10 90

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+8 t^2-17 t+23-17 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^6-4 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 77, 0 }
Jones polynomial [math]\displaystyle{ q^6-3 q^5+6 q^4-9 q^3+12 q^2-13 q+12-10 q^{-1} +7 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^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2+2 a^2+ a^{-4} -2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+7 a z^7+5 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +6 a^2 z^6-21 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -3 z^6+3 a^3 z^5-10 a z^5-15 z^5 a^{-1} -11 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4+15 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} -6 z^4-2 a^3 z^3+7 a z^3+9 z^3 a^{-1} +7 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+6 a^2 z^2-5 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +8 z^2-2 a z-2 z a^{-1} -z a^{-3} -z a^{-5} -2 a^2+ a^{-4} -2 }[/math]
The A2 invariant [math]\displaystyle{ q^{12}-q^{10}+2 q^8+2 q^6-2 q^4+2 q^2-3- q^{-6} +3 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+6 q^{58}-5 q^{56}+11 q^{52}-22 q^{50}+33 q^{48}-39 q^{46}+30 q^{44}-11 q^{42}-20 q^{40}+62 q^{38}-90 q^{36}+104 q^{34}-85 q^{32}+36 q^{30}+31 q^{28}-103 q^{26}+157 q^{24}-161 q^{22}+114 q^{20}-23 q^{18}-74 q^{16}+146 q^{14}-151 q^{12}+96 q^{10}-2 q^8-96 q^6+137 q^4-108 q^2+9+113 q^{-2} -199 q^{-4} +206 q^{-6} -126 q^{-8} -18 q^{-10} +157 q^{-12} -255 q^{-14} +265 q^{-16} -189 q^{-18} +54 q^{-20} +93 q^{-22} -195 q^{-24} +229 q^{-26} -176 q^{-28} +68 q^{-30} +46 q^{-32} -131 q^{-34} +148 q^{-36} -93 q^{-38} -4 q^{-40} +110 q^{-42} -162 q^{-44} +139 q^{-46} -48 q^{-48} -75 q^{-50} +172 q^{-52} -204 q^{-54} +161 q^{-56} -64 q^{-58} -48 q^{-60} +133 q^{-62} -159 q^{-64} +134 q^{-66} -69 q^{-68} + q^{-70} +45 q^{-72} -66 q^{-74} +59 q^{-76} -37 q^{-78} +16 q^{-80} +2 q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_90.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^9-2 q^7+4 q^5-3 q^3+2 q- q^{-1} - q^{-3} +3 q^{-5} -3 q^{-7} +3 q^{-9} -2 q^{-11} + q^{-13} }[/math]
2 [math]\displaystyle{ q^{26}-2 q^{24}+q^{22}+6 q^{20}-10 q^{18}+18 q^{14}-20 q^{12}-7 q^{10}+30 q^8-17 q^6-16 q^4+26 q^2-17 q^{-2} +5 q^{-4} +16 q^{-6} -9 q^{-8} -17 q^{-10} +22 q^{-12} +5 q^{-14} -29 q^{-16} +16 q^{-18} +17 q^{-20} -26 q^{-22} +3 q^{-24} +18 q^{-26} -12 q^{-28} -5 q^{-30} +8 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ q^{51}-2 q^{49}+q^{47}+3 q^{45}-q^{43}-7 q^{41}+q^{39}+15 q^{37}-5 q^{35}-28 q^{33}+11 q^{31}+48 q^{29}-7 q^{27}-86 q^{25}-3 q^{23}+124 q^{21}+31 q^{19}-151 q^{17}-78 q^{15}+160 q^{13}+125 q^{11}-133 q^9-160 q^7+85 q^5+169 q^3-21 q-155 q^{-1} -40 q^{-3} +123 