10 52

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-7 t^2+13 t-15+13 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+5 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 59, 2 }
Jones polynomial [math]\displaystyle{ -q^6+3 q^5-6 q^4+8 q^3-9 q^2+10 q-8+7 q^{-1} -4 q^{-2} +2 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +6 z^2-2 a^2- a^{-4} +4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+4 z^8 a^{-2} +6 z^8+a^3 z^7+a z^7+7 z^7 a^{-1} +7 z^7 a^{-3} -9 a^2 z^6-3 z^6 a^{-2} +8 z^6 a^{-4} -20 z^6-5 a^3 z^5-16 a z^5-28 z^5 a^{-1} -11 z^5 a^{-3} +6 z^5 a^{-5} +13 a^2 z^4-9 z^4 a^{-2} -12 z^4 a^{-4} +3 z^4 a^{-6} +19 z^4+8 a^3 z^3+24 a z^3+24 z^3 a^{-1} +2 z^3 a^{-3} -5 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+4 z^2 a^{-2} +6 z^2 a^{-4} -9 z^2-4 a^3 z-9 a z-7 z a^{-1} +2 z a^{-5} +2 a^2- a^{-4} +4 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}-q^8-q^6+2 q^4+3+2 q^{-2} +2 q^{-6} -2 q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} - q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{60}-q^{58}+4 q^{56}-6 q^{54}+6 q^{52}-6 q^{50}-q^{48}+12 q^{46}-25 q^{44}+33 q^{42}-31 q^{40}+12 q^{38}+14 q^{36}-46 q^{34}+64 q^{32}-64 q^{30}+38 q^{28}-46 q^{24}+71 q^{22}-71 q^{20}+46 q^{18}-4 q^{16}-32 q^{14}+53 q^{12}-47 q^{10}+21 q^8+18 q^6-42 q^4+56 q^2-36+2 q^{-2} +45 q^{-4} -75 q^{-6} +88 q^{-8} -62 q^{-10} +17 q^{-12} +40 q^{-14} -83 q^{-16} +100 q^{-18} -82 q^{-20} +39 q^{-22} +15 q^{-24} -58 q^{-26} +72 q^{-28} -57 q^{-30} +21 q^{-32} +17 q^{-34} -41 q^{-36} +39 q^{-38} -19 q^{-40} -13 q^{-42} +38 q^{-44} -48 q^{-46} +40 q^{-48} -15 q^{-50} -17 q^{-52} +41 q^{-54} -55 q^{-56} +50 q^{-58} -32 q^{-60} +8 q^{-62} +15 q^{-64} -33 q^{-66} +39 q^{-68} -36 q^{-70} +26 q^{-72} -9 q^{-74} -5 q^{-76} +12 q^{-78} -18 q^{-80} +16 q^{-82} -12 q^{-84} +8 q^{-86} - q^{-88} -2 q^{-90} +3 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_52.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^9+q^7-2 q^5+3 q^3-q+2 q^{-1} + q^{-3} - q^{-5} +2 q^{-7} -3 q^{-9} +2 q^{-11} - q^{-13} }[/math]
2 [math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+4 q^{22}-8 q^{18}+6 q^{16}+6 q^{14}-13 q^{12}+3 q^{10}+12 q^8-12 q^6-3 q^4+14 q^2-4-7 q^{-2} +8 q^{-4} +5 q^{-6} -8 q^{-8} -2 q^{-10} +11 q^{-12} -2 q^{-14} -12 q^{-16} +11 q^{-18} +3 q^{-20} -14 q^{-22} +8 q^{-24} +3 q^{-26} -8 q^{-28} +4 q^{-30} + q^{-32} -2 q^{-34} + q^{-36} }[/math]
3 [math]\displaystyle{ -q^{57}+q^{55}+2 q^{53}-5 q^{49}-2 q^{47}+8 q^{45}+8 q^{43}-10 q^{41}-17 q^{39}+6 q^{37}+28 q^{35}+4 q^{33}-35 q^{31}-20 q^{29}+33 q^{27}+38 q^{25}-26 q^{23}-49 q^{21}+8 q^{19}+57 q^{17}+7 q^{15}-54 q^{13}-24 q^{11}+50 q^9+36 q^7-37 q^5-47 q^3+28 q+52 q^{-1} -13 q^{-3} -56 q^{-5} +57 q^{-9} +14 q^{-11} -46 q^{-13} -29 q^{-15} +35 q^{-17} +41 q^{-19} -12 q^{-21} -50 q^{-23} -8 q^{-25} +46 q^{-27} +30 q^{-29} -43 q^{-31} -41 q^{-33} +30 q^{-35} +43 q^{-37} -21 q^{-39} -36 q^{-41} +14 q^{-43} +25 q^{-45} -9 q^{-47} -18 q^{-49} +8 q^{-51} +10 q^{-53} -6 q^{-55} -3 q^{-57} +3 q^{-59} +2 q^{-61} -2 q^{-63} - q^{-65} +2 q^{-67} - q^{-69} }[/math]
