9 27

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 49, 0 }
Jones polynomial	[math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+9-8 q^{-1} +7 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} -6 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+6 a^2 z^6+4 z^6 a^{-2} +7 z^6+a^5 z^5-4 a^3 z^5-8 a z^5+3 z^5 a^{-3} -7 a^4 z^4-17 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -16 z^4-2 a^5 z^3-2 a^3 z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+12 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +12 z^2+a^5 z+2 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{12}-q^{10}+2 q^8+2 q^2-1+2 q^{-2} -2 q^{-4} + 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}-4 q^{70}-6 q^{68}+19 q^{66}-29 q^{64}+33 q^{62}-29 q^{60}+6 q^{58}+22 q^{56}-50 q^{54}+65 q^{52}-56 q^{50}+32 q^{48}+6 q^{46}-42 q^{44}+61 q^{42}-58 q^{40}+33 q^{38}+3 q^{36}-32 q^{34}+41 q^{32}-25 q^{30}-q^{28}+36 q^{26}-53 q^{24}+50 q^{22}-24 q^{20}-20 q^{18}+64 q^{16}-89 q^{14}+88 q^{12}-53 q^{10}+8 q^8+45 q^6-81 q^4+87 q^2-67+26 q^{-2} +16 q^{-4} -47 q^{-6} +48 q^{-8} -25 q^{-10} -5 q^{-12} +31 q^{-14} -41 q^{-16} +24 q^{-18} + q^{-20} -35 q^{-22} +58 q^{-24} -59 q^{-26} +43 q^{-28} -9 q^{-30} -22 q^{-32} +45 q^{-34} -51 q^{-36} +45 q^{-38} -28 q^{-40} +7 q^{-42} +11 q^{-44} -23 q^{-46} +24 q^{-48} -18 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_27.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-2 q^7+2 q^5-q^3+q+2 q^{-1} -2 q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+7 q^{26}-2 q^{24}-9 q^{22}+11 q^{20}+2 q^{18}-15 q^{16}+9 q^{14}+7 q^{12}-13 q^{10}+3 q^8+8 q^6-4 q^4-4 q^2+5+9 q^{-2} -11 q^{-4} -3 q^{-6} +15 q^{-8} -10 q^{-10} -7 q^{-12} +13 q^{-14} -4 q^{-16} -5 q^{-18} +6 q^{-20} - q^{-22} -2 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-7 q^{55}+2 q^{53}+16 q^{51}+q^{49}-23 q^{47}-11 q^{45}+29 q^{43}+26 q^{41}-31 q^{39}-42 q^{37}+26 q^{35}+55 q^{33}-13 q^{31}-65 q^{29}-q^{27}+67 q^{25}+14 q^{23}-63 q^{21}-25 q^{19}+52 q^{17}+34 q^{15}-39 q^{13}-36 q^{11}+22 q^9+39 q^7-4 q^5-36 q^3-18 q+34 q^{-1} +42 q^{-3} -27 q^{-5} -55 q^{-7} +13 q^{-9} +68 q^{-11} -2 q^{-13} -69 q^{-15} -12 q^{-17} +62 q^{-19} +20 q^{-21} -48 q^{-23} -23 q^{-25} +34 q^{-27} +21 q^{-29} -21 q^{-31} -16 q^{-33} +11 q^{-35} +11 q^{-37} -7 q^{-39} -5 q^{-41} +3 q^{-43} +3 q^{-45} - q^{-47} -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}-9 q^{92}-16 q^{90}-q^{88}+11 q^{86}+41 q^{84}-2 q^{82}-47 q^{80}-41 q^{78}-7 q^{76}+100 q^{74}+62 q^{72}-42 q^{70}-117 q^{68}-108 q^{66}+117 q^{64}+173 q^{62}+62 q^{60}-141 q^{58}-259 q^{56}+16 q^{54}+223 q^{52}+230 q^{50}-46 q^{48}-346 q^{46}-148 q^{44}+156 q^{42}+333 q^{40}+100 q^{38}-305 q^{36}-255 q^{34}+30 q^{32}+318 q^{30}+195 q^{28}-190 q^{26}-261 q^{24}-69 q^{22}+225 q^{20}+214 q^{18}-58 q^{16}-208 q^{14}-135 q^{12}+100 q^{10}+195 q^8+88 q^6-124 q^4-197 q^2-61+154 q^{-2} +254 q^{-4} -2 q^{-6} -232 q^{-8} -236 q^{-10} +52 q^{-12} +355 q^{-14} +151 q^{-16} -171 q^{-18} -343 q^{-20} -94 q^{-22} +321 q^{-24} +243 q^{-26} -34 