q^{-5} +91 q^{-7} -83 q^{-9} -127 q^{-11} +42 q^{-13} +152 q^{-15} -170 q^{-19} -37 q^{-21} +171 q^{-23} +80 q^{-25} -159 q^{-27} -123 q^{-29} +131 q^{-31} +155 q^{-33} -82 q^{-35} -171 q^{-37} +25 q^{-39} +162 q^{-41} +28 q^{-43} -126 q^{-45} -64 q^{-47} +78 q^{-49} +73 q^{-51} -34 q^{-53} -59 q^{-55} +2 q^{-57} +37 q^{-59} +10 q^{-61} -17 q^{-63} -8 q^{-65} +5 q^{-67} +4 q^{-69} - q^{-71} -2 q^{-73} + q^{-75} }[/math]
4 [math]\displaystyle{ q^{84}-2 q^{82}+q^{80}+3 q^{78}-4 q^{76}+2 q^{74}-6 q^{72}+4 q^{70}+10 q^{68}-16 q^{66}+7 q^{64}-6 q^{62}+22 q^{60}+21 q^{58}-76 q^{56}-29 q^{54}+15 q^{52}+152 q^{50}+122 q^{48}-214 q^{46}-269 q^{44}-102 q^{42}+421 q^{40}+560 q^{38}-170 q^{36}-731 q^{34}-672 q^{32}+451 q^{30}+1245 q^{28}+442 q^{26}-844 q^{24}-1467 q^{22}-195 q^{20}+1414 q^{18}+1269 q^{16}-178 q^{14}-1617 q^{12}-1035 q^{10}+686 q^8+1430 q^6+703 q^4-882 q^2-1258-279 q^{-2} +843 q^{-4} +1071 q^{-6} +40 q^{-8} -903 q^{-10} -852 q^{-12} +175 q^{-14} +1020 q^{-16} +633 q^{-18} -525 q^{-20} -1116 q^{-22} -252 q^{-24} +928 q^{-26} +1047 q^{-28} -235 q^{-30} -1333 q^{-32} -676 q^{-34} +758 q^{-36} +1448 q^{-38} +259 q^{-40} -1309 q^{-42} -1202 q^{-44} +169 q^{-46} +1532 q^{-48} +976 q^{-50} -687 q^{-52} -1398 q^{-54} -715 q^{-56} +911 q^{-58} +1307 q^{-60} +290 q^{-62} -832 q^{-64} -1143 q^{-66} -63 q^{-68} +817 q^{-70} +769 q^{-72} +45 q^{-74} -733 q^{-76} -515 q^{-78} +59 q^{-80} +471 q^{-82} +393 q^{-84} -121 q^{-86} -292 q^{-88} -209 q^{-90} +50 q^{-92} +208 q^{-94} +78 q^{-96} -24 q^{-98} -96 q^{-100} -48 q^{-102} +33 q^{-104} +26 q^{-106} +19 q^{-108} -11 q^{-110} -14 q^{-112} +2 q^{-114} + q^{-116} +4 q^{-118} - q^{-120} -2 q^{-122} + q^{-124} }[/math]
5 [math]\displaystyle{ q^{125}-2 q^{123}+q^{121}+3 q^{119}-4 q^{117}-q^{115}+3 q^{113}-3 q^{111}-q^{109}+5 q^{107}-q^{105}+15 q^{101}-q^{99}-27 q^{97}-39 q^{95}-16 q^{93}+48 q^{91}+121 q^{89}+114 q^{87}-61 q^{85}-286 q^{83}-338 q^{81}-35 q^{79}+498 q^{77}+802 q^{75}+409 q^{73}-667 q^{71}-1577 q^{69}-1237 q^{67}+547 q^{65}+2514 q^{63}+2730 q^{61}+284 q^{59}-3369 q^{57}-4870 q^{55}-2063 q^{53}+3520 q^{51}+7226 q^{49}+4996 q^{47}-2436 q^{45}-9142 q^{43}-8627 q^{41}-136 q^{39}+9702 q^{37}+12097 q^{35}+4020 q^{33}-8385 q^{31}-14428 q^{29}-8283 q^{27}+5247 q^{25}+14680 q^{23}+11867 q^{21}-914 q^{19}-12781 q^{17}-13779 q^{15}-3448 q^{13}+9153 q^{11}+13615 q^9+6881 q^7-4783 q^5-11677 q^3-8780 q+692 q^{-1} +8712 q^{-3} +9167 q^{-5} +2435 q^{-7} -5583 q^{-9} -8546 q^{-11} -4429 q^{-13} +3030 q^{-15} +7576 q^{-17} +5501 q^{-19} -1302 q^{-21} -6829 q^{-23} -6153 q^{-25} +275 q^{-27} +6647 q^{-29} +6865 q^{-31} +376 q^{-33} -6928 q^{-35} -7942 q^{-37} -1212 q^{-39} +7358 q^{-41} +9501 q^{-43} +2569 q^{-45} -7451 q^{-47} -11203 q^{-49} -4686 