10 26

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+7 t^2-13 t+17-13 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 { 61, 0 }
Jones polynomial [math]\displaystyle{ q^6-2 q^5+4 q^4-7 q^3+9 q^2-10 q+10-8 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-6 z^2 a^{-2} +3 z^2 a^{-4} -2 z^2+a^2-3 a^{-2} +2 a^{-4} +1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8+5 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +2 z^7 a^{-5} +5 a^2 z^6-12 z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -z^6+3 a^3 z^5-7 a z^5-9 z^5 a^{-1} -6 z^5 a^{-3} -7 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+14 z^4 a^{-2} +4 z^4 a^{-4} -4 z^4 a^{-6} -2 z^4-3 a^3 z^3+5 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+4 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} +4 z^2 a^{-6} +z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -a^2+3 a^{-2} +2 a^{-4} +1 }[/math]
The A2 invariant [math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-q^4+3 q^2-1+ q^{-2} - q^{-4} -2 q^{-6} + q^{-8} -2 q^{-10} + q^{-12} + q^{-14} + q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+12 q^{52}-20 q^{50}+28 q^{48}-29 q^{46}+17 q^{44}+q^{42}-27 q^{40}+53 q^{38}-65 q^{36}+62 q^{34}-39 q^{32}+43 q^{28}-73 q^{26}+85 q^{24}-66 q^{22}+28 q^{20}+16 q^{18}-49 q^{16}+60 q^{14}-40 q^{12}+4 q^{10}+37 q^8-57 q^6+47 q^4-6 q^2-47+94 q^{-2} -108 q^{-4} +83 q^{-6} -27 q^{-8} -44 q^{-10} +103 q^{-12} -130 q^{-14} +114 q^{-16} -64 q^{-18} -4 q^{-20} +59 q^{-22} -92 q^{-24} +83 q^{-26} -49 q^{-28} - q^{-30} +38 q^{-32} -57 q^{-34} +41 q^{-36} - q^{-38} -41 q^{-40} +69 q^{-42} -70 q^{-44} +38 q^{-46} +11 q^{-48} -57 q^{-50} +88 q^{-52} -83 q^{-54} +59 q^{-56} -14 q^{-58} -29 q^{-60} +57 q^{-62} -62 q^{-64} +51 q^{-66} -27 q^{-68} +2 q^{-70} +16 q^{-72} -24 q^{-74} +24 q^{-76} -17 q^{-78} +9 q^{-80} - q^{-82} -4 q^{-84} +4 q^{-86} -5 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_26.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^9-2 q^7+3 q^5-2 q^3+2 q- q^{-3} +2 q^{-5} -3 q^{-7} +2 q^{-9} - q^{-11} + q^{-13} }[/math]
2 [math]\displaystyle{ q^{26}-2 q^{24}+6 q^{20}-7 q^{18}-3 q^{16}+15 q^{14}-10 q^{12}-9 q^{10}+19 q^8-7 q^6-12 q^4+13 q^2+2-8 q^{-2} - q^{-4} +10 q^{-6} - q^{-8} -13 q^{-10} +12 q^{-12} +8 q^{-14} -18 q^{-16} +6 q^{-18} +12 q^{-20} -14 q^{-22} +9 q^{-26} -6 q^{-28} -2 q^{-30} +4 q^{-32} - q^{-34} - q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ q^{51}-2 q^{49}+3 q^{45}+q^{43}-6 q^{41}-4 q^{39}+13 q^{37}+7 q^{35}-20 q^{33}-13 q^{31}+27 q^{29}+26 q^{27}-37 q^{25}-39 q^{23}+42 q^{21}+55 q^{19}-42 q^{17}-70 q^{15}+33 q^{13}+79 q^{11}-19 q^9-77 q^7+q^5+67 