q^{-28} -299 q^{-30} -185 q^{-32} +176 q^{-34} +204 q^{-36} +72 q^{-38} -160 q^{-40} -163 q^{-42} +51 q^{-44} +96 q^{-46} +80 q^{-48} -49 q^{-50} -85 q^{-52} +7 q^{-54} +23 q^{-56} +41 q^{-58} -9 q^{-60} -29 q^{-62} +4 q^{-64} +12 q^{-68} -2 q^{-70} -8 q^{-72} +3 q^{-74} +3 q^{-78} - q^{-80} -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}+9 q^{141}+16 q^{139}+q^{137}-20 q^{135}-29 q^{133}-19 q^{131}+20 q^{129}+61 q^{127}+62 q^{125}-11 q^{123}-97 q^{121}-121 q^{119}-47 q^{117}+108 q^{115}+220 q^{113}+161 q^{111}-78 q^{109}-309 q^{107}-326 q^{105}-59 q^{103}+348 q^{101}+541 q^{99}+291 q^{97}-284 q^{95}-722 q^{93}-609 q^{91}+62 q^{89}+811 q^{87}+963 q^{85}+295 q^{83}-745 q^{81}-1264 q^{79}-732 q^{77}+497 q^{75}+1431 q^{73}+1192 q^{71}-123 q^{69}-1433 q^{67}-1558 q^{65}-315 q^{63}+1258 q^{61}+1785 q^{59}+742 q^{57}-978 q^{55}-1843 q^{53}-1070 q^{51}+644 q^{49}+1750 q^{47}+1273 q^{45}-324 q^{43}-1556 q^{41}-1345 q^{39}+50 q^{37}+1315 q^{35}+1314 q^{33}+149 q^{31}-1043 q^{29}-1228 q^{27}-316 q^{25}+799 q^{23}+1121 q^{21}+450 q^{19}-541 q^{17}-1013 q^{15}-623 q^{13}+280 q^{11}+929 q^9+805 q^7+27 q^5-818 q^3-1032 q-389 q^{-1} +672 q^{-3} +1265 q^{-5} +782 q^{-7} -438 q^{-9} -1410 q^{-11} -1218 q^{-13} +110 q^{-15} +1477 q^{-17} +1587 q^{-19} +274 q^{-21} -1350 q^{-23} -1852 q^{-25} -701 q^{-27} +1098 q^{-29} +1925 q^{-31} +1056 q^{-33} -717 q^{-35} -1801 q^{-37} -1283 q^{-39} +301 q^{-41} +1502 q^{-43} +1336 q^{-45} +71 q^{-47} -1114 q^{-49} -1217 q^{-51} -315 q^{-53} +702 q^{-55} +984 q^{-57} +430 q^{-59} -366 q^{-61} -710 q^{-63} -417 q^{-65} +137 q^{-67} +449 q^{-69} +335 q^{-71} -3 q^{-73} -256 q^{-75} -239 q^{-77} -36 q^{-79} +127 q^{-81} +138 q^{-83} +47 q^{-85} -51 q^{-87} -81 q^{-89} -32 q^{-91} +23 q^{-93} +34 q^{-95} +17 q^{-97} -5 q^{-99} -13 q^{-101} -10 q^{-103} +3 q^{-105} +8 q^{-107} - q^{-109} -3 q^{-111} +3 q^{-119} - q^{-121} -2 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{12}-q^{10}+2 q^8+2 q^2-1+2 q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+52 q^{36}-86 q^{34}+130 q^{32}-180 q^{30}+222 q^{28}-248 q^{26}+258 q^{24}-236 q^{22}+176 q^{20}-88 q^{18}-24 q^{16}+144 q^{14}-267 q^{12}+376 q^{10}-450 q^8+492 q^6-483 q^4+440 q^2-352+244 q^{-2} -120 q^{-4} -4 q^{-6} +102 q^{-8} -176 q^{-10} +222 q^{-12} -238 q^{-14} +228 q^{-16} -196 q^{-18} +162 q^{-20} -126 q^{-22} +88 q^{-24} -58 q^{-26} +36 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}+4 q^{32}-3 q^{30}-3 q^{28}+4 q^{26}+4 q^{24}-5 q^{22}-5 q^{20}+5 q^{18}+2 q^{16}-6 q^{14}+6 q^{10}-2 q^8-q^6+6 q^4+q^2-2+3 q^{-2} +5 q^{-4} -7 q^{-6} -5 q^{-8} +7 q^{-10} +2 q^{-12} -7 q^{-14} +6 q^{-18} -3 q^{-22} +2 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-2 q^{32}+q^{30}+3 q^{28}-7 q^{26}+3 q^{24}+4 q^{22}-12 q^{20}+5 q^{18}+8 q^{16}-10 q^{14}+5 q^{12}+10 q^{10}-5 q^8-q^6+4 q^4+q^2-4-4 q^{-2} +9 q^{-4} -5 q^{-6} -8 q^{-8} +13 q^{-10} -2 q^{-12} -8 q^{-14} +10 q^{-16} -6 q^{-20} +4 q^{-22} -2 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}+q^{15}-q^{13}+3 q^{11}+2 q^7+q^3- q^{-1} + q^{-3} -2 