8 17

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+4 t^2-8 t+11-8 t^{-1} +4 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 37, 0 }
Jones polynomial	[math]\displaystyle{ q^4-3 q^3+5 q^2-6 q+7-6 q^{-1} +5 q^{-2} -3 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -4 z^4+2 a^2 z^2+2 z^2 a^{-2} -5 z^2+a^2+ a^{-2} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a z^7+2 z^7 a^{-1} +4 a^2 z^6+4 z^6 a^{-2} +8 z^6+3 a^3 z^5+2 a z^5+2 z^5 a^{-1} +3 z^5 a^{-3} +a^4 z^4-6 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} -14 z^4-4 a^3 z^3-6 a z^3-6 z^3 a^{-1} -4 z^3 a^{-3} -a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +8 z^2+a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^2- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-q^{10}+q^8-q^4+2 q^2-1+2 q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+4 q^{58}-2 q^{56}-4 q^{54}+14 q^{52}-21 q^{50}+26 q^{48}-20 q^{46}+3 q^{44}+17 q^{42}-36 q^{40}+47 q^{38}-38 q^{36}+17 q^{34}+10 q^{32}-32 q^{30}+41 q^{28}-29 q^{26}+6 q^{24}+18 q^{22}-31 q^{20}+23 q^{18}-2 q^{16}-24 q^{14}+44 q^{12}-47 q^{10}+33 q^8-5 q^6-27 q^4+53 q^2-63+53 q^{-2} -27 q^{-4} -5 q^{-6} +33 q^{-8} -47 q^{-10} +44 q^{-12} -24 q^{-14} -2 q^{-16} +23 q^{-18} -31 q^{-20} +18 q^{-22} +6 q^{-24} -29 q^{-26} +41 q^{-28} -32 q^{-30} +10 q^{-32} +17 q^{-34} -38 q^{-36} +47 q^{-38} -36 q^{-40} +17 q^{-42} +3 q^{-44} -20 q^{-46} +26 q^{-48} -21 q^{-50} +14 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_17.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-2 q^7+2 q^5-q^3+q+ q^{-1} - q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{26}-2 q^{24}-q^{22}+7 q^{20}-4 q^{18}-8 q^{16}+11 q^{14}-11 q^{10}+8 q^8+4 q^6-7 q^4+q^2+5+ q^{-2} -7 q^{-4} +4 q^{-6} +8 q^{-8} -11 q^{-10} +11 q^{-14} -8 q^{-16} -4 q^{-18} +7 q^{-20} - q^{-22} -2 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{51}-2 q^{49}-q^{47}+4 q^{45}+4 q^{43}-7 q^{41}-14 q^{39}+10 q^{37}+26 q^{35}-3 q^{33}-37 q^{31}-11 q^{29}+45 q^{27}+24 q^{25}-44 q^{23}-38 q^{21}+35 q^{19}+46 q^{17}-23 q^{15}-46 q^{13}+13 q^{11}+40 q^9+q^7-31 q^5-11 q^3+21 q+21 q^{-1} -11 q^{-3} -31 q^{-5} + q^{-7} +40 q^{-9} +13 q^{-11} -46 q^{-13} -23 q^{-15} +46 q^{-17} +35 q^{-19} -38 q^{-21} -44 q^{-23} +24 q^{-25} +45 q^{-27} -11 q^{-29} -37 q^{-31} -3 q^{-33} +26 q^{-35} +10 q^{-37} -14 q^{-39} -7 q^{-41} +4 q^{-43} +4 q^{-45} - q^{-47} -2 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{84}-2 q^{82}-q^{80}+4 q^{78}+q^{76}+q^{74}-13 q^{72}-8 q^{70}+18 q^{68}+22 q^{66}+21 q^{64}-44 q^{62}-66 q^{60}-q^{58}+67 q^{56}+119 q^{54}-15 