9 34

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+6 t^2-16 t+23-16 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 { 69, 0 }
Jones polynomial [math]\displaystyle{ q^4-4 q^3+8 q^2-10 q+12-12 q^{-1} +10 q^{-2} -7 q^{-3} +4 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+3 a^2 z^2+z^2 a^{-2} -4 z^2+a^2+ a^{-2} -1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 a^2 z^8+3 z^8+6 a^3 z^7+14 a z^7+8 z^7 a^{-1} +4 a^4 z^6+5 a^2 z^6+8 z^6 a^{-2} +9 z^6+a^5 z^5-11 a^3 z^5-26 a z^5-10 z^5 a^{-1} +4 z^5 a^{-3} -7 a^4 z^4-19 a^2 z^4-10 z^4 a^{-2} +z^4 a^{-4} -23 z^4-a^5 z^3+5 a^3 z^3+12 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +3 a^4 z^2+10 a^2 z^2+4 z^2 a^{-2} +11 z^2-a z-z a^{-1} -a^2- a^{-2} -1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-2 q^{10}+2 q^8-q^6-q^4+2 q^2-2+3 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-12 q^{70}-q^{68}+27 q^{66}-55 q^{64}+83 q^{62}-84 q^{60}+44 q^{58}+24 q^{56}-112 q^{54}+181 q^{52}-188 q^{50}+127 q^{48}-11 q^{46}-114 q^{44}+205 q^{42}-213 q^{40}+135 q^{38}-7 q^{36}-116 q^{34}+167 q^{32}-131 q^{30}+24 q^{28}+103 q^{26}-178 q^{24}+183 q^{22}-102 q^{20}-37 q^{18}+174 q^{16}-269 q^{14}+272 q^{12}-182 q^{10}+32 q^8+130 q^6-244 q^4+280 q^2-217+85 q^{-2} +58 q^{-4} -170 q^{-6} +191 q^{-8} -117 q^{-10} -6 q^{-12} +123 q^{-14} -165 q^{-16} +125 q^{-18} -16 q^{-20} -113 q^{-22} +195 q^{-24} -206 q^{-26} +141 q^{-28} -26 q^{-30} -88 q^{-32} +159 q^{-34} -165 q^{-36} +126 q^{-38} -55 q^{-40} -10 q^{-42} +48 q^{-44} -68 q^{-46} +60 q^{-48} -38 q^{-50} +18 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_34.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+3 q^9-3 q^7+3 q^5-2 q^3+2 q^{-1} -2 q^{-3} +4 q^{-5} -3 q^{-7} + q^{-9} }[/math]
2 [math]\displaystyle{ q^{32}-3 q^{30}-q^{28}+12 q^{26}-8 q^{24}-16 q^{22}+25 q^{20}+q^{18}-32 q^{16}+22 q^{14}+16 q^{12}-28 q^{10}+6 q^8+19 q^6-10 q^4-13 q^2+11+14 q^{-2} -25 q^{-4} -2 q^{-6} +33 q^{-8} -21 q^{-10} -15 q^{-12} +30 q^{-14} -7 q^{-16} -15 q^{-18} +10 q^{-20} + q^{-22} -3 q^{-24} + q^{-26} }[/math]