q^{-51} +6652 q^{-53} +12622 q^{-55} +7432 q^{-57} -4693 q^{-59} -13095 q^{-61} -10270 q^{-63} +1496 q^{-65} +12104 q^{-67} +12524 q^{-69} +2473 q^{-71} -9478 q^{-73} -13383 q^{-75} -6370 q^{-77} +5424 q^{-79} +12375 q^{-81} +9281 q^{-83} -798 q^{-85} -9546 q^{-87} -10368 q^{-89} -3345 q^{-91} +5490 q^{-93} +9409 q^{-95} +6061 q^{-97} -1303 q^{-99} -6839 q^{-101} -6794 q^{-103} -1963 q^{-105} +3538 q^{-107} +5770 q^{-109} +3679 q^{-111} -605 q^{-113} -3737 q^{-115} -3768 q^{-117} -1258 q^{-119} +1580 q^{-121} +2800 q^{-123} +1913 q^{-125} -51 q^{-127} -1513 q^{-129} -1629 q^{-131} -664 q^{-133} +462 q^{-135} +993 q^{-137} +739 q^{-139} +80 q^{-141} -418 q^{-143} -488 q^{-145} -226 q^{-147} +76 q^{-149} +230 q^{-151} +181 q^{-153} +28 q^{-155} -74 q^{-157} -82 q^{-159} -37 q^{-161} +6 q^{-163} +33 q^{-165} +22 q^{-167} -3 q^{-169} -8 q^{-171} -4 q^{-173} -2 q^{-175} + q^{-177} +4 q^{-179} - q^{-181} -2 q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{12}-q^{10}+2 q^8+2 q^6-2 q^4+2 q^2-3- q^{-6} +3 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{36}-4 q^{34}+10 q^{32}-20 q^{30}+42 q^{28}-74 q^{26}+120 q^{24}-180 q^{22}+264 q^{20}-358 q^{18}+454 q^{16}-556 q^{14}+643 q^{12}-692 q^{10}+672 q^8-584 q^6+404 q^4-130 q^2-206+588 q^{-2} -941 q^{-4} +1248 q^{-6} -1450 q^{-8} +1526 q^{-10} -1481 q^{-12} +1298 q^{-14} -1014 q^{-16} +658 q^{-18} -275 q^{-20} -94 q^{-22} +424 q^{-24} -660 q^{-26} +789 q^{-28} -818 q^{-30} +760 q^{-32} -640 q^{-34} +489 q^{-36} -344 q^{-38} +220 q^{-40} -126 q^{-42} +66 q^{-44} -30 q^{-46} +12 q^{-48} -4 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{32}-q^{30}+5 q^{26}-6 q^{22}+2 q^{20}+9 q^{18}-2 q^{16}-15 q^{14}+2 q^{12}+12 q^{10}-10 q^8-10 q^6+11 q^4+8 q^2-4+9 q^{-4} -2 q^{-6} -6 q^{-8} +7 q^{-10} - q^{-12} -11 q^{-14} +5 q^{-16} +11 q^{-18} -8 q^{-20} -8 q^{-22} +7 q^{-24} +8 q^{-26} -7 q^{-28} -8 q^{-30} +8 q^{-32} +4 q^{-34} -5 q^{-36} -3 q^{-38} +2 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }[/math]

A3 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-8 q^{22}+13 q^{20}-17 q^{18}+23 q^{16}-25 q^{14}+28 q^{12}-24 q^{10}+19 q^8-9 q^6-3 q^4+16 q^2-31+40 q^{-2} -50 q^{-4} +52 q^{-6} -51 q^{-8} +44 q^{-10} -33 q^{-12} +21 q^{-14} -6 q^{-16} -6 q^{-18} +17 q^{-20} -23 q^{-22} +27 q^{-24} -26 q^{-26} +24 q^{-28} -20 q^{-30} +14 q^{-32} -9 q^{-34} +5 q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0 [math]\displaystyle{ q^{46}-2 q^{42}-2 q^{40}+2 q^{38}+7 q^{36}+2 q^{34}-10 q^{32}-10 q^{30}+6 q^{28}+20 q^{26}+5 q^{24}-20 q^{22}-18 q^{20}+13 q^{18}+27 q^{16}+q^{14}-27 q^{12}-13 q^{10}+20 q^8+20 q^6-12 q^4-22 q^2+5+21 q^{-2} +2 q^{-4} -20 q^{-6} -6 q^{-8} +16 q^{-10} +9 q^{-12} -15 q^{-14} -12 q^{-16} +14 q^{-18} +17 