10 57

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Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^3-8 t^2+18 t-23+18 t^{-1} -8 t^{-2} +2 t^{-3} }
Conway polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^6+4 z^4+4 z^2+1}
2nd Alexander ideal (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature { 79, 2 }
Jones polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8+3 q^7-7 q^6+10 q^5-12 q^4+14 q^3-12 q^2+10 q-6+3 q^{-1} - q^{-2} }
HOMFLY-PT polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+4 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} -2 z^2+2 a^{-2} +2 a^{-4} -2 a^{-6} -1}
Kauffman polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +4 z^7 a^{-1} +9 z^7 a^{-3} +10 z^7 a^{-5} +5 z^7 a^{-7} +2 z^6 a^{-2} -7 z^6 a^{-4} -3 z^6 a^{-6} +3 z^6 a^{-8} +3 z^6+a z^5-5 z^5 a^{-1} -19 z^5 a^{-3} -23 z^5 a^{-5} -9 z^5 a^{-7} +z^5 a^{-9} -11 z^4 a^{-2} -z^4 a^{-4} -z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-2 a z^3+12 z^3 a^{-3} +18 z^3 a^{-5} +6 z^3 a^{-7} -2 z^3 a^{-9} +8 z^2 a^{-2} -2 z^2 a^{-6} +2 z^2 a^{-8} +4 z^2+a z+z a^{-1} -2 z a^{-3} -6 z a^{-5} -3 z a^{-7} +z a^{-9} -2 a^{-2} +2 a^{-4} +2 a^{-6} -1}
The A2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+q^4-q^2-1+3 q^{-2} -2 q^{-4} +3 q^{-6} + q^{-8} + q^{-10} +3 q^{-12} -2 q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-22} - q^{-24} }
The G2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+8 q^{24}-7 q^{22}-2 q^{20}+16 q^{18}-32 q^{16}+46 q^{14}-51 q^{12}+35 q^{10}-4 q^8-46 q^6+99 q^4-131 q^2+132-89 q^{-2} +5 q^{-4} +95 q^{-6} -177 q^{-8} +210 q^{-10} -173 q^{-12} +77 q^{-14} +42 q^{-16} -142 q^{-18} +181 q^{-20} -139 q^{-22} +45 q^{-24} +70 q^{-26} -141 q^{-28} +131 q^{-30} -45 q^{-32} -84 q^{-34} +200 q^{-36} -242 q^{-38} +197 q^{-40} -59 q^{-42} -109 q^{-44} +259 q^{-46} -323 q^{-48} +286 q^{-50} -155 q^{-52} -17 q^{-54} +169 q^{-56} -248 q^{-58} +240 q^{-60} -143 q^{-62} +13 q^{-64} +100 q^{-66} -156 q^{-68} +120 q^{-70} -25 q^{-72} -91 q^{-74} +166 q^{-76} -168 q^{-78} +91 q^{-80} +31 q^{-82} -152 q^{-84} +220 q^{-86} -214 q^{-88} +134 q^{-90} -21 q^{-92} -92 q^{-94} +155 q^{-96} -160 q^{-98} +122 q^{-100} -55 q^{-102} -6 q^{-104} +46 q^{-106} -64 q^{-108} +54 q^{-110} -34 q^{-112} +15 q^{-114} + q^{-116} -8 q^{-118} +9 q^{-120} -8 q^{-122} +5 q^{-124} -2 q^{-126} + q^{-128} }
Further Quantum Invariants
Further quantum knot invariants for 10_57.

A1 Invariants.

