q^{14}-q^{12}+5 q^{10}-6 q^8-5 q^6+2 q^4-3+7 q^{-2} +9 q^{-4} -3 q^{-6} -2 q^{-8} +9 q^{-10} + q^{-12} -12 q^{-14} - q^{-16} +6 q^{-18} - q^{-20} -5 q^{-22} +4 q^{-26} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+7 q^{28}-10 q^{26}+13 q^{24}-14 q^{22}+15 q^{20}-12 q^{18}+9 q^{16}-q^{14}-4 q^{12}+13 q^{10}-19 q^8+25 q^6-28 q^4+27 q^2-26+19 q^{-2} -13 q^{-4} +5 q^{-6} +2 q^{-8} -8 q^{-10} +13 q^{-12} -14 q^{-14} +15 q^{-16} -13 q^{-18} +11 q^{-20} -7 q^{-22} +4 q^{-24} -2 q^{-26} + q^{-28} }[/math]
1,0 [math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+5 q^{46}-2 q^{44}-9 q^{42}-4 q^{40}+10 q^{38}+10 q^{36}-6 q^{34}-15 q^{32}-q^{30}+16 q^{28}+10 q^{26}-11 q^{24}-12 q^{22}+5 q^{20}+14 q^{18}-11 q^{14}-2 q^{12}+10 q^{10}+4 q^8-9 q^6-6 q^4+7 q^2+9-6 q^{-2} -11 q^{-4} +3 q^{-6} +12 q^{-8} -14 q^{-12} -6 q^{-14} +13 q^{-16} +13 q^{-18} -7 q^{-20} -15 q^{-22} +14 q^{-26} +7 q^{-28} -7 q^{-30} -9 q^{-32} + q^{-34} +6 q^{-36} +2 q^{-38} -2 q^{-40} -2 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}+6 q^{38}-9 q^{36}+8 q^{34}-11 q^{32}+11 q^{30}-13 q^{28}+9 q^{26}-8 q^{24}+9 q^{22}-q^{20}+9 q^{16}-4 q^{14}+17 q^{12}-17 q^{10}+19 q^8-20 q^6+21 q^4-24 q^2+14-20 q^{-2} +12 q^{-4} -9 q^{-6} +2 q^{-8} -2 q^{-10} - q^{-12} +11 q^{-14} -7 q^{-16} +11 q^{-18} -11 q^{-20} +14 q^{-22} -9 q^{-24} +8 q^{-26} -9 q^{-28} +6 q^{-30} -3 q^{-32} +2 q^{-34} -2 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-5 q^{70}-5 q^{68}+19 q^{66}-31 q^{64}+39 q^{62}-34 q^{60}+11 q^{58}+19 q^{56}-55 q^{54}+79 q^{52}-79 q^{50}+49 q^{48}-2 q^{46}-48 q^{44}+83 q^{42}-84 q^{40}+60 q^{38}-9 q^{36}-36 q^{34}+61 q^{32}-52 q^{30}+14 q^{28}+39 q^{26}-69 q^{24}+75 q^{22}-40 q^{20}-15 q^{18}+77 q^{16}-118 q^{14}+120 q^{12}-84 q^{10}+16 q^8+55 q^6-109 q^4+123 q^2-97+43 q^{-2} +15 q^{-4} -64 q^{-6} +72 q^{-8} -52 q^{-10} +6 q^{-12} +39 q^{-14} -60 q^{-16} +48 q^{-18} -8 q^{-20} -41 q^{-22} +79 q^{-24} -87 q^{-26} +65 q^{-28} -21 q^{-30} -29 q^{-32} +67 q^{-34} -77 q^{-36} +67 q^{-38} -34 q^{-40} +5 q^{-42} +19 q^{-44} -33 q^{-46} +32 q^{-48} -23 q^{-50} +13 q^{-52} -2 q^{-54} -4 q^{-56} +5 q^{-58} -6 q^{-60} +4 q^{-62} -2 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 30"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+5 t^2-12 t+17-12 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]= { 53, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^4-3 q^3+6 q^2-8 q+9-9 q^{-1} +8 q^{-2} -5 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} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^4+4 a^2+2 a^{-2} -4 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+7 a z^7+4 z^7 a^{-1} +3 a^4 z^6+8 a^2 z^6+5 z^6 a^{-2} +10 z^6+a^5 z^5-3 a^3 z^5-9 a z^5-2 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4-22 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -23 z^4-2 a^5 z^3-3 a^3 z^3-2 z^3 a^{-1} -3 z^3 a^{-3} +5 a^4 z^2+16 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +17 z^2+a^5 z+2 a^3 z+a z+z a^{-1} +z a^{-3} -a^4-4 a^2-2 a^{-2} -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{82}{3} }[/math] [math]\displaystyle{ \frac{38}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{208}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{328}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ -\frac{2431}{30} }[/math] [math]\displaystyle{ \frac{1222}{15} }[/math] [math]\displaystyle{ -\frac{8342}{45} }[/math] [math]\displaystyle{ \frac{607}{18} }[/math] [math]\displaystyle{ -\frac{1471}{30} }[/math]