4 [math]\displaystyle{ q^{96}-q^{94}-2 q^{92}+q^{88}+7 q^{86}-8 q^{82}-8 q^{80}-6 q^{78}+22 q^{76}+20 q^{74}-3 q^{72}-28 q^{70}-48 q^{68}+14 q^{66}+56 q^{64}+58 q^{62}-2 q^{60}-110 q^{58}-73 q^{56}+19 q^{54}+133 q^{52}+128 q^{50}-68 q^{48}-163 q^{46}-145 q^{44}+71 q^{42}+247 q^{40}+114 q^{38}-88 q^{36}-278 q^{34}-136 q^{32}+180 q^{30}+256 q^{28}+125 q^{26}-222 q^{24}-289 q^{22}-21 q^{20}+223 q^{18}+284 q^{16}-48 q^{14}-286 q^{12}-180 q^{10}+103 q^8+313 q^6+86 q^4-214 q^2-243+15 q^{-2} +287 q^{-4} +157 q^{-6} -158 q^{-8} -277 q^{-10} -41 q^{-12} +253 q^{-14} +224 q^{-16} -74 q^{-18} -288 q^{-20} -139 q^{-22} +143 q^{-24} +280 q^{-26} +98 q^{-28} -196 q^{-30} -245 q^{-32} -92 q^{-34} +212 q^{-36} +282 q^{-38} +40 q^{-40} -222 q^{-42} -320 q^{-44} +10 q^{-46} +303 q^{-48} +245 q^{-50} -51 q^{-52} -350 q^{-54} -156 q^{-56} +162 q^{-58} +248 q^{-60} +82 q^{-62} -213 q^{-64} -146 q^{-66} +40 q^{-68} +124 q^{-70} +84 q^{-72} -91 q^{-74} -60 q^{-76} +11 q^{-78} +34 q^{-80} +36 q^{-82} -39 q^{-84} -10 q^{-86} +8 q^{-88} +3 q^{-90} +13 q^{-92} -15 q^{-94} + q^{-96} +4 q^{-98} -3 q^{-100} +3 q^{-102} -4 q^{-104} +2 q^{-106} + q^{-108} -2 q^{-110} + q^{-112} }[/math]
5 [math]\displaystyle{ -q^{145}+q^{143}+2 q^{141}-q^{137}-3 q^{135}-5 q^{133}+10 q^{129}+10 q^{127}+3 q^{125}-8 q^{123}-24 q^{121}-23 q^{119}+6 q^{117}+40 q^{115}+50 q^{113}+24 q^{111}-38 q^{109}-96 q^{107}-86 q^{105}+2 q^{103}+120 q^{101}+175 q^{99}+101 q^{97}-83 q^{95}-254 q^{93}-264 q^{91}-56 q^{89}+254 q^{87}+430 q^{85}+302 q^{83}-97 q^{81}-509 q^{79}-598 q^{77}-222 q^{75}+401 q^{73}+807 q^{71}+646 q^{69}-47 q^{67}-815 q^{65}-1046 q^{63}-472 q^{61}+536 q^{59}+1230 q^{57}+1048 q^{55}+18 q^{53}-1134 q^{51}-1478 q^{49}-684 q^{47}+689 q^{45}+1620 q^{43}+1335 q^{41}-39 q^{39}-1442 q^{37}-1752 q^{35}-690 q^{33}+971 q^{31}+1897 q^{29}+1304 q^{27}-353 q^{25}-1742 q^{23}-1715 q^{21}-259 q^{19}+1399 q^{17}+1862 q^{15}+756 q^{13}-964 q^{11}-1823 q^9-1061 q^7+596 q^5+1638 q^3+1185 q-312 q^{-1} -1455 q^{-3} -1182 q^{-5} +185 q^{-7} +1311 q^{-9} +1128 q^{-11} -141 q^{-13} -1248 q^{-15} -1124 q^{-17} +117 q^{-19} +1264 q^{-21} +1196 q^{-23} -30 q^{-25} -1249 q^{-27} -1354 q^{-29} -222 q^{-31} +1149 q^{-33} +1534 q^{-35} +617 q^{-37} -825 q^{-39} -1622 q^{-41} -1134 q^{-43} +275 q^{-45} +1513 q^{-47} +1612 q^{-49} +475 q^{-51} -1123 q^{-53} -1936 q^{-55} -1249 q^{-57} +469 q^{-59} +1937 q^{-61} +1926 q^{-63} +302 q^{-65} -1639 q^{-67} -2272 q^{-69} -1034 q^{-71} +1073 q^{-73} +2285 q^{-75} +1547 q^{-77} -464 q^{-79} -1967 q^{-81} -1729 q^{-83} -82 