3 [math]\displaystyle{ -q^{63}+3 q^{61}+q^{59}-8 q^{57}-7 q^{55}+16 q^{53}+30 q^{51}-24 q^{49}-64 q^{47}+6 q^{45}+107 q^{43}+40 q^{41}-138 q^{39}-106 q^{37}+135 q^{35}+178 q^{33}-96 q^{31}-232 q^{29}+37 q^{27}+250 q^{25}+28 q^{23}-234 q^{21}-86 q^{19}+198 q^{17}+119 q^{15}-144 q^{13}-137 q^{11}+85 q^9+150 q^7-25 q^5-151 q^3-42 q+150 q^{-1} +110 q^{-3} -132 q^{-5} -180 q^{-7} +98 q^{-9} +234 q^{-11} -44 q^{-13} -255 q^{-15} -22 q^{-17} +239 q^{-19} +81 q^{-21} -187 q^{-23} -113 q^{-25} +121 q^{-27} +105 q^{-29} -53 q^{-31} -82 q^{-33} +15 q^{-35} +48 q^{-37} -2 q^{-39} -18 q^{-41} -3 q^{-43} +7 q^{-45} + q^{-47} -3 q^{-49} + q^{-51} }[/math]
4 [math]\displaystyle{ q^{104}-3 q^{102}-q^{100}+8 q^{98}+3 q^{96}-q^{94}-30 q^{92}-20 q^{90}+45 q^{88}+67 q^{86}+55 q^{84}-123 q^{82}-211 q^{80}-29 q^{78}+231 q^{76}+437 q^{74}+26 q^{72}-539 q^{70}-606 q^{68}-39 q^{66}+965 q^{64}+866 q^{62}-237 q^{60}-1287 q^{58}-1138 q^{56}+669 q^{54}+1720 q^{52}+979 q^{50}-1012 q^{48}-2141 q^{46}-546 q^{44}+1536 q^{42}+2020 q^{40}+111 q^{38}-2061 q^{36}-1539 q^{34}+565 q^{32}+2042 q^{30}+996 q^{28}-1229 q^{26}-1668 q^{24}-264 q^{22}+1418 q^{20}+1232 q^{18}-399 q^{16}-1348 q^{14}-721 q^{12}+758 q^{10}+1234 q^8+313 q^6-1013 q^4-1169 q^2+46+1278 q^{-2} +1203 q^{-4} -513 q^{-6} -1679 q^{-8} -981 q^{-10} +1003 q^{-12} +2097 q^{-14} +482 q^{-16} -1631 q^{-18} -2013 q^{-20} +11 q^{-22} +2191 q^{-24} +1535 q^{-26} -628 q^{-28} -2103 q^{-30} -1083 q^{-32} +1160 q^{-34} +1627 q^{-36} +495 q^{-38} -1100 q^{-40} -1210 q^{-42} +49 q^{-44} +779 q^{-46} +690 q^{-48} -151 q^{-50} -563 q^{-52} -227 q^{-54} +109 q^{-56} +291 q^{-58} +71 q^{-60} -111 q^{-62} -74 q^{-64} -24 q^{-66} +53 q^{-68} +22 q^{-70} -13 q^{-72} -6 q^{-74} -6 q^{-76} +7 q^{-78} + q^{-80} -3 q^{-82} + q^{-84} }[/math]
5 [math]\displaystyle{ -q^{155}+3 q^{153}+q^{151}-8 q^{149}-3 q^{147}+5 q^{145}+15 q^{143}+20 q^{141}-q^{139}-59 q^{137}-86 q^{135}-15 q^{133}+123 q^{131}+240 q^{129}+177 q^{127}-129 q^{125}-545 q^{123}-631 q^{121}-100 q^{119}+802 q^{117}+1419 q^{115}+982 q^{113}-623 q^{111}-2384 q^{109}-2611 q^{107}-542 q^{105}+2761 q^{103}+4754 q^{101}+3135 q^{99}-1778 q^{97}-6538 q^{95}-6794 q^{93}-1213 q^{91}+6706 q^{89}+10559 q^{87}+6079 q^{85}-4446 q^{83}-12971 q^{81}-11708 q^{79}-320 q^{77}+12798 q^{75}+16578 q^{73}+6682 q^{71}-9776 q^{69}-19250 q^{67}-13013 q^{65}+4536 q^{63}+19015 q^{61}+17848 q^{59}+1556 q^{57}-16212 q^{55}-20243 q^{53}-7017 q^{51}+11839 q^{49}+20042 q^{47}+10881 q^{45}-7083 q^{43}-17954 q^{41}-12766 q^{39}+2989 q^{37}+14851 q^{35}+12931 q^{33}+56 q^{31}-11650 q^{29}-12114 q^{27}-1995 q^{25}+8933 q^{23}+11006 q^{21}+3244 q^{19}-6807 q^{17}-10249 q^{15}-4442 q^{13}+5184 q^{11}+10107 