9 29

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+12 t-15+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	{ 51, -2 }
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+5 q-7+9 q^{-1} -8 q^{-2} +8 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-2 z^4-2 a^4 z^2+7 a^2 z^2+z^2 a^{-2} -5 z^2-2 a^4+5 a^2+ a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^2 z^8+2 z^8+6 a^3 z^7+9 a z^7+3 z^7 a^{-1} +8 a^4 z^6+6 a^2 z^6+z^6 a^{-2} -z^6+6 a^5 z^5-8 a^3 z^5-24 a z^5-10 z^5 a^{-1} +3 a^6 z^4-13 a^4 z^4-24 a^2 z^4-3 z^4 a^{-2} -11 z^4+a^7 z^3-5 a^5 z^3-a^3 z^3+14 a z^3+9 z^3 a^{-1} +8 a^4 z^2+17 a^2 z^2+3 z^2 a^{-2} +12 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} -2 a^4-5 a^2- a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{18}+q^{16}-2 q^{14}-q^{12}+2 q^{10}+4 q^6+q^2-2 q^{-2} + q^{-4} - q^{-6} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+3 q^{92}-2 q^{90}-q^{88}+8 q^{86}-12 q^{84}+16 q^{82}-19 q^{80}+13 q^{78}-6 q^{76}-9 q^{74}+28 q^{72}-39 q^{70}+44 q^{68}-37 q^{66}+16 q^{64}+13 q^{62}-46 q^{60}+63 q^{58}-61 q^{56}+31 q^{54}+5 q^{52}-41 q^{50}+57 q^{48}-42 q^{46}+9 q^{44}+30 q^{42}-59 q^{40}+56 q^{38}-21 q^{36}-30 q^{34}+78 q^{32}-91 q^{30}+77 q^{28}-24 q^{26}-29 q^{24}+77 q^{22}-96 q^{20}+88 q^{18}-50 q^{16}+47 q^{12}-69 q^{10}+69 q^8-36 q^6-5 q^4+36 q^2-55+41 q^{-2} -7 q^{-4} -39 q^{-6} +68 q^{-8} -70 q^{-10} +39 q^{-12} +10 q^{-14} -56 q^{-16} +77 q^{-18} -70 q^{-20} +40 q^{-22} -3 q^{-24} -32 q^{-26} +47 q^{-28} -40 q^{-30} +27 q^{-32} -6 q^{-34} -7 q^{-36} +11 q^{-38} -10 q^{-40} +6 q^{-42} -2 q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_29.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+2 q^{11}-3 q^9+2 q^7+q^3+2 q-2 q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{36}-2 q^{34}+q^{32}+4 q^{30}-9 q^{28}+3 q^{26}+11 q^{24}-15 q^{22}-q^{20}+14 q^{18}-9 q^{16}-6 q^{14}+10 q^{12}+4 q^{10}-8 q^8+11 q^4-5 q^2-10+13 q^{-2} + q^{-4} -15 q^{-6} +9 q^{-8} +8 q^{-10} -11 q^{-12} + q^{-14} +7 q^{-16} -3 q^{-18} -2 q^{-20} + q^{-22} }[/math]