q^{-85} +1471 q^{-87} +1629 q^{-89} +416 q^{-91} -965 q^{-93} -1316 q^{-95} -521 q^{-97} +538 q^{-99} +936 q^{-101} +481 q^{-103} -265 q^{-105} -602 q^{-107} -339 q^{-109} +118 q^{-111} +335 q^{-113} +209 q^{-115} -50 q^{-117} -176 q^{-119} -107 q^{-121} +33 q^{-123} +87 q^{-125} +35 q^{-127} -19 q^{-129} -37 q^{-131} -15 q^{-133} +15 q^{-135} +19 q^{-137} + q^{-139} -12 q^{-141} -4 q^{-143} +3 q^{-145} + q^{-147} + q^{-149} +2 q^{-151} -3 q^{-153} - q^{-155} +4 q^{-157} -2 q^{-159} - q^{-161} +2 q^{-163} - q^{-165} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{12}-q^8-q^6+2 q^4+3+2 q^{-2} +2 q^{-6} -2 q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} - q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{36}-2 q^{34}+8 q^{32}-18 q^{30}+35 q^{28}-66 q^{26}+104 q^{24}-148 q^{22}+190 q^{20}-230 q^{18}+248 q^{16}-238 q^{14}+201 q^{12}-142 q^{10}+52 q^8+58 q^6-163 q^4+266 q^2-354+420 q^{-2} -440 q^{-4} +432 q^{-6} -382 q^{-8} +316 q^{-10} -213 q^{-12} +118 q^{-14} -24 q^{-16} -60 q^{-18} +115 q^{-20} -154 q^{-22} +168 q^{-24} -178 q^{-26} +166 q^{-28} -146 q^{-30} +126 q^{-32} -110 q^{-34} +87 q^{-36} -62 q^{-38} +48 q^{-40} -36 q^{-42} +22 q^{-44} -12 q^{-46} +8 q^{-48} -4 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{34}-q^{30}+2 q^{26}-4 q^{22}-2 q^{20}+4 q^{18}-6 q^{14}-q^{12}+4 q^{10}+q^8-6 q^6+2 q^4+7 q^2+6 q^{-4} + q^{-6} -2 q^{-8} +4 q^{-10} +4 q^{-12} -3 q^{-14} - q^{-16} +7 q^{-18} -10 q^{-22} +4 q^{-26} -5 q^{-28} -5 q^{-30} +2 q^{-32} +3 q^{-34} - q^{-36} - q^{-38} +2 q^{-40} - q^{-44} + q^{-46} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{26}-q^{24}+2 q^{22}+q^{20}-4 q^{18}+3 q^{16}-3 q^{14}-10 q^{12}+4 q^{10}-3 q^8-8 q^6+13 q^4+4 q^2-2+14 q^{-2} +6 q^{-4} -5 q^{-6} +2 q^{-8} +2 q^{-10} - q^{-12} -8 q^{-14} + q^{-16} +6 q^{-18} -11 q^{-20} +10 q^{-24} -10 q^{-26} -2 q^{-28} +10 q^{-30} -6 q^{-32} -3 q^{-34} +6 q^{-36} - q^{-38} -2 q^{-40} + q^{-42} }[/math]
1,0,0 [math]\displaystyle{ -q^{15}-2 q^{11}-2 q^7+2 q^5+4 q+2 q^{-1} +3 q^{-3} + q^{-5} + q^{-9} -2 q^{-11} + q^{-13} -2 q^{-15} + q^{-17} -2 q^{-19} + q^{-21} - q^{-23} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{32}+q^{28}+3 q^{26}+q^{24}-q^{22}+2 q^{20}-3 q^{18}-10 q^{16}-7 q^{14}-6 q^{12}-11 q^{10}-10 q^8+7 q^6+11 q^4+3 q^2+13+26 q^{-2} +10 q^{-4} - q^{-6} +9 q^{-8} +4 q^{-10} -10 q^{-12} -7 q^{-14} + q^{-16} -5 q^{-18} -11 q^{-20} +3 q^{-22} +3 q^{-24} -9 q^{-26} -2 q^{-28} +8 q^{-30} -2 q^{-32} -7 q^{-34} +3 q^{-36} +5 q^{-38} -2 q^{-40} -4 q^{-42} +3 q^{-44} +3 q^{-46} -2 q^{-48} - q^{-50} + q^{-52} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{18}-2 q^{14}-q^{12}-q^{10}-2 q^8+2 q^6+4 q^2+3+3 q^{-2} +3 q^{-4} + q^{-6} + q^{-8} - q^{-10} + q^{-12} -2 q^{-14} + q^{-16} -2 q^{-18} -2 q^{-24} + q^{-26} - q^{-28} }[/math]

B2 Invariants.