Weight Invariant
1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+2 q^3-3 q+4 q^{-1} -2 q^{-3} +2 q^{-5} +2 q^{-7} -2 q^{-9} +3 q^{-11} -4 q^{-13} +2 q^{-15} - q^{-17} }
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{16}-2 q^{14}-q^{12}+7 q^{10}-7 q^8-7 q^6+20 q^4-10 q^2-20+32 q^{-2} -2 q^{-4} -30 q^{-6} +25 q^{-8} +11 q^{-10} -22 q^{-12} +3 q^{-14} +16 q^{-16} - q^{-18} -21 q^{-20} +12 q^{-22} +19 q^{-24} -32 q^{-26} +2 q^{-28} +30 q^{-30} -25 q^{-32} -8 q^{-34} +24 q^{-36} -9 q^{-38} -9 q^{-40} +8 q^{-42} -2 q^{-46} + q^{-48} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{33}+2 q^{31}+q^{29}-3 q^{27}-4 q^{25}+7 q^{23}+10 q^{21}-14 q^{19}-20 q^{17}+20 q^{15}+39 q^{13}-26 q^{11}-69 q^9+25 q^7+108 q^5-11 q^3-147 q-25 q^{-1} +183 q^{-3} +68 q^{-5} -190 q^{-7} -121 q^{-9} +178 q^{-11} +165 q^{-13} -136 q^{-15} -185 q^{-17} +84 q^{-19} +182 q^{-21} -18 q^{-23} -162 q^{-25} -39 q^{-27} +126 q^{-29} +92 q^{-31} -80 q^{-33} -142 q^{-35} +34 q^{-37} +173 q^{-39} +18 q^{-41} -201 q^{-43} -65 q^{-45} +198 q^{-47} +117 q^{-49} -182 q^{-51} -153 q^{-53} +141 q^{-55} +174 q^{-57} -90 q^{-59} -167 q^{-61} +34 q^{-63} +142 q^{-65} +7 q^{-67} -104 q^{-69} -25 q^{-71} +59 q^{-73} +32 q^{-75} -28 q^{-77} -23 q^{-79} +10 q^{-81} +11 q^{-83} -2 q^{-85} -4 q^{-87} +2 q^{-91} - q^{-93} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{56}-2 q^{54}-q^{52}+3 q^{50}+4 q^{46}-10 q^{44}-5 q^{42}+15 q^{40}+5 q^{38}+12 q^{36}-41 q^{34}-29 q^{32}+49 q^{30}+48 q^{28}+46 q^{26}-127 q^{24}-136 q^{22}+78 q^{20}+194 q^{18}+219 q^{16}-235 q^{14}-440 q^{12}-70 q^{10}+403 q^8+688 q^6-124 q^4-862 q^2-607+353 q^{-2} +1321 q^{-4} +458 q^{-6} -974 q^{-8} -1342 q^{-10} -221 q^{-12} +1572 q^{-14} +1244 q^{-16} -471 q^{-18} -1664 q^{-20} -993 q^{-22} +1125 q^{-24} +1593 q^{-26} +308 q^{-28} -1282 q^{-30} -1366 q^{-32} +300 q^{-34} +1297 q^{-36} +849 q^{-38} -538 q^{-40} -1225 q^{-42} -438 q^{-44} +696 q^{-46} +1065 q^{-48} +196 q^{-50} -852 q^{-52} -1015 q^{-54} +67 q^{-56} +1141 q^{-58} +862 q^{-60} -411 q^{-62} -1466 q^{-64} -590 q^{-66} +1028 q^{-68} +1439 q^{-70} +206 q^{-72} -1598 q^{-74} -1235 q^{-76} +526 q^{-78} +1647 q^{-80} +928 q^{-82} -1152 q^{-84} -1518 q^{-86} -253 q^{-88} +1219 q^{-90} +1318 q^{-92} -326 q^{-94} -1156 q^{-96} -757 q^{-98} +413 q^{-100} +1059 q^{-102} +270 q^{-104} -448 q^{-106} -663 q^{-108} -134 q^{-110} +471 q^{-112} +313 q^{-114} +14 q^{-116} -278 q^{-118} -196 q^{-120} +85 q^{-122} +117 q^{-124} +86 q^{-126} -47 q^{-128} -73 q^{-130} -2 q^{-132} +10 q^{-134} +28 q^{-136} -12 q^{-140} -2 q^{-144} +4 q^{-146} -2 q^{-150} + q^{-152} }
5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{85}+2 q^{83}+q^{81}-3 q^{79}-q^{73}+5 q^{71}+4 q^{69}-11 q^{67}-8 q^{65}+6 q^{63}+13 q^{61}+19 q^{59}-43 q^{55}-56 q^{53}+6 q^{51}+97 q^{49}+121 q^{47}+23 q^{45}-171 q^{43}-284 q^{41}-126 q^{39}+276 q^{37}+563 q^{35}+367 q^{33}-310 q^{31}-976 q^{29}-898 q^{27}+169 q^{25}+1494 q^{23}+1771 q^{21}+348 q^{19}-1918 q^{17}-3020 q^{15}-1465 q^{13}+1989 q^{11}+4488 q^9+3296 q^7-1380 q^5-5823 q^3-5720 q-226 q^{-1} +6544 q^{-3} +8462 q^{-5} +2799 q^{-7} -6276 q^{-9} -10816 q^{-11} -6085 q^{-13} +4694 q^{-15} +12319 q^{-17} +9499 q^{-19} -2066 q^{-21} -12438 q^{-23} -12278 q^{-25} -1267 q^{-27} +11155 q^{-29} +13928 q^{-31} +4554 q^{-33} -8711 q^{-35} -14147 q^{-37} -7210 q^{-39} +5650 q^{-41} +13040 q^{-43} +8886 q^{-45} -2517 q^{-47} -11013 q^{-49} -9553 q^{-51} -189 q^{-53} +8510 q^{-55} +9383 q^{-57} +2399 q^{-59} -6004 q^{-61} -8807 q^{-63} -4102 q^{-65} +3733 q^{-67} +8085 q^{-69} +5531 q^{-71} -1679 q^{-73} -7485 q^{-75} -6949 q^{-77} -240 q^{-79} +7027 q^{-81} +8404 q^{-83} +2269 q^{-85} -6459 q^{-87} -10013 q^{-89} -4531 q^{-91} +5644 q^{-93} +11399 q^{-95} +7055 q^{-97} -4159 q^{-99} -12354 q^{-101} -9674 q^{-103} +2021 q^{-105} +12377 q^{-107} +11974 q^{-109} +827 q^{-111} -11296 q^{-113} -13511 q^{-115} -3913 q^{-117} +8970 q^{-119} +13880 q^{-121} +6769 q^{-123} -5791 q^{-125} -12831 q^{-127} -8755 q^{-129} +2218 q^{-131} +10530 q^{-133} +9539 q^{-135} +957 q^{-137} -7424 q^{-139} -8926 q^{-141} -3237 q^{-143} +4153 q^{-145} +7271 q^{-147} +4321 q^{-149} -1433 q^{-151} -5062 q^{-153} -4222 q^{-155} -426 q^{-157} +2882 q^{-159} +3401 q^{-161} +1304 q^{-163} -1247 q^{-165} -2240 q^{-167} -1396 q^{-169} +205 q^{-171} +1218 q^{-173} +1095 q^{-175} +230 q^{-177} -522 q^{-179} -660 q^{-181} -304 q^{-183} +139 q^{-185} +320 q^{-187} +227 q^{-189} +6 q^{-191} -132 q^{-193} -115 q^{-195} -28 q^{-197} +35 q^{-199} +44 q^{-201} +28 q^{-203} -9 q^{-205} -21 q^{-207} -5 q^{-209} +2 q^{-211} + q^{-213} +4 q^{-215} +2 q^{-217} -4 q^{-219} +2 q^{-223} - q^{-225} }