q^{52}-145 q^{50}-121 q^{48}+26 q^{46}+234 q^{44}+127 q^{42}-120 q^{40}-247 q^{38}-123 q^{36}+222 q^{34}+253 q^{32}+15 q^{30}-243 q^{28}-236 q^{26}+104 q^{24}+247 q^{22}+119 q^{20}-138 q^{18}-224 q^{16}-8 q^{14}+152 q^{12}+140 q^{10}-32 q^8-149 q^6-77 q^4+56 q^2+133+56 q^{-2} -77 q^{-4} -149 q^{-6} -32 q^{-8} +140 q^{-10} +152 q^{-12} -8 q^{-14} -224 q^{-16} -138 q^{-18} +119 q^{-20} +247 q^{-22} +104 q^{-24} -236 q^{-26} -243 q^{-28} +15 q^{-30} +253 q^{-32} +222 q^{-34} -123 q^{-36} -247 q^{-38} -120 q^{-40} +127 q^{-42} +234 q^{-44} +26 q^{-46} -121 q^{-48} -145 q^{-50} -15 q^{-52} +119 q^{-54} +67 q^{-56} - q^{-58} -66 q^{-60} -44 q^{-62} +21 q^{-64} +22 q^{-66} +18 q^{-68} -8 q^{-70} -13 q^{-72} + q^{-74} + q^{-76} +4 q^{-78} - q^{-80} -2 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ q^{125}-2 q^{123}-q^{121}+4 q^{119}+q^{117}-2 q^{115}-5 q^{113}-7 q^{111}+21 q^{107}+28 q^{105}+q^{103}-40 q^{101}-69 q^{99}-42 q^{97}+44 q^{95}+147 q^{93}+146 q^{91}-8 q^{89}-211 q^{87}-306 q^{85}-148 q^{83}+207 q^{81}+502 q^{79}+414 q^{77}-77 q^{75}-620 q^{73}-749 q^{71}-235 q^{69}+595 q^{67}+1066 q^{65}+660 q^{63}-387 q^{61}-1227 q^{59}-1104 q^{57}+2 q^{55}+1203 q^{53}+1449 q^{51}+437 q^{49}-989 q^{47}-1594 q^{45}-839 q^{43}+651 q^{41}+1548 q^{39}+1096 q^{37}-285 q^{35}-1347 q^{33}-1181 q^{31}-25 q^{29}+1052 q^{27}+1126 q^{25}+245 q^{23}-747 q^{21}-987 q^{19}-360 q^{17}+474 q^{15}+807 q^{13}+432 q^{11}-255 q^9-657 q^7-476 q^5+83 q^3+543 q+543 q^{-1} +83 q^{-3} -476 q^{-5} -657 q^{-7} -255 q^{-9} +432 q^{-11} +807 q^{-13} +474 q^{-15} -360 q^{-17} -987 q^{-19} -747 q^{-21} +245 q^{-23} +1126 q^{-25} +1052 q^{-27} -25 q^{-29} -1181 q^{-31} -1347 q^{-33} -285 q^{-35} +1096 q^{-37} +1548 q^{-39} +651 q^{-41} -839 q^{-43} -1594 q^{-45} -989 q^{-47} +437 q^{-49} +1449 q^{-51} +1203 q^{-53} +2 q^{-55} -1104 q^{-57} -1227 q^{-59} -387 q^{-61} +660 q^{-63} +1066 q^{-65} +595 q^{-67} -235 q^{-69} -749 q^{-71} -620 q^{-73} -77 q^{-75} +414 q^{-77} +502 q^{-79} +207 q^{-81} -148 q^{-83} -306 q^{-85} -211 q^{-87} -8 q^{-89} +146 q^{-91} +147 q^{-93} +44 q^{-95} -42 q^{-97} -69 q^{-99} -40 q^{-101} + q^{-103} +28 q^{-105} +21 q^{-107} -7 q^{-111} -5 q^{-113} -2 q^{-115} + q^{-117} +4 q^{-119} - q^{-121} -2 q^{-123} + q^{-125} }[/math]