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

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{34}-q^{32}+3 q^{30}-3 q^{28}+6 q^{26}-7 q^{24}+7 q^{22}-12 q^{20}+8 q^{18}-16 q^{16}+6 q^{14}-14 q^{12}+8 q^{10}-7 q^8+5 q^6+5 q^4+4 q^2+16-6 q^{-2} +21 q^{-4} -14 q^{-6} +22 q^{-8} -22 q^{-10} +21 q^{-12} -22 q^{-14} +17 q^{-16} -18 q^{-18} +11 q^{-20} -9 q^{-22} +4 q^{-24} -2 q^{-26} -5 q^{-28} +5 q^{-30} -8 q^{-32} +10 q^{-34} -12 q^{-36} +9 q^{-38} -10 q^{-40} +12 q^{-42} -8 q^{-44} +5 q^{-46} -6 q^{-48} +6 q^{-50} -2 q^{-52} + q^{-54} -2 q^{-56} + q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{60}-q^{58}+4 q^{56}-6 q^{54}+6 q^{52}-6 q^{50}-q^{48}+12 q^{46}-25 q^{44}+33 q^{42}-31 q^{40}+12 q^{38}+14 q^{36}-46 q^{34}+64 q^{32}-64 q^{30}+38 q^{28}-46 q^{24}+71 q^{22}-71 q^{20}+46 q^{18}-4 q^{16}-32 q^{14}+53 q^{12}-47 q^{10}+21 q^8+18 q^6-42 q^4+56 q^2-36+2 q^{-2} +45 q^{-4} -75 q^{-6} +88 q^{-8} -62 q^{-10} +17 q^{-12} +40 q^{-14} -83 q^{-16} +100 q^{-18} -82 q^{-20} +39 q^{-22} +15 q^{-24} -58 q^{-26} +72 q^{-28} -57 q^{-30} +21 q^{-32} +17 q^{-34} -41 q^{-36} +39 q^{-38} -19 q^{-40} -13 q^{-42} +38 q^{-44} -48 q^{-46} +40 q^{-48} -15 q^{-50} -17 q^{-52} +41 q^{-54} -55 q^{-56} +50 q^{-58} -32 q^{-60} +8 q^{-62} +15 q^{-64} -33 q^{-66} +39 q^{-68} -36 q^{-70} +26 q^{-72} -9 q^{-74} -5 q^{-76} +12 q^{-78} -18 q^{-80} +16 q^{-82} -12 q^{-84} +8 q^{-86} - q^{-88} -2 q^{-90} +3 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/math]

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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 52"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-7 t^2+13 t-15+13 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+5 z^4+3 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]= { 59, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^6+3 q^5-6 q^4+8 q^3-9 q^2+10 q-8+7 q^{-1} -4 q^{-2} +2 q^{-3} - q^{-4} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +6 z^2-2 a^2- a^{-4} +4 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+4 z^8 a^{-2} +6 z^8+a^3 z^7+a z^7+7 z^7 a^{-1} +7 z^7 a^{-3} -9 a^2 z^6-3 z^6 a^{-2} +8 z^6 a^{-4} -20 z^6-5 a^3 z^5-16 a z^5-28 z^5 a^{-1} -11 z^5 a^{-3} +6 z^5 a^{-5} +13 a^2 z^4-9 z^4 a^{-2} -12 z^4 a^{-4} +3 z^4 a^{-6} +19 z^4+8 a^3 z^3+24 a z^3+24 z^3 a^{-1} +2 z^3 a^{-3} -5 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+4 z^2 a^{-2} +6 z^2 a^{-4} -9 z^2-4 a^3 z-9 a z-7 z a^{-1} +2 z a^{-5} +2 a^2- a^{-4} +4 }[/math]
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