q^9+6046 q^7-3522 q^5-10451 q^3-8507 q+1299 q^{-1} +10908 q^{-3} +11682 q^{-5} +1980 q^{-7} -10674 q^{-9} -15162 q^{-11} -6449 q^{-13} +9048 q^{-15} +18039 q^{-17} +11723 q^{-19} -5574 q^{-21} -19230 q^{-23} -16847 q^{-25} +368 q^{-27} +17919 q^{-29} +20551 q^{-31} +5661 q^{-33} -13978 q^{-35} -21629 q^{-37} -11107 q^{-39} +8117 q^{-41} +19635 q^{-43} +14545 q^{-45} -1807 q^{-47} -15140 q^{-49} -15083 q^{-51} -3294 q^{-53} +9385 q^{-55} +12979 q^{-57} +6171 q^{-59} -4122 q^{-61} -9312 q^{-63} -6523 q^{-65} +394 q^{-67} +5411 q^{-69} +5270 q^{-71} +1385 q^{-73} -2442 q^{-75} -3377 q^{-77} -1635 q^{-79} +677 q^{-81} +1733 q^{-83} +1211 q^{-85} +37 q^{-87} -739 q^{-89} -650 q^{-91} -151 q^{-93} +224 q^{-95} +283 q^{-97} +113 q^{-99} -60 q^{-101} -110 q^{-103} -41 q^{-105} +20 q^{-107} +27 q^{-109} +11 q^{-111} - q^{-113} -9 q^{-115} -6 q^{-117} +7 q^{-119} + q^{-121} -3 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-2 q^{10}+2 q^8-q^6-q^4+2 q^2-2+3 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} }[/math]
1,1 [math]\displaystyle{ q^{44}-6 q^{42}+20 q^{40}-50 q^{38}+107 q^{36}-204 q^{34}+344 q^{32}-514 q^{30}+701 q^{28}-864 q^{26}+950 q^{24}-938 q^{22}+791 q^{20}-502 q^{18}+98 q^{16}+374 q^{14}-840 q^{12}+1276 q^{10}-1606 q^8+1788 q^6-1806 q^4+1650 q^2-1352+934 q^{-2} -456 q^{-4} -16 q^{-6} +432 q^{-8} -726 q^{-10} +888 q^{-12} -918 q^{-14} +850 q^{-16} -712 q^{-18} +538 q^{-20} -376 q^{-22} +248 q^{-24} -148 q^{-26} +77 q^{-28} -38 q^{-30} +18 q^{-32} -6 q^{-34} + q^{-36} }[/math]
2,0 [math]\displaystyle{ q^{42}-q^{40}-3 q^{38}+7 q^{34}+5 q^{32}-10 q^{30}-7 q^{28}+9 q^{26}+7 q^{24}-13 q^{22}-7 q^{20}+13 q^{18}+8 q^{16}-11 q^{14}-q^{12}+13 q^{10}-5 q^8-3 q^6+4 q^4-q^2-7+5 q^{-2} +7 q^{-4} -13 q^{-6} -4 q^{-8} +16 q^{-10} +6 q^{-12} -17 q^{-14} + q^{-16} +14 q^{-18} -9 q^{-22} -3 q^{-24} +6 q^{-26} -2 q^{-30} + q^{-32} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+12 q^{28}-18 q^{26}+24 q^{24}-25 q^{22}+26 q^{20}-22 q^{18}+15 q^{16}-4 q^{14}-9 q^{12}+22 q^{10}-35 q^8+44 q^6-49 q^4+49 q^2-44+34 q^{-2} -20 q^{-4} +8 q^{-6} +5 q^{-8} -15 q^{-10} +23 q^{-12} -26 q^{-14} +26 q^{-16} -23 q^{-18} +18 q^{-20} -11 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28} }[/math]
