10 118

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-5 t^3+12 t^2-19 t+23-19 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+3 z^6+2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 97, 0 }
Jones polynomial [math]\displaystyle{ -q^5+4 q^4-8 q^3+12 q^2-15 q+17-15 q^{-1} +12 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +5 z^6-3 a^2 z^4-3 z^4 a^{-2} +8 z^4-2 a^2 z^2-2 z^2 a^{-2} +4 z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 a z^9+3 z^9 a^{-1} +7 a^2 z^8+7 z^8 a^{-2} +14 z^8+7 a^3 z^7+6 a z^7+6 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +4 z^6 a^{-4} -30 z^6+a^5 z^5-12 a^3 z^5-20 a z^5-20 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+6 a^2 z^4+6 z^4 a^{-2} -6 z^4 a^{-4} +24 z^4-a^5 z^3+5 a^3 z^3+15 a z^3+15 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -6 z^2-a^3 z-3 a z-3 z a^{-1} -z a^{-3} +1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{14}+2 q^{12}-2 q^{10}+2 q^8-2 q^4+4 q^2-3+4 q^{-2} -2 q^{-4} +2 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-13 q^{70}+2 q^{68}+22 q^{66}-48 q^{64}+77 q^{62}-93 q^{60}+75 q^{58}-27 q^{56}-60 q^{54}+162 q^{52}-237 q^{50}+259 q^{48}-188 q^{46}+27 q^{44}+173 q^{42}-342 q^{40}+401 q^{38}-319 q^{36}+115 q^{34}+129 q^{32}-313 q^{30}+362 q^{28}-237 q^{26}+12 q^{24}+212 q^{22}-326 q^{20}+260 q^{18}-55 q^{16}-209 q^{14}+412 q^{12}-458 q^{10}+343 q^8-79 q^6-234 q^4+477 q^2-571+477 q^{-2} -234 q^{-4} -79 q^{-6} +343 q^{-8} -458 q^{-10} +412 q^{-12} -209 q^{-14} -55 q^{-16} +260 q^{-18} -326 q^{-20} +212 q^{-22} +12 q^{-24} -237 q^{-26} +362 q^{-28} -313 q^{-30} +129 q^{-32} +115 q^{-34} -319 q^{-36} +401 q^{-38} -342 q^{-40} +173 q^{-42} +27 q^{-44} -188 q^{-46} +259 q^{-48} -237 q^{-50} +162 q^{-52} -60 q^{-54} -27 q^{-56} +75 q^{-58} -93 q^{-60} +77 q^{-62} -48 q^{-64} +22 q^{-66} +2 q^{-68} -13 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_118.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+3 q^9-4 q^7+4 q^5-3 q^3+2 q+2 q^{-1} -3 q^{-3} +4 q^{-5} -4 q^{-7} +3 q^{-9} - q^{-11} }[/math]
2 [math]\displaystyle{ q^{32}-3 q^{30}+10 q^{26}-13 q^{24}-8 q^{22}+33 q^{20}-16 q^{18}-32 q^{16}+46 q^{14}-47 q^{10}+31 q^8+19 q^6-33 q^4+27-33 q^{-4} +19 q^{-6} +31 q^{-8} -47 q^{-10} +46 q^{-14} -32 q^{-16} -16 q^{-18} +33 q^{-20} -8 q^{-22} -13 q^{-24} +10 q^{-26} -3 q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{63}+3 q^{61}-6 q^{57}-q^{55}+13 q^{53}+7 q^{51}-35 q^{49}-18 q^{47}+63 q^{45}+58 q^{43}-88 q^{41}-135 q^{39}+89 q^{37}+232 q^{35}-40 q^{33}-321 q^{31}-62 