V2,1 through V6,9 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 30. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
9         11
7        2 -2
5       41 3
3      42  -2
1     54   1
-1    55    0
-3   34     -1
-5  25      3
-7 13       -2
-9 2        2
-111         -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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 30]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 30]]
Out[3]=  
PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], 

 X[14, 8, 15, 7], X[18, 15, 1, 16], X[16, 11, 17, 12], 



  X[12, 17, 13, 18], X[6, 14, 7, 13]]
In[4]:=
GaussCode[Knot[9, 30]]
Out[4]=  
GaussCode[1, -4, 3, -1, 2, -9, 5, -3, 4, -2, 7, -8, 9, -5, 6, -7, 8, -6]
In[5]:=
BR[Knot[9, 30]]
Out[5]=  
BR[4, {-1, -1, 2, 2, -1, 2, -3, 2, -3}]
In[6]:=
alex = Alexander[Knot[9, 30]][t]
Out[6]=  
      -3   5    12             2    3

17 - t   + -- - -- - 12 t + 5 t  - t



           2   t


           t
In[7]:=
Conway[Knot[9, 30]][z]
Out[7]=  
     2    4    6
1 - z  - z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 30], Knot[11, NonAlternating, 130]}
In[9]:=
{KnotDet[Knot[9, 30]], KnotSignature[Knot[9, 30]]}
Out[9]=  
{53, 0}
In[10]:=
J=Jones[Knot[9, 30]][q]
Out[10]=  
     -5   3    5    8    9            2      3    4

9 - q   + -- - -- + -- - - - 8 q + 6 q  - 3 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, 30], Knot[11, NonAlternating, 114]}
In[12]:=
A2Invariant[Knot[9, 30]][q]
Out[12]=  
      -16    -12    -10   3     -6    -2    2      4    6      8

-3 - q    + q    - q    + -- + q   + q   + q  - 2 q  + q  + 2 q  - 



                          8
                         q

  10    12


  q   + q
In[13]:=
Kauffman[Knot[9, 30]][a, z]
Out[13]=  
                                                              2

    2       2    4   z    z            3      5         2   z



-4 - -- - 4 a  - a  + -- + - + a z + 2 a  z + a  z + 17 z  - -- + 



     2                3   a                                  4
    a                a                                      a

    2                           3      3
 5 z        2  2      4  2   3 z    2 z       3  3      5  3       4
 ---- + 16 a  z  + 5 a  z  - ---- - ---- - 3 a  z  - 2 a  z  - 23 z  + 
   2                           3     a
  a                           a

  4      4                           5      5
 z    7 z        2  4      4  4   3 z    2 z         5      3  5
 -- - ---- - 22 a  z  - 7 a  z  + ---- - ---- - 9 a z  - 3 a  z  + 
  4     2                           3     a
 a     a                           a

                    6                          7
  5  5       6   5 z       2  6      4  6   4 z         7      3  7
 a  z  + 10 z  + ---- + 8 a  z  + 3 a  z  + ---- + 7 a z  + 3 a  z  + 
                   2                         a
                  a

  8    2  8


  z  + a  z
In[14]:=
{Vassiliev[2][Knot[9, 30]], Vassiliev[3][Knot[9, 30]]}
Out[14]=  
{0, -1}
In[15]:=
Kh[Knot[9, 30]][q, t]
Out[15]=  
5           1        2       1       3       2       5       3

- + 5 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

  4      5               3        3  2      5  2    5  3      7  3
 ---- + --- + 4 q t + 4 q  t + 2 q  t  + 4 q  t  + q  t  + 2 q  t  + 
  3     q t
 q  t

  9  4


  q  t
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