A2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+q^4-q^2-1+3 q^{-2} -2 q^{-4} +3 q^{-6} + q^{-8} + q^{-10} +3 q^{-12} -2 q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-22} - q^{-24} }
1,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-4 q^{18}+12 q^{16}-28 q^{14}+58 q^{12}-108 q^{10}+182 q^8-288 q^6+423 q^4-582 q^2+744-896 q^{-2} +992 q^{-4} -1002 q^{-6} +900 q^{-8} -664 q^{-10} +312 q^{-12} +162 q^{-14} -668 q^{-16} +1184 q^{-18} -1616 q^{-20} +1938 q^{-22} -2086 q^{-24} +2048 q^{-26} -1838 q^{-28} +1462 q^{-30} -998 q^{-32} +470 q^{-34} +36 q^{-36} -480 q^{-38} +814 q^{-40} -1004 q^{-42} +1065 q^{-44} -1012 q^{-46} +876 q^{-48} -696 q^{-50} +506 q^{-52} -344 q^{-54} +214 q^{-56} -120 q^{-58} +62 q^{-60} -28 q^{-62} +12 q^{-64} -4 q^{-66} + q^{-68} }
2,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-q^{16}-2 q^{14}+3 q^{12}+3 q^{10}-5 q^8-5 q^6+7 q^4+5 q^2-13-6 q^{-2} +16 q^{-4} +2 q^{-6} -15 q^{-8} +3 q^{-10} +14 q^{-12} - q^{-14} -6 q^{-16} +11 q^{-18} +8 q^{-20} -8 q^{-22} +6 q^{-24} +7 q^{-26} -11 q^{-28} -7 q^{-30} +12 q^{-32} - q^{-34} -15 q^{-36} +12 q^{-40} -3 q^{-42} -14 q^{-44} +4 q^{-46} +8 q^{-48} -3 q^{-50} -5 q^{-52} + q^{-54} +4 q^{-56} - q^{-60} + q^{-62} }

A3 Invariants.