q^{29}+371 q^{27}+187 q^{25}-357 q^{23}-300 q^{21}+286 q^{19}+371 q^{17}-180 q^{15}-389 q^{13}+66 q^{11}+357 q^9+45 q^7-301 q^5-135 q^3+223 q+223 q^{-1} -135 q^{-3} -301 q^{-5} +45 q^{-7} +357 q^{-9} +66 q^{-11} -389 q^{-13} -180 q^{-15} +371 q^{-17} +286 q^{-19} -300 q^{-21} -357 q^{-23} +187 q^{-25} +371 q^{-27} -62 q^{-29} -321 q^{-31} -40 q^{-33} +232 q^{-35} +89 q^{-37} -135 q^{-39} -88 q^{-41} +58 q^{-43} +63 q^{-45} -18 q^{-47} -35 q^{-49} +7 q^{-51} +13 q^{-53} - q^{-55} -6 q^{-57} +3 q^{-61} - q^{-63} }[/math]
5 [math]\displaystyle{ -q^{155}+3 q^{153}-6 q^{149}+3 q^{147}+3 q^{145}-2 q^{143}-2 q^{141}-4 q^{139}-7 q^{137}+21 q^{135}+50 q^{133}+8 q^{131}-79 q^{129}-142 q^{127}-87 q^{125}+120 q^{123}+392 q^{121}+416 q^{119}-74 q^{117}-793 q^{115}-1126 q^{113}-495 q^{111}+1047 q^{109}+2435 q^{107}+2087 q^{105}-554 q^{103}-3869 q^{101}-5003 q^{99}-1882 q^{97}+4322 q^{95}+8943 q^{93}+6977 q^{91}-2088 q^{89}-12194 q^{87}-14473 q^{85}-4453 q^{83}+12310 q^{81}+22517 q^{79}+15307 q^{77}-6984 q^{75}-27793 q^{73}-28484 q^{71}-4602 q^{69}+27215 q^{67}+40348 q^{65}+20561 q^{63}-19257 q^{61}-46869 q^{59}-37138 q^{57}+4964 q^{55}+45725 q^{53}+49989 q^{51}+12146 q^{49}-37040 q^{47}-55883 q^{45}-27875 q^{43}+23334 q^{41}+54096 q^{39}+38750 q^{37}-8360 q^{35}-46136 q^{33}-43230 q^{31}-4552 q^{29}+34875 q^{27}+41972 q^{25}+13541 q^{23}-23425 q^{21}-37066 q^{19}-18291 q^{17}+13799 q^{15}+30926 q^{13}+20188 q^{11}-6807 q^9-25762 q^7-20934 q^5+2016 q^3+22419 q+22419 q^{-1} +2016 q^{-3} -20934 q^{-5} -25762 q^{-7} -6807 q^{-9} +20188 q^{-11} +30926 q^{-13} +13799 q^{-15} -18291 q^{-17} -37066 q^{-19} -23425 q^{-21} +13541 q^{-23} +41972 q^{-25} +34875 q^{-27} -4552 q^{-29} -43230 q^{-31} -46136 q^{-33} -8360 q^{-35} +38750 q^{-37} +54096 q^{-39} +23334 q^{-41} -27875 q^{-43} -55883 q^{-45} -37040 q^{-47} +12146 q^{-49} +49989 q^{-51} +45725 q^{-53} +4964 q^{-55} -37138 q^{-57} -46869 q^{-59} -19257 q^{-61} +20561 q^{-63} +40348 q^{-65} +27215 q^{-67} -4602 q^{-69} -28484 q^{-71} -27793 q^{-73} -6984 q^{-75} +15307 q^{-77} +22517 q^{-79} +12310 q^{-81} -4453 q^{-83} -14473 q^{-85} -12194 q^{-87} -2088 q^{-89} +6977 q^{-91} +8943 q^{-93} +4322 q^{-95} -1882 q^{-97} -5003 q^{-99} -3869 q^{-101} -554 q^{-103} +2087 q^{-105} +2435 q^{-107} +1047 q^{-109} -495 q^{-111} -1126 q^{-113} -793 q^{-115} -74 q^{-117} +416 q^{-119} +392 q^{-121} +120 q^{-123} -87 q^{-125} -142 q^{-127} -79 q^{-129} +8 q^{-131} +50 q^{-133} +21 q^{-135} -7 q^{-137} -4 q^{-139} -2 q^{-141} -2 q^{-143} +3 q^{-145} +3 q^{-147} -6 q^{-149} +3 q^{-153} - q^{-155} }[/math]