q^3+19 q-48 q^{-1} -35 q^{-3} +26 q^{-5} +51 q^{-7} -4 q^{-9} -55 q^{-11} -19 q^{-13} +62 q^{-15} +37 q^{-17} -60 q^{-19} -55 q^{-21} +49 q^{-23} +66 q^{-25} -36 q^{-27} -75 q^{-29} +19 q^{-31} +76 q^{-33} -67 q^{-37} -17 q^{-39} +57 q^{-41} +26 q^{-43} -38 q^{-45} -30 q^{-47} +22 q^{-49} +26 q^{-51} -9 q^{-53} -19 q^{-55} + q^{-57} +12 q^{-59} + q^{-61} -6 q^{-63} -2 q^{-65} +3 q^{-67} + q^{-69} - q^{-71} - q^{-73} + q^{-75} }[/math]
4 [math]\displaystyle{ q^{84}-2 q^{82}+3 q^{78}-2 q^{76}+2 q^{74}-7 q^{72}+q^{70}+12 q^{68}-5 q^{66}+4 q^{64}-23 q^{62}+37 q^{58}-2 q^{56}-2 q^{54}-64 q^{52}-4 q^{50}+92 q^{48}+31 q^{46}-15 q^{44}-160 q^{42}-46 q^{40}+180 q^{38}+143 q^{36}+3 q^{34}-302 q^{32}-175 q^{30}+216 q^{28}+307 q^{26}+126 q^{24}-365 q^{22}-348 q^{20}+108 q^{18}+374 q^{16}+291 q^{14}-246 q^{12}-401 q^{10}-89 q^8+250 q^6+356 q^4-14 q^2-276-231 q^{-2} +30 q^{-4} +276 q^{-6} +187 q^{-8} -75 q^{-10} -280 q^{-12} -157 q^{-14} +147 q^{-16} +308 q^{-18} +93 q^{-20} -277 q^{-22} -278 q^{-24} +22 q^{-26} +373 q^{-28} +227 q^{-30} -234 q^{-32} -353 q^{-34} -115 q^{-36} +365 q^{-38} +340 q^{-40} -106 q^{-42} -355 q^{-44} -279 q^{-46} +237 q^{-48} +387 q^{-50} +86 q^{-52} -235 q^{-54} -374 q^{-56} +21 q^{-58} +289 q^{-60} +222 q^{-62} -27 q^{-64} -305 q^{-66} -133 q^{-68} +98 q^{-70} +196 q^{-72} +111 q^{-74} -133 q^{-76} -129 q^{-78} -35 q^{-80} +78 q^{-82} +107 q^{-84} -15 q^{-86} -48 q^{-88} -49 q^{-90} +2 q^{-92} +46 q^{-94} +9 q^{-96} -2 q^{-98} -20 q^{-100} -9 q^{-102} +12 q^{-104} +2 q^{-106} +4 q^{-108} -4 q^{-110} -4 q^{-112} +3 q^{-114} + q^{-118} - q^{-120} - q^{-122} + q^{-124} }[/math]
5 [math]\displaystyle{ q^{125}-2 q^{123}+3 q^{119}-2 q^{117}-q^{115}+q^{113}-2 q^{111}+7 q^{107}+q^{105}-8 q^{103}-3 q^{101}-q^{99}+5 q^{97}+10 q^{95}+5 q^{93}-16 q^{91}-25 q^{89}+5 q^{87}+36 q^{85}+41 q^{83}-3 q^{81}-74 q^{79}-100 q^{77}-9 q^{75}+156 q^{73}+208 q^{71}+46 q^{69}-247 q^{67}-395 q^{65}-176 q^{63}+334 q^{61}+689 q^{59}+411 q^{57}-375 q^{55}-1006 q^{53}-802 q^{51}+268 q^{49}+1330 q^{47}+1305 q^{45}-1520 q^{41}-1829 q^{39}-461 q^{37}+1502 q^{35}+2271 q^{33}+1028 q^{31}-1234 q^{29}-2494 q^{27}-1578 q^{25}+741 q^{23}+2422 q^{21}+1992 q^{19}-128 q^{17}-2069 q^{15}-2173 q^{13}-467 q^{11}+1491 q^9+2089 q^7+961 q^5-828 q^3-1793 q-1275 q^{-1} +193 q^{-3} +1376 q^{-5} +1409 q^{-7} +361 q^{-9} -938 q^{-11} -1448 q^{-13} -760 q^{-15} +559 q^{-17} +1413 q^{-19} +1068 q^{-21} -247 q^{-23} -1416 q^{-25} -1305 q^{-27} +39 q^{-29} +1416 q^{-31} +1528 q^{-33} +167 q^{-35} -1453 q^{-37} -1766 q^{-39} -382 q^{-41} +1451 q^{-43} +2007 q^{-45} +674 q^{-47} -1363 q^{-49} -2207 q^{-51} -1050 