6	[math]\displaystyle{ q^{174}-2 q^{172}-q^{170}+4 q^{168}+q^{166}-2 q^{164}-8 q^{162}+q^{160}+q^{158}+3 q^{156}+27 q^{154}+15 q^{152}-13 q^{150}-56 q^{148}-52 q^{146}-33 q^{144}+25 q^{142}+149 q^{140}+178 q^{138}+92 q^{136}-132 q^{134}-295 q^{132}-403 q^{130}-261 q^{128}+227 q^{126}+676 q^{124}+850 q^{122}+432 q^{120}-289 q^{118}-1202 q^{116}-1594 q^{114}-892 q^{112}+527 q^{110}+2013 q^{108}+2457 q^{106}+1602 q^{104}-748 q^{102}-3103 q^{100}-3788 q^{98}-2200 q^{96}+1151 q^{94}+4210 q^{92}+5345 q^{90}+2858 q^{88}-1786 q^{86}-5772 q^{84}-6553 q^{82}-3159 q^{80}+2485 q^{78}+7387 q^{76}+7555 q^{74}+2942 q^{72}-3800 q^{70}-8443 q^{68}-7853 q^{66}-2296 q^{64}+5237 q^{62}+9135 q^{60}+7293 q^{58}+755 q^{56}-6163 q^{54}-9009 q^{52}-6069 q^{50}+992 q^{48}+6768 q^{46}+8011 q^{44}+3991 q^{42}-2258 q^{40}-6645 q^{38}-6452 q^{36}-1892 q^{34}+3183 q^{32}+5820 q^{30}+4377 q^{28}+376 q^{26}-3479 q^{24}-4663 q^{22}-2564 q^{20}+750 q^{18}+3287 q^{16}+3344 q^{14}+1332 q^{12}-1415 q^{10}-2963 q^8-2417 q^6-392 q^4+1877 q^2+2757+1877 q^{-2} -392 q^{-4} -2417 q^{-6} -2963 q^{-8} -1415 q^{-10} +1332 q^{-12} +3344 q^{-14} +3287 q^{-16} +750 q^{-18} -2564 q^{-20} -4663 q^{-22} -3479 q^{-24} +376 q^{-26} +4377 q^{-28} +5820 q^{-30} +3183 q^{-32} -1892 q^{-34} -6452 q^{-36} -6645 q^{-38} -2258 q^{-40} +3991 q^{-42} +8011 q^{-44} +6768 q^{-46} +992 q^{-48} -6069 q^{-50} -9009 q^{-52} -6163 q^{-54} +755 q^{-56} +7293 q^{-58} +9135 q^{-60} +5237 q^{-62} -2296 q^{-64} -7853 q^{-66} -8443 q^{-68} -3800 q^{-70} +2942 q^{-72} +7555 q^{-74} +7387 q^{-76} +2485 q^{-78} -3159 q^{-80} -6553 q^{-82} -5772 q^{-84} -1786 q^{-86} +2858 q^{-88} +5345 q^{-90} +4210 q^{-92} +1151 q^{-94} -2200 q^{-96} -3788 q^{-98} -3103 q^{-100} -748 q^{-102} +1602 q^{-104} +2457 q^{-106} +2013 q^{-108} +527 q^{-110} -892 q^{-112} -1594 q^{-114} -1202 q^{-116} -289 q^{-118} +432 q^{-120} +850 q^{-122} +676 q^{-124} +227 q^{-126} -261 q^{-128} -403 q^{-130} -295 q^{-132} -132 q^{-134} +92 q^{-136} +178 q^{-138} +149 q^{-140} +25 q^{-142} -33 q^{-144} -52 q^{-146} -56 q^{-148} -13 q^{-150} +15 q^{-152} +27 q^{-154} +3 q^{-156} + q^{-158} + q^{-160} -8 q^{-162} -2 q^{-164} + q^{-166} +4 q^{-168} - q^{-170} -2 q^{-172} + q^{-174} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-q^{10}+q^8-q^4+2 q^2-1+2 q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{36}-4 q^{34}+10 q^{32}-20 q^{30}+38 q^{28}-62 q^{26}+92 q^{24}-122 q^{22}+144 q^{20}-156 q^{18}+144 q^{16}-110 q^{14}+55 q^{12}+22 q^{10}-98 q^8+176 q^6-237 q^4+278 q^2-294+278 q^{-2} -237 q^{-4} +176 q^{-6} -98 q^{-8} +22 q^{-10} +55 q^{-12} -110 q^{-14} +144 q^{-16} -156 q^{-18} +144 q^{-20} -122 q^{-22} +92 q^{-24} -62 q^{-26} +38 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+3 q^{26}+q^{24}-3 q^{22}-2 q^{20}+3 q^{18}+2 q^{16}-6 q^{14}+q^{12}+5 q^{10}-2 q^8-2 q^6+3 q^4+2 q^2-2+2 q^{-2} +3 q^{-4} -2 q^{-6} -2 q^{-8} +5 