q^{-20} -10 q^{-22} -24 q^{-24} +3 q^{-26} +28 q^{-28} +11 q^{-30} -24 q^{-32} -23 q^{-34} +13 q^{-36} +28 q^{-38} + q^{-40} -23 q^{-42} -12 q^{-44} +14 q^{-46} +15 q^{-48} -4 q^{-50} -11 q^{-52} -2 q^{-54} +6 q^{-56} +3 q^{-58} -2 q^{-60} -2 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}+7 q^{30}-10 q^{28}+10 q^{26}-12 q^{24}+20 q^{22}-19 q^{20}+19 q^{18}-19 q^{16}+24 q^{14}-15 q^{12}+11 q^{10}-9 q^8+2 q^6+9 q^4-16 q^2+19-31 q^{-2} +35 q^{-4} -38 q^{-6} +38 q^{-8} -43 q^{-10} +38 q^{-12} -32 q^{-14} +29 q^{-16} -23 q^{-18} +14 q^{-20} -4 q^{-22} +7 q^{-26} -13 q^{-28} +20 q^{-30} -19 q^{-32} +21 q^{-34} -21 q^{-36} +21 q^{-38} -17 q^{-40} +14 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-48} +3 q^{-50} -2 q^{-52} + q^{-54} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+6 q^{58}-5 q^{56}+11 q^{52}-22 q^{50}+33 q^{48}-39 q^{46}+30 q^{44}-11 q^{42}-20 q^{40}+62 q^{38}-90 q^{36}+104 q^{34}-85 q^{32}+36 q^{30}+31 q^{28}-103 q^{26}+157 q^{24}-161 q^{22}+114 q^{20}-23 q^{18}-74 q^{16}+146 q^{14}-151 q^{12}+96 q^{10}-2 q^8-96 q^6+137 q^4-108 q^2+9+113 q^{-2} -199 q^{-4} +206 q^{-6} -126 q^{-8} -18 q^{-10} +157 q^{-12} -255 q^{-14} +265 q^{-16} -189 q^{-18} +54 q^{-20} +93 q^{-22} -195 q^{-24} +229 q^{-26} -176 q^{-28} +68 q^{-30} +46 q^{-32} -131 q^{-34} +148 q^{-36} -93 q^{-38} -4 q^{-40} +110 q^{-42} -162 q^{-44} +139 q^{-46} -48 q^{-48} -75 q^{-50} +172 q^{-52} -204 q^{-54} +161 q^{-56} -64 q^{-58} -48 q^{-60} +133 q^{-62} -159 q^{-64} +134 q^{-66} -69 q^{-68} + q^{-70} +45 q^{-72} -66 q^{-74} +59 q^{-76} -37 q^{-78} +16 q^{-80} +2 q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 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 90"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+8 t^2-17 t+23-17 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^6-4 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]= { 77, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-3 q^5+6 q^4-9 q^3+12 q^2-13 q+12-10 q^{-1} +7 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^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2+2 a^2+ a^{-4} -2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+7 a z^7+5 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +6 a^2 z^6-21 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -3 z^6+3 a^3 z^5-10 a z^5-15 z^5 a^{-1} -11 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4+15 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} -6 z^4-2 a^3 z^3+7 a z^3+9 z^3 a^{-1} +7 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+6 a^2 z^2-5 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +8 z^2-2 a z-2 z a^{-1} -z a^{-3} -z a^{-5} -2 a^2+ a^{-4} -2 }[/math]
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