9 25

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -3 t^2+12 t-17+12 t^{-1} -3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 1-3 z^4 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 47, -2 }
Jones polynomial [math]\displaystyle{ q-2+5 q^{-1} -7 q^{-2} +8 q^{-3} -8 q^{-4} +7 q^{-5} -5 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^8+3 z^2 a^6+3 a^6-2 z^4 a^4-4 z^2 a^4-3 a^4-z^4 a^2+a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+4 z^2 a^8-a^8+3 z^7 a^7-4 z^5 a^7-2 z^3 a^7+z a^7+z^8 a^6+6 z^6 a^6-18 z^4 a^6+13 z^2 a^6-3 a^6+6 z^7 a^5-10 z^5 a^5+5 z^3 a^5-z a^5+z^8 a^4+6 z^6 a^4-15 z^4 a^4+13 z^2 a^4-3 a^4+3 z^7 a^3-3 z^5 a^3+3 z^3 a^3-z a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}+q^{18}+2 q^{16}-2 q^{14}-2 q^{10}+q^6-q^4+3 q^2+ q^{-4} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-4 q^{118}-6 q^{116}+19 q^{114}-29 q^{112}+34 q^{110}-28 q^{108}+4 q^{106}+23 q^{104}-50 q^{102}+63 q^{100}-55 q^{98}+26 q^{96}+12 q^{94}-46 q^{92}+60 q^{90}-48 q^{88}+22 q^{86}+15 q^{84}-38 q^{82}+41 q^{80}-18 q^{78}-14 q^{76}+48 q^{74}-60 q^{72}+50 q^{70}-13 q^{68}-30 q^{66}+68 q^{64}-87 q^{62}+79 q^{60}-45 q^{58}-6 q^{56}+48 q^{54}-78 q^{52}+76 q^{50}-50 q^{48}+8 q^{46}+26 q^{44}-46 q^{42}+38 q^{40}-14 q^{38}-20 q^{36}+43 q^{34}-43 q^{32}+21 q^{30}+13 q^{28}-44 q^{26}+63 q^{24}-54 q^{22}+31 q^{20}-q^{18}-27 q^{16}+43 q^{14}-43 q^{12}+34 q^{10}-14 q^8+11 q^4-16 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_25.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+2 q^{15}-2 q^{13}+2 q^{11}-q^9+q^5-2 q^3+3 q- q^{-1} + q^{-3} }[/math]
2 [math]\displaystyle{ q^{48}-2 q^{46}-2 q^{44}+7 q^{42}-2 q^{40}-9 q^{38}+11 q^{36}+3 q^{34}-15 q^{32}+7 q^{30}+8 q^{28}-12 q^{26}+q^{24}+8 q^{22}-3 q^{20}-6 q^{18}+3 q^{16}+9 q^{14}-10 q^{12}-3 q^{10}+16 q^8-8 q^6-8 q^4+12 q^2-2-5 q^{-2} +4 q^{-4} - q^{-8} + q^{-10} }[/math]
3 [math]\displaystyle{ -q^{93}+2 q^{91}+2 q^{89}-3 q^{87}-7 q^{85}+2 q^{83}+16 q^{81}+q^{79}-23 q^{77}-12 q^{75}+29 q^{73}+29 q^{71}-29 q^{69}-46 q^{67}+19 q^{65}+59 q^{63}-2 q^{61}-68 q^{59}-15 q^{57}+66 q^{55}+30 q^{53}-55 q^{51}-40 q^{49}+44 q^{47}+44 q^{45}-29 q^{43}-44 q^{41}+11 q^{39}+40 q^{37}+7 q^{35}-36 q^{33}-28 q^{31}+29 q^{29}+46 q^{27}-17 q^{25}-61 q^{23}+3 q^{21}+70 q^{19}+14 q^{17}-69 q^{15}-25 q^{13}+55 q^{11}+36 q^9-39 q^7-35 q^5+24 q^3+27 q-8 q^{-1} -18 q^{-3} +3 q^{-5} +9 q^{-7} - q^{-9} -4 q^{-11} + q^{-13} + q^{-15} - q^{-19} + q^{-21} }[/math]
4 [math]\displaystyle{ q^{152}-2 q^{150}-2 q^{148}+3 q^{146}+3 q^{144}+7 q^{142}-9 q^{140}-16 q^{138}-q^{136}+11 q^{134}+41 q^{132}-q^{130}-47 q^{128}-44 q^{126}-10 q^{124}+101 q^{122}+72 q^{120}-33 q^{118}-123 q^{116}-129 q^{114}+98 q^{112}+190 q^{110}+105 q^{108}-121 q^{106}-295 q^{104}-53 q^{102}+211 q^{100}+299 q^{98}+33 q^{96}-349 q^{94}-252 q^{92}+79 q^{90}+374 q^{88}+218 q^{86}-243 q^{84}-338 q^{82}-85 q^{80}+301 q^{78}+291 q^{76}-93 