3	[math]\displaystyle{ -q^{69}+2 q^{67}-q^{65}-2 q^{63}+3 q^{61}+3 q^{59}-6 q^{57}-8 q^{55}+17 q^{53}+16 q^{51}-30 q^{49}-29 q^{47}+36 q^{45}+50 q^{43}-36 q^{41}-71 q^{39}+25 q^{37}+78 q^{35}-76 q^{31}-23 q^{29}+55 q^{27}+45 q^{25}-32 q^{23}-53 q^{21}+6 q^{19}+58 q^{17}+17 q^{15}-56 q^{13}-32 q^{11}+52 q^9+46 q^7-46 q^5-56 q^3+32 q+70 q^{-1} -19 q^{-3} -74 q^{-5} -4 q^{-7} +74 q^{-9} +28 q^{-11} -61 q^{-13} -45 q^{-15} +40 q^{-17} +54 q^{-19} -14 q^{-21} -50 q^{-23} -7 q^{-25} +34 q^{-27} +17 q^{-29} -16 q^{-31} -18 q^{-33} +4 q^{-35} +10 q^{-37} +2 q^{-39} -3 q^{-41} -2 q^{-43} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{112}-2 q^{110}+q^{108}+2 q^{106}-5 q^{104}+3 q^{102}+6 q^{98}-3 q^{96}-26 q^{94}+13 q^{92}+27 q^{90}+31 q^{88}-28 q^{86}-108 q^{84}+116 q^{80}+152 q^{78}-38 q^{76}-292 q^{74}-141 q^{72}+179 q^{70}+390 q^{68}+125 q^{66}-400 q^{64}-409 q^{62}+9 q^{60}+492 q^{58}+406 q^{56}-200 q^{54}-488 q^{52}-290 q^{50}+251 q^{48}+476 q^{46}+141 q^{44}-244 q^{42}-395 q^{40}-96 q^{38}+269 q^{36}+317 q^{34}+61 q^{32}-296 q^{30}-286 q^{28}+46 q^{26}+330 q^{24}+214 q^{22}-185 q^{20}-349 q^{18}-75 q^{16}+332 q^{14}+294 q^{12}-117 q^{10}-407 q^8-186 q^6+320 q^4+392 q^2+25-417 q^{-2} -365 q^{-4} +167 q^{-6} +440 q^{-8} +273 q^{-10} -247 q^{-12} -476 q^{-14} -129 q^{-16} +271 q^{-18} +433 q^{-20} +77 q^{-22} -324 q^{-24} -318 q^{-26} -57 q^{-28} +298 q^{-30} +267 q^{-32} -10 q^{-34} -206 q^{-36} -224 q^{-38} +24 q^{-40} +162 q^{-42} +130 q^{-44} +8 q^{-46} -122 q^{-48} -76 q^{-50} +57 q^{-54} +56 q^{-56} -8 q^{-58} -24 q^{-60} -23 q^{-62} -3 q^{-64} +13 q^{-66} +5 q^{-68} +2 q^{-70} -3 q^{-72} -2 q^{-74} + q^{-76} }[/math]
5	[math]\displaystyle{ -q^{165}+2 q^{163}-q^{161}-2 q^{159}+5 q^{157}-q^{155}-6 q^{153}+5 q^{149}+9 q^{147}+6 q^{145}-16 q^{143}-38 q^{141}-10 q^{139}+59 q^{137}+89 q^{135}+14 q^{133}-130 q^{131}-200 q^{129}-67 q^{127}+254 q^{125}+441 q^{123}+189 q^{121}-403 q^{119}-816 q^{117}-496 q^{115}+483 q^{113}+1329 q^{111}+1073 q^{109}-382 q^{107}-1849 q^{105}-1866 q^{103}-78 q^{101}+2147 q^{99}+2768 q^{97}+911 q^{95}-2033 q^{93}-3470 q^{91}-1957 q^{89}+1380 q^{87}+3686 q^{85}+2946 q^{83}-303 q^{81}-3318 q^{79}-3528 q^{77}-872 q^{75}+2349 q^{73}+3541 q^{71}+1880 q^{69}-1126 q^{67}-3004 q^{65}-2413 q^{63}-87 q^{61}+2092 q^{59}+2513 q^{57}+1006 q^{55}-1127 q^{53}-2239 q^{51}-1560 q^{49}+319 q^{47}+1843 q^{45}+1785 q^{43}+217 q^{41}-1502 q^{39}-1837 q^{37}-484 q^{35}+1323 q^{33}+1861 q^{31}+586 q^{29}-1311 q^{27}-1969 q^{25}-675 