1,0 [math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+9 q^{46}-q^{44}-15 q^{42}-8 q^{40}+18 q^{38}+19 q^{36}-10 q^{34}-27 q^{32}-3 q^{30}+28 q^{28}+15 q^{26}-21 q^{24}-22 q^{22}+10 q^{20}+23 q^{18}-q^{16}-20 q^{14}-3 q^{12}+18 q^{10}+7 q^8-16 q^6-11 q^4+13 q^2+16-10 q^{-2} -20 q^{-4} +7 q^{-6} +23 q^{-8} + q^{-10} -25 q^{-12} -10 q^{-14} +23 q^{-16} +22 q^{-18} -14 q^{-20} -27 q^{-22} + q^{-24} +23 q^{-26} +10 q^{-28} -13 q^{-30} -14 q^{-32} +3 q^{-34} +10 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}-3 q^{44}+4 q^{42}-6 q^{40}+10 q^{38}-15 q^{36}+17 q^{34}-18 q^{32}+22 q^{30}-21 q^{28}+18 q^{26}-15 q^{24}+14 q^{22}-5 q^{20}-5 q^{18}+10 q^{16}-14 q^{14}+25 q^{12}-33 q^{10}+33 q^8-35 q^6+40 q^4-37 q^2+31-29 q^{-2} +25 q^{-4} -12 q^{-6} +4 q^{-8} -2 q^{-10} -5 q^{-12} +16 q^{-14} -17 q^{-16} +18 q^{-18} -22 q^{-20} +23 q^{-22} -16 q^{-24} +14 q^{-26} -14 q^{-28} +10 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-12 q^{70}-q^{68}+27 q^{66}-55 q^{64}+83 q^{62}-84 q^{60}+44 q^{58}+24 q^{56}-112 q^{54}+181 q^{52}-188 q^{50}+127 q^{48}-11 q^{46}-114 q^{44}+205 q^{42}-213 q^{40}+135 q^{38}-7 q^{36}-116 q^{34}+167 q^{32}-131 q^{30}+24 q^{28}+103 q^{26}-178 q^{24}+183 q^{22}-102 q^{20}-37 q^{18}+174 q^{16}-269 q^{14}+272 q^{12}-182 q^{10}+32 q^8+130 q^6-244 q^4+280 q^2-217+85 q^{-2} +58 q^{-4} -170 q^{-6} +191 q^{-8} -117 q^{-10} -6 q^{-12} +123 q^{-14} -165 q^{-16} +125 q^{-18} -16 q^{-20} -113 q^{-22} +195 q^{-24} -206 q^{-26} +141 q^{-28} -26 q^{-30} -88 q^{-32} +159 q^{-34} -165 q^{-36} +126 q^{-38} -55 q^{-40} -10 q^{-42} +48 q^{-44} -68 q^{-46} +60 q^{-48} -38 q^{-50} +18 q^{-52} + q^{-54} -8 q^{-56} +10 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 34"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+6 t^2-16 t+23-16 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]= { 69, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^4-4 q^3+8 q^2-10 q+12-12 q^{-1} +10 q^{-2} -7 q^{-3} +4 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+3 a^2 z^2+z^2 a^{-2} -4 z^2+a^2+ a^{-2} -1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 a^2 z^8+3 z^8+6 a^3 z^7+14 a z^7+8 z^7 a^{-1} +4 a^4 z^6+5 a^2 z^6+8 z^6 a^{-2} +9 z^6+a^5 z^5-11 a^3 z^5-26 a z^5-10 z^5 a^{-1} +4 z^5 a^{-3} -7 a^4 z^4-19 a^2 z^4-10 z^4 a^{-2} +z^4 a^{-4} -23 z^4-a^5 z^3+5 a^3 z^3+12 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +3 a^4 z^2+10 a^2 z^2+4 z^2 a^{-2} +11 z^2-a z-z a^{-1} -a^2- a^{-2} -1 }[/math]
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