10 73

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-7 t^2+20 t-27+20 t^{-1} -7 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 { 83, -2 }
Jones polynomial [math]\displaystyle{ -q^2+4 q-7+11 q^{-1} -13 q^{-2} +14 q^{-3} -13 q^{-4} +10 q^{-5} -6 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^6-3 z^4 a^4-6 z^2 a^4-4 a^4+z^6 a^2+3 z^4 a^2+5 z^2 a^2+3 a^2-z^4-z^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+4 z^7 a^7-5 z^5 a^7+z^3 a^7+3 z^8 a^6+3 z^6 a^6-14 z^4 a^6+12 z^2 a^6-3 a^6+z^9 a^5+10 z^7 a^5-21 z^5 a^5+14 z^3 a^5-3 z a^5+7 z^8 a^4-2 z^6 a^4-17 z^4 a^4+17 z^2 a^4-4 a^4+z^9 a^3+12 z^7 a^3-26 z^5 a^3+16 z^3 a^3-3 z a^3+4 z^8 a^2+2 z^6 a^2-16 z^4 a^2+12 z^2 a^2-3 a^2+6 z^7 a-10 z^5 a+4 z^3 a-z a+4 z^6-7 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1} }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}+3 q^{16}-3 q^{14}-q^{10}-q^8+3 q^6-2 q^4+4 q^2- q^{-2} +2 q^{-4} - q^{-6} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+8 q^{120}-7 q^{118}-2 q^{116}+16 q^{114}-31 q^{112}+46 q^{110}-52 q^{108}+38 q^{106}-8 q^{104}-43 q^{102}+98 q^{100}-138 q^{98}+143 q^{96}-104 q^{94}+19 q^{92}+89 q^{90}-183 q^{88}+235 q^{86}-206 q^{84}+111 q^{82}+22 q^{80}-145 q^{78}+206 q^{76}-180 q^{74}+88 q^{72}+41 q^{70}-135 q^{68}+154 q^{66}-84 q^{64}-50 q^{62}+184 q^{60}-258 q^{58}+227 q^{56}-101 q^{54}-87 q^{52}+259 q^{50}-356 q^{48}+339 q^{46}-215 q^{44}+20 q^{42}+165 q^{40}-285 q^{38}+298 q^{36}-206 q^{34}+58 q^{32}+88 q^{30}-170 q^{28}+162 q^{26}-69 q^{24}-56 q^{22}+161 q^{20}-188 q^{18}+126 q^{16}+2 q^{14}-141 q^{12}+238 q^{10}-247 q^8+177 q^6-51 q^4-82 q^2+173-199 q^{-2} +165 q^{-4} -88 q^{-6} +7 q^{-8} +53 q^{-10} -83 q^{-12} +78 q^{-14} -52 q^{-16} +26 q^{-18} - q^{-20} -12 q^{-22} +14 q^{-24} -13 q^{-26} +7 q^{-28} -3 q^{-30} + q^{-32} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_73.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+2 q^{15}-3 q^{13}+4 q^{11}-3 q^9+q^7+q^5-2 q^3+4 q-3 q^{-1} +3 q^{-3} - q^{-5} }[/math]
2 [math]\displaystyle{ q^{48}-2 q^{46}-q^{44}+7 q^{42}-7 q^{40}-6 q^{38}+21 q^{36}-12 q^{34}-20 q^{32}+34 q^{30}-6 q^{28}-32 q^{26}+29 q^{24}+8 q^{22}-26 q^{20}+5 q^{18}+16 q^{16}-4 q^{14}-20 q^{12}+16 q^{10}+19 q^8-34 q^6+6 q^4+32 q^2-29-6 q^{-2} +27 q^{-4} -12 q^{-6} -9 q^{-8} +11 q^{-10} - q^{-12} -3 q^{-14} + q^{-16} }[/math]