q^{23}+1392 q^{21}+2211 q^{19}+855 q^{17}-1456 q^{15}-2516 q^{13}-1226 q^{11}+1355 q^9+2841 q^7+1764 q^5-1010 q^3-2984 q-2389 q^{-1} +331 q^{-3} +2866 q^{-5} +2967 q^{-7} +552 q^{-9} -2323 q^{-11} -3273 q^{-13} -1548 q^{-15} +1403 q^{-17} +3166 q^{-19} +2365 q^{-21} -246 q^{-23} -2544 q^{-25} -2779 q^{-27} -907 q^{-29} +1522 q^{-31} +2650 q^{-33} +1737 q^{-35} -353 q^{-37} -1991 q^{-39} -2039 q^{-41} -653 q^{-43} +1031 q^{-45} +1796 q^{-47} +1222 q^{-49} -105 q^{-51} -1161 q^{-53} -1270 q^{-55} -531 q^{-57} +437 q^{-59} +947 q^{-61} +741 q^{-63} +97 q^{-65} -465 q^{-67} -608 q^{-69} -348 q^{-71} +74 q^{-73} +346 q^{-75} +327 q^{-77} +114 q^{-79} -100 q^{-81} -192 q^{-83} -147 q^{-85} -20 q^{-87} +72 q^{-89} +85 q^{-91} +44 q^{-93} -2 q^{-95} -30 q^{-97} -31 q^{-99} -8 q^{-101} +6 q^{-103} +8 q^{-105} +5 q^{-107} +2 q^{-109} -3 q^{-111} -2 q^{-113} + q^{-115} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{18}+q^{16}-2 q^{14}-q^{12}+2 q^{10}+4 q^6+q^2-2 q^{-2} + q^{-4} - q^{-6} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{52}-4 q^{50}+8 q^{48}-12 q^{46}+22 q^{44}-36 q^{42}+48 q^{40}-64 q^{38}+93 q^{36}-118 q^{34}+142 q^{32}-176 q^{30}+194 q^{28}-204 q^{26}+174 q^{24}-128 q^{22}+52 q^{20}+62 q^{18}-170 q^{16}+292 q^{14}-375 q^{12}+438 q^{10}-454 q^8+428 q^6-372 q^4+276 q^2-162+38 q^{-2} +70 q^{-4} -164 q^{-6} +234 q^{-8} -258 q^{-10} +250 q^{-12} -212 q^{-14} +168 q^{-16} -114 q^{-18} +66 q^{-20} -36 q^{-22} +14 q^{-24} -4 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{46}-q^{44}+q^{42}+2 q^{40}-2 q^{38}-2 q^{36}+3 q^{34}+3 q^{32}-8 q^{30}-8 q^{28}+3 q^{26}+3 q^{24}-7 q^{22}+2 q^{20}+11 q^{18}+4 q^{16}+q^{14}+3 q^{12}+2 q^{10}-4 q^8-q^6+2 q^4-6 q^2-4+5 q^{-2} +2 q^{-4} -6 q^{-6} + q^{-8} +7 q^{-10} +2 q^{-12} -4 q^{-14} - q^{-16} +5 q^{-18} + q^{-20} -3 q^{-22} -2 q^{-24} + q^{-28} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}+2 q^{40}-3 q^{38}+6 q^{36}-9 q^{34}+10 q^{32}-14 q^{30}+13 q^{28}-12 q^{26}+8 q^{24}-3 q^{22}-2 q^{20}+10 q^{18}-15 q^{16}+22 q^{14}-23 q^{12}+27 q^{10}-23 q^8+20 q^6-13 q^4+7 q^2-1-6 q^{-2} +10 q^{-4} -13 q^{-6} +13 q^{-8} -13 q^{-10} +10 q^{-12} -8 q^{-14} +6 q^{-16} -2 q^{-18} + q^{-20} }[/math]