q^{-5} + q^{-7} - q^{-9} +2 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}-3 q^{34}-3 q^{32}+2 q^{30}-8 q^{26}-2 q^{24}+9 q^{22}+q^{20}-5 q^{18}+9 q^{16}+11 q^{14}-3 q^{12}-2 q^{10}+7 q^8-10 q^4+2 q^2+4-9 q^{-2} -4 q^{-4} +10 q^{-6} -2 q^{-8} -7 q^{-10} +7 q^{-12} +8 q^{-14} -4 q^{-16} -4 q^{-18} +6 q^{-20} +2 q^{-22} -5 q^{-24} - q^{-26} +3 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}+3 q^{14}+q^{12}+2 q^{10}+2 q^8+q^4-q^2-2 q^{-2} + q^{-4} -2 q^{-6} + q^{-8} +2 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}-9 q^{26}+11 q^{24}-12 q^{22}+12 q^{20}-9 q^{18}+6 q^{16}-5 q^{12}+12 q^{10}-17 q^8+21 q^6-22 q^4+23 q^2-20+16 q^{-2} -9 q^{-4} +3 q^{-6} +2 q^{-8} -7 q^{-10} +10 q^{-12} -12 q^{-14} +12 q^{-16} -10 q^{-18} +8 q^{-20} -6 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}-8 q^{42}-3 q^{40}+9 q^{38}+8 q^{36}-7 q^{34}-13 q^{32}+13 q^{28}+7 q^{26}-9 q^{24}-9 q^{22}+5 q^{20}+11 q^{18}-8 q^{14}-q^{12}+8 q^{10}+3 q^8-7 q^6-5 q^4+7 q^2+7-5 q^{-2} -9 q^{-4} +4 q^{-6} +10 q^{-8} - q^{-10} -12 q^{-12} -4 q^{-14} +11 q^{-16} +9 q^{-18} -6 q^{-20} -12 q^{-22} +11 q^{-26} +6 q^{-28} -5 q^{-30} -7 q^{-32} +5 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}-8 q^{36}+7 q^{34}-10 q^{32}+9 q^{30}-11 q^{28}+6 q^{26}-6 q^{24}+6 q^{22}-q^{18}+9 q^{16}-4 q^{14}+15 q^{12}-14 q^{10}+16 q^8-16 q^6+18 q^4-19 q^2+12-15 q^{-2} +10 q^{-4} -6 q^{-6} + q^{-8} - q^{-10} -2 q^{-12} +9 q^{-14} -6 q^{-16} +8 q^{-18} -9 q^{-20} +11 q^{-22} -7 q^{-24} +6 q^{-26} -7 q^{-28} +5 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}-4 q^{70}-6 q^{68}+19 q^{66}-29 q^{64}+33 q^{62}-29 q^{60}+6 q^{58}+22 q^{56}-50 q^{54}+65 q^{52}-56 q^{50}+32 q^{48}+6 q^{46}-42 q^{44}+61 q^{42}-58 q^{40}+33 q^{38}+3 q^{36}-32 q^{34}+41 q^{32}-25 q^{30}-q^{28}+36 q^{26}-53 q^{24}+50 q^{22}-24 q^{20}-20 q^{18}+64 q^{16}-89 q^{14}+88 q^{12}-53 q^{10}+8 q^8+45 q^6-81 q^4+87 q^2-67+26 q^{-2} +16 q^{-4} -47 q^{-6} +48 q^{-8} -25 q^{-10} -5 q^{-12} +31 q^{-14} -41 q^{-16} +24 q^{-18} + q^{-20} -35 q^{-22} +58 q^{-24} -59 q^{-26} +43 q^{-28} -9 q^{-30} -22 q^{-32} +45 q^{-34} -51 q^{-36} +45 q^{-38} -28 q^{-40} +7 q^{-42} +11 q^{-44} -23 q^{-46} +24 q^{-48} -18 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 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 27"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-z^4+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]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+9-8 q^{-1} +7 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} -6 z^2-a^4+3 a^2+ a^{-2} -2 }[/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+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+6 a^2 z^6+4 z^6 a^{-2} +7 z^6+a^5 z^5-4 a^3 z^5-8 a z^5+3 z^5 a^{-3} -7 a^4 z^4-17 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -16 z^4-2 a^5 z^3-2 a^3 z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+12 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +12 z^2+a^5 z+2 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^4-3 a^2- a^{-2} -2 }[/math]