3 [math]\displaystyle{ -q^{93}+2 q^{91}+q^{89}-3 q^{87}-4 q^{85}+7 q^{83}+9 q^{81}-15 q^{79}-18 q^{77}+24 q^{75}+38 q^{73}-34 q^{71}-69 q^{69}+39 q^{67}+111 q^{65}-31 q^{63}-158 q^{61}+2 q^{59}+203 q^{57}+40 q^{55}-224 q^{53}-94 q^{51}+218 q^{49}+147 q^{47}-189 q^{45}-179 q^{43}+132 q^{41}+190 q^{39}-62 q^{37}-182 q^{35}-7 q^{33}+154 q^{31}+75 q^{29}-117 q^{27}-135 q^{25}+70 q^{23}+187 q^{21}-22 q^{19}-221 q^{17}-33 q^{15}+237 q^{13}+88 q^{11}-223 q^9-136 q^7+191 q^5+168 q^3-138 q-173 q^{-1} +75 q^{-3} +160 q^{-5} -26 q^{-7} -124 q^{-9} -8 q^{-11} +82 q^{-13} +23 q^{-15} -44 q^{-17} -24 q^{-19} +22 q^{-21} +13 q^{-23} -6 q^{-25} -7 q^{-27} + q^{-29} +3 q^{-31} - q^{-33} }[/math]
4 [math]\displaystyle{ q^{152}-2 q^{150}-q^{148}+3 q^{146}+4 q^{142}-10 q^{140}-4 q^{138}+16 q^{136}+3 q^{134}+8 q^{132}-44 q^{130}-23 q^{128}+61 q^{126}+49 q^{124}+30 q^{122}-150 q^{120}-128 q^{118}+124 q^{116}+220 q^{114}+181 q^{112}-321 q^{110}-456 q^{108}+43 q^{106}+518 q^{104}+655 q^{102}-331 q^{100}-994 q^{98}-452 q^{96}+641 q^{94}+1412 q^{92}+153 q^{90}-1329 q^{88}-1285 q^{86}+208 q^{84}+1910 q^{82}+1010 q^{80}-1006 q^{78}-1852 q^{76}-622 q^{74}+1662 q^{72}+1604 q^{70}-189 q^{68}-1684 q^{66}-1238 q^{64}+821 q^{62}+1548 q^{60}+591 q^{58}-975 q^{56}-1357 q^{54}-113 q^{52}+1058 q^{50}+1082 q^{48}-144 q^{46}-1164 q^{44}-926 q^{42}+446 q^{40}+1384 q^{38}+658 q^{36}-833 q^{34}-1601 q^{32}-244 q^{30}+1456 q^{28}+1405 q^{26}-260 q^{24}-1958 q^{22}-1006 q^{20}+1069 q^{18}+1834 q^{16}+551 q^{14}-1675 q^{12}-1521 q^{10}+235 q^8+1602 q^6+1187 q^4-828 q^2-1389-510 q^{-2} +814 q^{-4} +1180 q^{-6} -20 q^{-8} -735 q^{-10} -671 q^{-12} +96 q^{-14} +673 q^{-16} +253 q^{-18} -152 q^{-20} -380 q^{-22} -146 q^{-24} +210 q^{-26} +150 q^{-28} +47 q^{-30} -110 q^{-32} -91 q^{-34} +32 q^{-36} +33 q^{-38} +33 q^{-40} -14 q^{-42} -23 q^{-44} +2 q^{-46} +2 q^{-48} +7 q^{-50} - q^{-52} -3 q^{-54} + q^{-56} }[/math]
5 [math]\displaystyle{ -q^{225}+2 q^{223}+q^{221}-3 q^{219}-q^{213}+5 q^{211}+3 q^{209}-12 q^{207}-6 q^{205}+10 q^{203}+16 q^{201}+17 q^{199}-10 q^{197}-55 q^{195}-55 q^{193}+27 q^{191}+123 q^{189}+126 q^{187}-14 q^{185}-229 q^{183}-306 q^{181}-61 q^{179}+398 q^{177}+625 q^{175}+264 q^{173}-543 q^{171}-1137 q^{169}-763 q^{167}+587 q^{165}+1857 q^{163}+1653 q^{161}-320 q^{159}-2637 q^{157}-3055 q^{155}-535 q^{153}+3253 q^{151}+4911 q^{149}+2191 q^{147}-3332 q^{145}-6933 q^{143}-4701 q^{141}+2443 q^{139}+8665 q^{137}+7895 q^{135}-411 q^{133}-9580 q^{131}-11166 q^{129}-2720 q^{127}+9170 q^{125}+13925 q^{123}+6507 q^{121}-7380 q^{119}-15506 q^{117}-10190 q^{115}+4392 q^{113}+15509 q^{111}+13140 q^{109}-786 