q^{74}-289 q^{72}-165 q^{70}+170 q^{68}+258 q^{66}+32 q^{64}-191 q^{62}-191 q^{60}+36 q^{58}+196 q^{56}+152 q^{54}-78 q^{52}-215 q^{50}-129 q^{48}+113 q^{46}+288 q^{44}+76 q^{42}-208 q^{40}-305 q^{38}-32 q^{36}+357 q^{34}+254 q^{32}-95 q^{30}-388 q^{28}-210 q^{26}+269 q^{24}+329 q^{22}+82 q^{20}-286 q^{18}-287 q^{16}+78 q^{14}+232 q^{12}+171 q^{10}-100 q^8-202 q^6-36 q^4+74 q^2+119+7 q^{-2} -73 q^{-4} -33 q^{-6} -2 q^{-8} +40 q^{-10} +13 q^{-12} -14 q^{-14} -4 q^{-16} -7 q^{-18} +7 q^{-20} +2 q^{-22} -3 q^{-24} +2 q^{-26} -2 q^{-28} + q^{-30} - q^{-34} + q^{-36} }[/math]
5 [math]\displaystyle{ -q^{225}+2 q^{223}+2 q^{221}-3 q^{219}-3 q^{217}-3 q^{215}+9 q^{211}+16 q^{209}+q^{207}-20 q^{205}-29 q^{203}-19 q^{201}+19 q^{199}+61 q^{197}+65 q^{195}-8 q^{193}-97 q^{191}-128 q^{189}-59 q^{187}+101 q^{185}+232 q^{183}+192 q^{181}-50 q^{179}-314 q^{177}-380 q^{175}-132 q^{173}+317 q^{171}+608 q^{169}+433 q^{167}-179 q^{165}-764 q^{163}-819 q^{161}-168 q^{159}+769 q^{157}+1205 q^{155}+673 q^{153}-549 q^{151}-1463 q^{149}-1236 q^{147}+92 q^{145}+1502 q^{143}+1753 q^{141}+499 q^{139}-1299 q^{137}-2079 q^{135}-1103 q^{133}+880 q^{131}+2172 q^{129}+1600 q^{127}-375 q^{125}-2039 q^{123}-1894 q^{121}-105 q^{119}+1723 q^{117}+1983 q^{115}+488 q^{113}-1347 q^{111}-1887 q^{109}-723 q^{107}+973 q^{105}+1668 q^{103}+846 q^{101}-645 q^{99}-1424 q^{97}-882 q^{95}+374 q^{93}+1185 q^{91}+903 q^{89}-124 q^{87}-979 q^{85}-964 q^{83}-132 q^{81}+798 q^{79}+1067 q^{77}+447 q^{75}-597 q^{73}-1215 q^{71}-837 q^{69}+336 q^{67}+1349 q^{65}+1278 q^{63}+33 q^{61}-1400 q^{59}-1722 q^{57}-511 q^{55}+1301 q^{53}+2085 q^{51}+1038 q^{49}-1011 q^{47}-2255 q^{45}-1542 q^{43}+537 q^{41}+2191 q^{39}+1913 q^{37}-10 q^{35}-1840 q^{33}-2044 q^{31}-521 q^{29}+1324 q^{27}+1924 q^{25}+878 q^{23}-749 q^{21}-1559 q^{19}-1024 q^{17}+235 q^{15}+1098 q^{13}+964 q^{11}+102 q^9-649 q^7-735 q^5-264 q^3+292 q+489 q^{-1} +271 q^{-3} -86 q^{-5} -266 q^{-7} -199 q^{-9} -12 q^{-11} +122 q^{-13} +116 q^{-15} +34 q^{-17} -43 q^{-19} -59 q^{-21} -23 q^{-23} +13 q^{-25} +20 q^{-27} +12 q^{-29} +3 q^{-31} -10 q^{-33} -6 q^{-35} +3 q^{-37} +3 q^{-43} - q^{-45} -2 q^{-47} + q^{-49} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}+q^{18}+2 q^{16}-2 q^{14}-2 q^{10}+q^6-q^4+3 q^2+ q^{-4} }[/math]
1,1 [math]\displaystyle{ q^{68}-4 q^{66}+12 q^{64}-28 q^{62}+52 q^{60}-86 q^{58}+130 q^{56}-176 q^{54}+212 q^{52}-234 q^{50}+232 q^{48}-194 q^{46}+126 q^{44}-30 q^{42}-80 q^{40}+196 q^{38}-303 q^{36}+380 q^{34}-432 q^{32}+438 q^{30}-410 q^{28}+342 q^{26}-244 q^{24}+138 q^{22}-19 q^{20}-76 q^{18}+154 q^{16}-198 q^{14}+214 q^{12}-206 q^{10}+174 q^8-142 q^6+107 q^4-74 q^2+50-30 q^{-2} +21 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{66}+q^{64}-q^{62}-4 q^{60}-2 q^{58}+4 q^{56}+2 q^{54}-3 q^{52}+8 q^{48}+5 q^{46}-8 q^{44}-5 q^{42}+4 q^{40}-q^{38}-8 q^{36}-3 q^{34}+6 q^{32}+q^{30}-q^{28}+4 q^{26}+q^{24}-2 q^{22}+5 