q^{-53} +1121 q^{-55} +2324 q^{-57} +1451 q^{-59} -712 q^{-61} -2258 q^{-63} -1829 q^{-65} +156 q^{-67} +1975 q^{-69} +2084 q^{-71} +451 q^{-73} -1485 q^{-75} -2108 q^{-77} -998 q^{-79} +833 q^{-81} +1889 q^{-83} +1383 q^{-85} -173 q^{-87} -1447 q^{-89} -1484 q^{-91} -401 q^{-93} +875 q^{-95} +1360 q^{-97} +753 q^{-99} -335 q^{-101} -1019 q^{-103} -865 q^{-105} -101 q^{-107} +620 q^{-109} +770 q^{-111} +335 q^{-113} -258 q^{-115} -550 q^{-117} -394 q^{-119} +5 q^{-121} +316 q^{-123} +332 q^{-125} +112 q^{-127} -129 q^{-129} -217 q^{-131} -134 q^{-133} +18 q^{-135} +114 q^{-137} +103 q^{-139} +27 q^{-141} -44 q^{-143} -63 q^{-145} -29 q^{-147} +9 q^{-149} +27 q^{-151} +24 q^{-153} +3 q^{-155} -14 q^{-157} -10 q^{-159} -2 q^{-161} +6 q^{-165} +4 q^{-167} -3 q^{-169} -2 q^{-171} + q^{-173} + q^{-179} - q^{-181} - q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-q^4+3 q^2-1+ q^{-2} - q^{-4} -2 q^{-6} + q^{-8} -2 q^{-10} + q^{-12} + q^{-14} + q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{36}-4 q^{34}+10 q^{32}-20 q^{30}+38 q^{28}-64 q^{26}+100 q^{24}-144 q^{22}+197 q^{20}-246 q^{18}+290 q^{16}-322 q^{14}+326 q^{12}-292 q^{10}+226 q^8-124 q^6-12 q^4+172 q^2-332+474 q^{-2} -590 q^{-4} +656 q^{-6} -672 q^{-8} +630 q^{-10} -537 q^{-12} +410 q^{-14} -250 q^{-16} +98 q^{-18} +56 q^{-20} -180 q^{-22} +272 q^{-24} -322 q^{-26} +329 q^{-28} -318 q^{-30} +274 q^{-32} -220 q^{-34} +166 q^{-36} -118 q^{-38} +78 q^{-40} -46 q^{-42} +27 q^{-44} -14 q^{-46} +6 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{32}-q^{30}-q^{28}+3 q^{26}+q^{24}-3 q^{22}+6 q^{18}-8 q^{14}+3 q^{12}+8 q^{10}-5 q^8-7 q^6+6 q^4+2 q^2-6- q^{-2} +5 q^{-4} -3 q^{-6} -4 q^{-8} +7 q^{-10} +2 q^{-12} -5 q^{-14} +5 q^{-16} +9 q^{-18} -3 q^{-20} -4 q^{-22} +4 q^{-24} +3 q^{-26} -7 q^{-28} -5 q^{-30} +3 q^{-32} + q^{-34} -3 q^{-36} - q^{-38} +2 q^{-40} + q^{-42} + q^{-48} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+10 q^{20}-13 q^{18}+16 q^{16}-16 q^{14}+16 q^{12}-12 q^{10}+8 q^8-6 q^4+16 q^2-23+28 q^{-2} -31 q^{-4} +30 q^{-6} -29 q^{-8} +22 q^{-10} -16 q^{-12} +7 q^{-14} -7 q^{-18} +12 q^{-20} -14 q^{-22} +16 q^{-24} -14 q^{-26} +13 q^{-28} -10 q^{-30} +7 q^{-32} -5 q^{-34} +3 q^{-36} - q^{-38} + q^{-40} }[/math]
1,0 [math]\displaystyle{ q^{46}-2 q^{42}-2 q^{40}+2 q^{38}+6 q^{36}+q^{34}-8 q^{32}-7 q^{30}+6 q^{28}+14 q^{26}+q^{24}-15 q^{22}-10 q^{20}+11 q^{18}+16 q^{16}-2 q^{14}-16 q^{12}-4 q^{10}+13 q^8+10 q^6-9 q^4-10 q^2+6+11 q^{-2} -2 q^{-4} -11 q^{-6} - q^{-8} +9 q^{-10} +2 q^{-12} -11 q^{-14} -5 q^{-16} +10 q^{-18} +8 q^{-20} -10 q^{-22} -14 q^{-24} +5 q^{-26} +18 q^{-28} +4 q^{-30} -15 q^{-32} -11 