q^{107}-14023 q^{105}-14803 q^{103}-2703 q^{101}+11307 q^{99}+15010 q^{97}+5640 q^{95}-8001 q^{93}-13975 q^{91}-7660 q^{89}+4641 q^{87}+12053 q^{85}+8802 q^{83}-1525 q^{81}-9779 q^{79}-9322 q^{77}-1139 q^{75}+7490 q^{73}+9474 q^{71}+3499 q^{69}-5335 q^{67}-9620 q^{65}-5685 q^{63}+3310 q^{61}+9789 q^{59}+7910 q^{57}-1225 q^{55}-9931 q^{53}-10218 q^{51}-1116 q^{49}+9775 q^{47}+12500 q^{45}+3829 q^{43}-9018 q^{41}-14397 q^{39}-6894 q^{37}+7358 q^{35}+15572 q^{33}+9970 q^{31}-4761 q^{29}-15524 q^{27}-12623 q^{25}+1377 q^{23}+14091 q^{21}+14345 q^{19}+2231 q^{17}-11319 q^{15}-14647 q^{13}-5480 q^{11}+7621 q^9+13462 q^7+7753 q^5-3742 q^3-10984 q-8606 q^{-1} +308 q^{-3} +7784 q^{-5} +8132 q^{-7} +2075 q^{-9} -4603 q^{-11} -6592 q^{-13} -3225 q^{-15} +1961 q^{-17} +4625 q^{-19} +3290 q^{-21} -200 q^{-23} -2761 q^{-25} -2676 q^{-27} -646 q^{-29} +1331 q^{-31} +1794 q^{-33} +867 q^{-35} -438 q^{-37} -1058 q^{-39} -702 q^{-41} +39 q^{-43} +494 q^{-45} +450 q^{-47} +117 q^{-49} -203 q^{-51} -254 q^{-53} -90 q^{-55} +60 q^{-57} +107 q^{-59} +60 q^{-61} -5 q^{-63} -47 q^{-65} -33 q^{-67} +5 q^{-69} +15 q^{-71} +8 q^{-73} +2 q^{-75} -2 q^{-77} -7 q^{-79} + q^{-81} +3 q^{-83} - q^{-85} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}+3 q^{16}-3 q^{14}-q^{10}-q^8+3 q^6-2 q^4+4 q^2- q^{-2} +2 q^{-4} - q^{-6} }[/math]
2,0 [math]\displaystyle{ q^{66}+q^{64}-q^{62}-4 q^{60}-q^{58}+5 q^{56}+q^{54}-6 q^{52}+13 q^{48}+3 q^{46}-16 q^{44}-5 q^{42}+15 q^{40}-q^{38}-19 q^{36}+q^{34}+17 q^{32}-11 q^{28}+9 q^{26}+5 q^{24}-10 q^{22}+4 q^{20}+6 q^{18}-11 q^{16}-5 q^{14}+16 q^{12}+q^{10}-18 q^8+3 q^6+21 q^4-4 q^2-16+8 q^{-2} +12 q^{-4} -5 q^{-6} -7 q^{-8} +2 q^{-10} +5 q^{-12} -2 q^{-14} -2 q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{54}-2 q^{52}+q^{50}+4 q^{48}-9 q^{46}+2 q^{44}+12 q^{42}-19 q^{40}+2 q^{38}+24 q^{36}-25 q^{34}-q^{32}+27 q^{30}-20 q^{28}-6 q^{26}+17 q^{24}-4 q^{22}-9 q^{20}-q^{18}+13 q^{16}-5 q^{14}-18 q^{12}+23 q^{10}+4 q^8-27 q^6+23 q^4+8 q^2-22+15 q^{-2} +5 q^{-4} -12 q^{-6} +6 q^{-8} + q^{-10} -3 q^{-12} + q^{-14} }[/math]
1,0,0 [math]\displaystyle{ -q^{35}-q^{33}-q^{31}+2 q^{29}+3 q^{25}+3 q^{21}-4 q^{19}-3 q^{15}-q^{13}+q^9+3 q^7-q^5+4 q^3-q+2 q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{54}+2 q^{52}-5 q^{50}+8 q^{48}-13 q^{46}+18 q^{44}-24 q^{42}+29 q^{40}-30 q^{38}+30 q^{36}-23 q^{34}+15 q^{32}-q^{30}-14 q^{28}+30 q^{26}-45 q^{24}+54 q^{22}-61 q^{20}+59 q^{18}-55 q^{16}+43 q^{14}-28 q^{12}+13 q^{10}+4 q^8-15 q^6+27 q^4-30 q^2+32-29 q^{-2} +25 q^{-4} -18 q^{-6} +12 q^{-8} -7 q^{-10} +3 q^{-12} - q^{-14} }[/math]