q^{20}+4 q^{18}-7 q^{16}-2 q^{14}+7 q^{12}+q^{10}-9 q^8-2 q^6+9 q^4-5+ q^{-2} +4 q^{-4} + q^{-6} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{54}-2 q^{52}+q^{50}+3 q^{48}-7 q^{46}+3 q^{44}+4 q^{42}-11 q^{40}+6 q^{38}+7 q^{36}-10 q^{34}+4 q^{32}+7 q^{30}-5 q^{28}-q^{26}+3 q^{24}+2 q^{22}-4 q^{20}-4 q^{18}+8 q^{16}-5 q^{14}-8 q^{12}+12 q^{10}-3 q^8-7 q^6+10 q^4-3+4 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0 [math]\displaystyle{ -q^{35}-q^{33}-q^{31}+2 q^{29}+3 q^{25}+q^{23}+2 q^{21}-2 q^{19}-q^{17}-2 q^{15}-2 q^{13}+2 q^7-q^5+3 q^3+ q^{-1} + q^{-5} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{54}+2 q^{52}-5 q^{50}+7 q^{48}-9 q^{46}+11 q^{44}-12 q^{42}+11 q^{40}-8 q^{38}+5 q^{36}+2 q^{34}-6 q^{32}+13 q^{30}-17 q^{28}+21 q^{26}-23 q^{24}+20 q^{22}-18 q^{20}+12 q^{18}-8 q^{16}+q^{14}+4 q^{12}-8 q^{10}+11 q^8-11 q^6+12 q^4-8 q^2+7-4 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0 [math]\displaystyle{ q^{88}-2 q^{84}-2 q^{82}+3 q^{80}+5 q^{78}-2 q^{76}-8 q^{74}-3 q^{72}+9 q^{70}+8 q^{68}-7 q^{66}-12 q^{64}+q^{62}+13 q^{60}+6 q^{58}-10 q^{56}-8 q^{54}+5 q^{52}+9 q^{50}-2 q^{48}-8 q^{46}+8 q^{42}+2 q^{40}-8 q^{38}-3 q^{36}+7 q^{34}+6 q^{32}-6 q^{30}-8 q^{28}+4 q^{26}+10 q^{24}-2 q^{22}-12 q^{20}-3 q^{18}+11 q^{16}+8 q^{14}-7 q^{12}-11 q^{10}+q^8+11 q^6+4 q^4-4 q^2-5+ q^{-2} +4 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-4 q^{118}-6 q^{116}+19 q^{114}-29 q^{112}+34 q^{110}-28 q^{108}+4 q^{106}+23 q^{104}-50 q^{102}+63 q^{100}-55 q^{98}+26 q^{96}+12 q^{94}-46 q^{92}+60 q^{90}-48 q^{88}+22 q^{86}+15 q^{84}-38 q^{82}+41 q^{80}-18 q^{78}-14 q^{76}+48 q^{74}-60 q^{72}+50 q^{70}-13 q^{68}-30 q^{66}+68 q^{64}-87 q^{62}+79 q^{60}-45 q^{58}-6 q^{56}+48 q^{54}-78 q^{52}+76 q^{50}-50 q^{48}+8 q^{46}+26 q^{44}-46 q^{42}+38 q^{40}-14 q^{38}-20 q^{36}+43 q^{34}-43 q^{32}+21 q^{30}+13 q^{28}-44 q^{26}+63 q^{24}-54 q^{22}+31 q^{20}-q^{18}-27 q^{16}+43 q^{14}-43 q^{12}+34 q^{10}-14 q^8+11 q^4-16 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/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 25"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -3 t^2+12 t-17+12 t^{-1} -3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 1-3 z^4 }[/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]= { 47, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q-2+5 q^{-1} -7 q^{-2} +8 q^{-3} -8 q^{-4} +7 q^{-5} -5 q^{-6} +3 q^{-7} - q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^8+3 z^2 a^6+3 a^6-2 z^4 a^4-4 z^2 a^4-3 a^4-z^4 a^2+a^2+z^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+4 z^2 a^8-a^8+3 z^7 a^7-4 z^5 a^7-2 z^3 a^7+z a^7+z^8 a^6+6 z^6 a^6-18 z^4 a^6+13 z^2 a^6-3 a^6+6 z^7 a^5-10 z^5 a^5+5 z^3 a^5-z a^5+z^8 a^4+6 z^6 a^4-15 z^4 a^4+13 z^2 a^4-3 a^4+3 z^7 a^3-3 z^5 a^3+3 z^3 a^3-z a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1 }[/math]
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