Weight Invariant
0,1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}-2 q^{12}+q^{10}+4 q^8-9 q^6+2 q^4+11 q^2-19+2 q^{-2} +21 q^{-4} -24 q^{-6} + q^{-8} +25 q^{-10} -16 q^{-12} -2 q^{-14} +18 q^{-16} + q^{-18} -3 q^{-20} + q^{-22} +15 q^{-24} -6 q^{-26} -20 q^{-28} +17 q^{-30} -2 q^{-32} -28 q^{-34} +19 q^{-36} +5 q^{-38} -20 q^{-40} +14 q^{-42} +3 q^{-44} -9 q^{-46} +5 q^{-48} + q^{-50} -2 q^{-52} + q^{-54} }
1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+q^5-2 q^3+q-2 q^{-1} +3 q^{-3} -2 q^{-5} +3 q^{-7} + q^{-9} +2 q^{-11} +2 q^{-13} + q^{-15} +3 q^{-17} -2 q^{-19} +2 q^{-21} -3 q^{-23} -3 q^{-27} + q^{-29} - q^{-31} }

A4 Invariants.

Weight Invariant
0,1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{16}-q^{14}+3 q^{10}-q^8-5 q^6+2 q^4+4 q^2-8-9 q^{-2} +9 q^{-4} +8 q^{-6} -17 q^{-8} -3 q^{-10} +21 q^{-12} -2 q^{-14} -19 q^{-16} +12 q^{-18} +18 q^{-20} -7 q^{-22} + q^{-24} +27 q^{-26} +9 q^{-28} -11 q^{-30} +14 q^{-32} +12 q^{-34} -25 q^{-36} -13 q^{-38} +12 q^{-40} -13 q^{-42} -25 q^{-44} +4 q^{-46} +13 q^{-48} -9 q^{-50} -8 q^{-52} +12 q^{-54} +7 q^{-56} -7 q^{-58} +5 q^{-62} - q^{-64} - q^{-66} + q^{-68} }
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8+q^6-2 q^4-2 q^{-2} +3 q^{-4} -2 q^{-6} +3 q^{-8} + q^{-10} +2 q^{-12} +2 q^{-14} +2 q^{-16} +2 q^{-18} + q^{-20} +3 q^{-22} -2 q^{-24} +2 q^{-26} -3 q^{-28} - q^{-30} - q^{-32} -3 q^{-34} + q^{-36} - q^{-38} }

B2 Invariants.

Weight Invariant
0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{14}+2 q^{12}-5 q^{10}+8 q^8-13 q^6+18 q^4-23 q^2+27-28 q^{-2} +27 q^{-4} -20 q^{-6} +11 q^{-8} +3 q^{-10} -16 q^{-12} +32 q^{-14} -42 q^{-16} +53 q^{-18} -55 q^{-20} +55 q^{-22} -47 q^{-24} +36 q^{-26} -22 q^{-28} +7 q^{-30} +6 q^{-32} -18 q^{-34} +25 q^{-36} -29 q^{-38} +28 q^{-40} -26 q^{-42} +21 q^{-44} -15 q^{-46} +9 q^{-48} -5 q^{-50} +2 q^{-52} - q^{-54} }
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-2 q^{20}-2 q^{18}+3 q^{16}+6 q^{14}-q^{12}-11 q^{10}-7 q^8+11 q^6+17 q^4-4 q^2-25-11 q^{-2} +22 q^{-4} +25 q^{-6} -11 q^{-8} -31 q^{-10} -4 q^{-12} +29 q^{-14} +16 q^{-16} -18 q^{-18} -19 q^{-20} +13 q^{-22} +22 q^{-24} -3 q^{-26} -19 q^{-28} +3 q^{-30} +21 q^{-32} +5 q^{-34} -18 q^{-36} -7 q^{-38} +18 q^{-40} +13 q^{-42} -18 q^{-44} -22 q^{-46} +10 q^{-48} +26 q^{-50} -2 q^{-52} -32 q^{-54} -16 q^{-56} +23 q^{-58} +26 q^{-60} -10 q^{-62} -27 q^{-64} -5 q^{-66} +20 q^{-68} +13 q^{-70} -8 q^{-72} -12 q^{-74} +7 q^{-78} +3 q^{-80} -2 q^{-82} -2 q^{-84} + q^{-88} }

D4 Invariants.