q^{-34} +10 q^{-36} +16 q^{-38} -12 q^{-42} -6 q^{-44} +7 q^{-46} +7 q^{-48} -2 q^{-50} -6 q^{-52} -2 q^{-54} +3 q^{-56} +2 q^{-58} - q^{-60} - q^{-62} + q^{-66} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{38}-2 q^{36}+2 q^{34}-3 q^{32}+6 q^{30}-8 q^{28}+8 q^{26}-9 q^{24}+14 q^{22}-13 q^{20}+11 q^{18}-11 q^{16}+13 q^{14}-6 q^{12}+3 q^{10}-q^8-q^6+12 q^4-12 q^2+17-20 q^{-2} +24 q^{-4} -23 q^{-6} +21 q^{-8} -27 q^{-10} +18 q^{-12} -19 q^{-14} +11 q^{-16} -13 q^{-18} +3 q^{-20} + q^{-22} -2 q^{-24} +8 q^{-26} -7 q^{-28} +15 q^{-30} -9 q^{-32} +13 q^{-34} -11 q^{-36} +11 q^{-38} -9 q^{-40} +6 q^{-42} -7 q^{-44} +4 q^{-46} -3 q^{-48} +2 q^{-50} - q^{-52} + q^{-54} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+12 q^{52}-20 q^{50}+28 q^{48}-29 q^{46}+17 q^{44}+q^{42}-27 q^{40}+53 q^{38}-65 q^{36}+62 q^{34}-39 q^{32}+43 q^{28}-73 q^{26}+85 q^{24}-66 q^{22}+28 q^{20}+16 q^{18}-49 q^{16}+60 q^{14}-40 q^{12}+4 q^{10}+37 q^8-57 q^6+47 q^4-6 q^2-47+94 q^{-2} -108 q^{-4} +83 q^{-6} -27 q^{-8} -44 q^{-10} +103 q^{-12} -130 q^{-14} +114 q^{-16} -64 q^{-18} -4 q^{-20} +59 q^{-22} -92 q^{-24} +83 q^{-26} -49 q^{-28} - q^{-30} +38 q^{-32} -57 q^{-34} +41 q^{-36} - q^{-38} -41 q^{-40} +69 q^{-42} -70 q^{-44} +38 q^{-46} +11 q^{-48} -57 q^{-50} +88 q^{-52} -83 q^{-54} +59 q^{-56} -14 q^{-58} -29 q^{-60} +57 q^{-62} -62 q^{-64} +51 q^{-66} -27 q^{-68} +2 q^{-70} +16 q^{-72} -24 q^{-74} +24 q^{-76} -17 q^{-78} +9 q^{-80} - q^{-82} -4 q^{-84} +4 q^{-86} -5 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }[/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 26"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+7 t^2-13 t+17-13 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]= { 61, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-2 q^5+4 q^4-7 q^3+9 q^2-10 q+10-8 q^{-1} +6 q^{-2} -3 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^4-4 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-6 z^2 a^{-2} +3 z^2 a^{-4} -2 z^2+a^2-3 a^{-2} +2 a^{-4} +1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8+5 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +2 z^7 a^{-5} +5 a^2 z^6-12 z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -z^6+3 a^3 z^5-7 a z^5-9 z^5 a^{-1} -6 z^5 a^{-3} -7 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+14 z^4 a^{-2} +4 z^4 a^{-4} -4 z^4 a^{-6} -2 z^4-3 a^3 z^3+5 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+4 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} +4 z^2 a^{-6} +z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -a^2+3 a^{-2} +2 a^{-4} +1 }[/math]
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