1,0	[math]\displaystyle{ q^{68}-2 q^{64}-2 q^{62}+q^{60}+6 q^{58}+3 q^{56}-6 q^{54}-8 q^{52}+12 q^{48}+5 q^{46}-10 q^{44}-11 q^{42}+4 q^{40}+13 q^{38}-13 q^{34}-6 q^{32}+10 q^{30}+7 q^{28}-7 q^{26}-9 q^{24}+5 q^{22}+11 q^{20}+q^{18}-8 q^{16}+2 q^{14}+10 q^{12}+3 q^{10}-9 q^8-4 q^6+9 q^4+9 q^2-7-15 q^{-2} +14 q^{-6} +6 q^{-8} -11 q^{-10} -11 q^{-12} +6 q^{-14} +12 q^{-16} -8 q^{-20} -4 q^{-22} +5 q^{-24} +4 q^{-26} -2 q^{-28} -2 q^{-30} + q^{-34} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{58}-2 q^{56}+q^{54}-2 q^{52}+6 q^{50}-6 q^{48}+5 q^{46}-8 q^{44}+12 q^{42}-10 q^{40}+8 q^{38}-11 q^{36}+7 q^{34}-6 q^{32}-2 q^{28}-8 q^{26}+9 q^{24}-11 q^{22}+17 q^{20}-15 q^{18}+24 q^{16}-14 q^{14}+24 q^{12}-14 q^{10}+16 q^8-10 q^6+7 q^4-7 q^2-2+2 q^{-2} -8 q^{-4} +6 q^{-6} -11 q^{-8} +12 q^{-10} -10 q^{-12} +10 q^{-14} -8 q^{-16} +8 q^{-18} -5 q^{-20} +4 q^{-22} -2 q^{-24} + q^{-26} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+3 q^{92}-2 q^{90}-q^{88}+8 q^{86}-12 q^{84}+16 q^{82}-19 q^{80}+13 q^{78}-6 q^{76}-9 q^{74}+28 q^{72}-39 q^{70}+44 q^{68}-37 q^{66}+16 q^{64}+13 q^{62}-46 q^{60}+63 q^{58}-61 q^{56}+31 q^{54}+5 q^{52}-41 q^{50}+57 q^{48}-42 q^{46}+9 q^{44}+30 q^{42}-59 q^{40}+56 q^{38}-21 q^{36}-30 q^{34}+78 q^{32}-91 q^{30}+77 q^{28}-24 q^{26}-29 q^{24}+77 q^{22}-96 q^{20}+88 q^{18}-50 q^{16}+47 q^{12}-69 q^{10}+69 q^8-36 q^6-5 q^4+36 q^2-55+41 q^{-2} -7 q^{-4} -39 q^{-6} +68 q^{-8} -70 q^{-10} +39 q^{-12} +10 q^{-14} -56 q^{-16} +77 q^{-18} -70 q^{-20} +40 q^{-22} -3 q^{-24} -32 q^{-26} +47 q^{-28} -40 q^{-30} +27 q^{-32} -6 q^{-34} -7 q^{-36} +11 q^{-38} -10 q^{-40} +6 q^{-42} -2 q^{-44} + q^{-46} }[/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 29"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-5 t^2+12 t-15+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]=

{ 51, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-2 z^4-2 a^4 z^2+7 a^2 z^2+z^2 a^{-2} -5 z^2-2 a^4+5 a^2+ a^{-2} -3 }[/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+6 a^3 z^7+9 a z^7+3 z^7 a^{-1} +8 a^4 z^6+6 a^2 z^6+z^6 a^{-2} -z^6+6 a^5 z^5-8 a^3 z^5-24 a z^5-10 z^5 a^{-1} +3 a^6 z^4-13 a^4 z^4-24 a^2 z^4-3 z^4 a^{-2} -11 z^4+a^7 z^3-5 a^5 z^3-a^3 z^3+14 a z^3+9 z^3 a^{-1} +8 a^4 z^2+17 a^2 z^2+3 z^2 a^{-2} +12 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} -2 a^4-5 a^2- a^{-2} -3 }[/math]