Weight Invariant
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-2 q^{16}+3 q^{14}-4 q^{12}+7 q^{10}-11 q^8+11 q^6-15 q^4+18 q^2-23+19 q^{-2} -21 q^{-4} +23 q^{-6} -18 q^{-8} +11 q^{-10} -5 q^{-12} +4 q^{-14} +12 q^{-16} -18 q^{-18} +25 q^{-20} -27 q^{-22} +44 q^{-24} -38 q^{-26} +44 q^{-28} -39 q^{-30} +45 q^{-32} -32 q^{-34} +26 q^{-36} -27 q^{-38} +10 q^{-40} -6 q^{-42} -7 q^{-44} +3 q^{-46} -18 q^{-48} +21 q^{-50} -21 q^{-52} +22 q^{-54} -23 q^{-56} +22 q^{-58} -16 q^{-60} +14 q^{-62} -12 q^{-64} +8 q^{-66} -4 q^{-68} +3 q^{-70} -2 q^{-72} + q^{-74} }

G2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+8 q^{24}-7 q^{22}-2 q^{20}+16 q^{18}-32 q^{16}+46 q^{14}-51 q^{12}+35 q^{10}-4 q^8-46 q^6+99 q^4-131 q^2+132-89 q^{-2} +5 q^{-4} +95 q^{-6} -177 q^{-8} +210 q^{-10} -173 q^{-12} +77 q^{-14} +42 q^{-16} -142 q^{-18} +181 q^{-20} -139 q^{-22} +45 q^{-24} +70 q^{-26} -141 q^{-28} +131 q^{-30} -45 q^{-32} -84 q^{-34} +200 q^{-36} -242 q^{-38} +197 q^{-40} -59 q^{-42} -109 q^{-44} +259 q^{-46} -323 q^{-48} +286 q^{-50} -155 q^{-52} -17 q^{-54} +169 q^{-56} -248 q^{-58} +240 q^{-60} -143 q^{-62} +13 q^{-64} +100 q^{-66} -156 q^{-68} +120 q^{-70} -25 q^{-72} -91 q^{-74} +166 q^{-76} -168 q^{-78} +91 q^{-80} +31 q^{-82} -152 q^{-84} +220 q^{-86} -214 q^{-88} +134 q^{-90} -21 q^{-92} -92 q^{-94} +155 q^{-96} -160 q^{-98} +122 q^{-100} -55 q^{-102} -6 q^{-104} +46 q^{-106} -64 q^{-108} +54 q^{-110} -34 q^{-112} +15 q^{-114} + q^{-116} -8 q^{-118} +9 q^{-120} -8 q^{-122} +5 q^{-124} -2 q^{-126} + q^{-128} }

.

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 57"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^3-8 t^2+18 t-23+18 t^{-1} -8 t^{-2} +2 t^{-3} }
In[5]:= Conway[K][z]
Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^6+4 z^4+4 z^2+1}
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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 79, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8+3 q^7-7 q^6+10 q^5-12 q^4+14 q^3-12 q^2+10 q-6+3 q^{-1} - q^{-2} }
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+4 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} -2 z^2+2 a^{-2} +2 a^{-4} -2 a^{-6} -1}
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +4 z^7 a^{-1} +9 z^7 a^{-3} +10 z^7 a^{-5} +5 z^7 a^{-7} +2 z^6 a^{-2} -7 z^6 a^{-4} -3 z^6 a^{-6} +3 z^6 a^{-8} +3 z^6+a z^5-5 z^5 a^{-1} -19 z^5 a^{-3} -23 z^5 a^{-5} -9 z^5 a^{-7} +z^5 a^{-9} -11 z^4 a^{-2} -z^4 a^{-4} -z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-2 a z^3+12 z^3 a^{-3} +18 z^3 a^{-5} +6 z^3 a^{-7} -2 z^3 a^{-9} +8 z^2 a^{-2} -2 z^2 a^{-6} +2 z^2 a^{-8} +4 z^2+a z+z a^{-1} -2 z a^{-3} -6 z a^{-5} -3 z a^{-7} +z a^{-9} -2 a^{-2} +2 a^{-4} +2 a^{-6} -1}
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