1,0 [math]\displaystyle{ q^{88}-2 q^{84}-2 q^{82}+3 q^{80}+6 q^{78}-q^{76}-11 q^{74}-7 q^{72}+11 q^{70}+18 q^{68}-3 q^{66}-26 q^{64}-12 q^{62}+23 q^{60}+28 q^{58}-10 q^{56}-34 q^{54}-7 q^{52}+30 q^{50}+19 q^{48}-20 q^{46}-24 q^{44}+10 q^{42}+24 q^{40}-3 q^{38}-22 q^{36}-2 q^{34}+21 q^{32}+6 q^{30}-19 q^{28}-11 q^{26}+18 q^{24}+16 q^{22}-16 q^{20}-23 q^{18}+10 q^{16}+31 q^{14}+2 q^{12}-32 q^{10}-18 q^8+26 q^6+30 q^4-8 q^2-30-7 q^{-2} +22 q^{-4} +17 q^{-6} -9 q^{-8} -15 q^{-10} - q^{-12} +9 q^{-14} +4 q^{-16} -3 q^{-18} -3 q^{-20} + q^{-24} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+8 q^{120}-7 q^{118}-2 q^{116}+16 q^{114}-31 q^{112}+46 q^{110}-52 q^{108}+38 q^{106}-8 q^{104}-43 q^{102}+98 q^{100}-138 q^{98}+143 q^{96}-104 q^{94}+19 q^{92}+89 q^{90}-183 q^{88}+235 q^{86}-206 q^{84}+111 q^{82}+22 q^{80}-145 q^{78}+206 q^{76}-180 q^{74}+88 q^{72}+41 q^{70}-135 q^{68}+154 q^{66}-84 q^{64}-50 q^{62}+184 q^{60}-258 q^{58}+227 q^{56}-101 q^{54}-87 q^{52}+259 q^{50}-356 q^{48}+339 q^{46}-215 q^{44}+20 q^{42}+165 q^{40}-285 q^{38}+298 q^{36}-206 q^{34}+58 q^{32}+88 q^{30}-170 q^{28}+162 q^{26}-69 q^{24}-56 q^{22}+161 q^{20}-188 q^{18}+126 q^{16}+2 q^{14}-141 q^{12}+238 q^{10}-247 q^8+177 q^6-51 q^4-82 q^2+173-199 q^{-2} +165 q^{-4} -88 q^{-6} +7 q^{-8} +53 q^{-10} -83 q^{-12} +78 q^{-14} -52 q^{-16} +26 q^{-18} - q^{-20} -12 q^{-22} +14 q^{-24} -13 q^{-26} +7 q^{-28} -3 q^{-30} + q^{-32} }[/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["10 73"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-7 t^2+20 t-27+20 t^{-1} -7 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]= { 83, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^2+4 q-7+11 q^{-1} -13 q^{-2} +14 q^{-3} -13 q^{-4} +10 q^{-5} -6 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^6-3 z^4 a^4-6 z^2 a^4-4 a^4+z^6 a^2+3 z^4 a^2+5 z^2 a^2+3 a^2-z^4-z^2 }[/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^8-6 z^4 a^8+4 z^2 a^8-a^8+4 z^7 a^7-5 z^5 a^7+z^3 a^7+3 z^8 a^6+3 z^6 a^6-14 z^4 a^6+12 z^2 a^6-3 a^6+z^9 a^5+10 z^7 a^5-21 z^5 a^5+14 z^3 a^5-3 z a^5+7 z^8 a^4-2 z^6 a^4-17 z^4 a^4+17 z^2 a^4-4 a^4+z^9 a^3+12 z^7 a^3-26 z^5 a^3+16 z^3 a^3-3 z a^3+4 z^8 a^2+2 z^6 a^2-16 z^4 a^2+12 z^2 a^2-3 a^2+6 z^7 a-10 z^5 a+4 z^3 a-z a+4 z^6-7 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1} }[/math]
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