q^{-10} + q^{-12} -6 q^{-14} +2 q^{-16} +3 q^{-18} -2 q^{-20} -3 q^{-22} + q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-2 q^{26}+4 q^{22}-6 q^{20}+q^{18}+8 q^{16}-8 q^{14}+2 q^{12}+8 q^{10}-6 q^8-q^6+4 q^4-q^2-2- q^{-2} +4 q^{-4} - q^{-6} -6 q^{-8} +8 q^{-10} +2 q^{-12} -8 q^{-14} +8 q^{-16} + q^{-18} -6 q^{-20} +4 q^{-22} -2 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{15}-q^{13}+2 q^{11}-q^9+q^7-q^5+q^3+ q^{-3} - q^{-5} + q^{-7} - q^{-9} +2 q^{-11} - q^{-13} + q^{-15} }[/math]
1,0,1	[math]\displaystyle{ q^{46}-4 q^{44}+8 q^{42}-8 q^{40}-2 q^{38}+25 q^{36}-46 q^{34}+49 q^{32}-14 q^{30}-51 q^{28}+112 q^{26}-137 q^{24}+93 q^{22}+3 q^{20}-115 q^{18}+190 q^{16}-182 q^{14}+107 q^{12}+14 q^{10}-99 q^8+122 q^6-78 q^4+13 q^2+13+13 q^{-2} -78 q^{-4} +122 q^{-6} -99 q^{-8} +14 q^{-10} +107 q^{-12} -182 q^{-14} +190 q^{-16} -115 q^{-18} +3 q^{-20} +93 q^{-22} -137 q^{-24} +112 q^{-26} -51 q^{-28} -14 q^{-30} +49 q^{-32} -46 q^{-34} +25 q^{-36} -2 q^{-38} -8 q^{-40} +8 q^{-42} -4 q^{-44} + q^{-46} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+4 q^{58}-2 q^{56}-4 q^{54}+14 q^{52}-21 q^{50}+26 q^{48}-20 q^{46}+3 q^{44}+17 q^{42}-36 q^{40}+47 q^{38}-38 q^{36}+17 q^{34}+10 q^{32}-32 q^{30}+41 q^{28}-29 q^{26}+6 q^{24}+18 q^{22}-31 q^{20}+23 q^{18}-2 q^{16}-24 q^{14}+44 q^{12}-47 q^{10}+33 q^8-5 q^6-27 q^4+53 q^2-63+53 q^{-2} -27 q^{-4} -5 q^{-6} +33 q^{-8} -47 q^{-10} +44 q^{-12} -24 q^{-14} -2 q^{-16} +23 q^{-18} -31 q^{-20} +18 q^{-22} +6 q^{-24} -29 q^{-26} +41 q^{-28} -32 q^{-30} +10 q^{-32} +17 q^{-34} -38 q^{-36} +47 q^{-38} -36 q^{-40} +17 q^{-42} +3 q^{-44} -20 q^{-46} +26 q^{-48} -21 q^{-50} +14 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/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["8 17"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -t^3+4 t^2-8 t+11-8 t^{-1} +4 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-2 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]=

{ 37, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^4-3 q^3+5 q^2-6 q+7-6 q^{-1} +5 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^4+z^4 a^{-2} -4 z^4+2 a^2 z^2+2 z^2 a^{-2} -5 z^2+a^2+ a^{-2} -1 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ 2 a z^7+2 z^7 a^{-1} +4 a^2 z^6+4 z^6 a^{-2} +8 z^6+3 a^3 z^5+2 a z^5+2 z^5 a^{-1} +3 z^5 a^{-3} +a^4 z^4-6 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} -14 z^4-4 a^3 z^3-6 a z^3-6 z^3 a^{-1} -4 z^3 a^{-3} -a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +8 z^2+a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^2- a^{-2} -1 }[/math]