6 [math]\displaystyle{ q^{216}-3 q^{214}+6 q^{210}-3 q^{208}-3 q^{206}-2 q^{204}+16 q^{202}-q^{200}-21 q^{198}+5 q^{196}-31 q^{194}-23 q^{192}+26 q^{190}+135 q^{188}+115 q^{186}-35 q^{184}-129 q^{182}-370 q^{180}-394 q^{178}-57 q^{176}+715 q^{174}+1160 q^{172}+903 q^{170}+84 q^{168}-1724 q^{166}-3109 q^{164}-2846 q^{162}+192 q^{160}+4246 q^{158}+7066 q^{156}+6743 q^{154}+532 q^{152}-8833 q^{150}-16186 q^{148}-14323 q^{146}-2320 q^{144}+16041 q^{142}+31650 q^{140}+30246 q^{138}+7750 q^{136}-28523 q^{134}-56380 q^{132}-56893 q^{130}-19585 q^{128}+43960 q^{126}+95825 q^{124}+99536 q^{122}+37744 q^{120}-62744 q^{118}-149946 q^{116}-160730 q^{114}-67673 q^{112}+88593 q^{110}+221448 q^{108}+237152 q^{106}+106692 q^{104}-119529 q^{102}-308474 q^{100}-329701 q^{98}-144970 q^{96}+159969 q^{94}+403519 q^{92}+426370 q^{90}+177213 q^{88}-211477 q^{86}-505167 q^{84}-507961 q^{82}-189595 q^{80}+271458 q^{78}+598132 q^{76}+563336 q^{74}+173244 q^{72}-342966 q^{70}-662971 q^{68}-575952 q^{66}-127186 q^{64}+412211 q^{62}+691710 q^{60}+539184 q^{58}+50324 q^{56}-461414 q^{54}-672910 q^{52}-458530 q^{50}+38248 q^{48}+484300 q^{46}+606625 q^{44}+342381 q^{42}-117159 q^{40}-471582 q^{38}-504579 q^{36}-216335 q^{34}+178004 q^{32}+425404 q^{30}+380058 q^{28}+101732 q^{26}-212475 q^{24}-357505 q^{22}-256363 q^{20}-4192 q^{18}+224252 q^{16}+279709 q^{14}+146025 q^{12}-74126 q^{10}-223914 q^8-207137 q^6-46900 q^4+141254 q^2+221433+141254 q^{-2} -46900 q^{-4} -207137 q^{-6} -223914 q^{-8} -74126 q^{-10} +146025 q^{-12} +279709 q^{-14} +224252 q^{-16} -4192 q^{-18} -256363 q^{-20} -357505 q^{-22} -212475 q^{-24} +101732 q^{-26} +380058 q^{-28} +425404 q^{-30} +178004 q^{-32} -216335 q^{-34} -504579 q^{-36} -471582 q^{-38} -117159 q^{-40} +342381 q^{-42} +606625 q^{-44} +484300 q^{-46} +38248 q^{-48} -458530 q^{-50} -672910 q^{-52} -461414 q^{-54} +50324 q^{-56} +539184 q^{-58} +691710 q^{-60} +412211 q^{-62} -127186 q^{-64} -575952 q^{-66} -662971 q^{-68} -342966 q^{-70} +173244 q^{-72} +563336 q^{-74} +598132 q^{-76} +271458 q^{-78} -189595 q^{-80} -507961 q^{-82} -505167 q^{-84} -211477 q^{-86} +177213 q^{-88} +426370 q^{-90} +403519 q^{-92} +159969 q^{-94} -144970 q^{-96} -329701 q^{-98} -308474 q^{-100} -119529 q^{-102} +106692 q^{-104} +237152 q^{-106} +221448 q^{-108} +88593 q^{-110} -67673 q^{-112} -160730 q^{-114} -149946 q^{-116} -62744 q^{-118} +37744 q^{-120} +99536 q^{-122} +95825 q^{-124} +43960 q^{-126} -19585 q^{-128} -56893 q^{-130} -56380 q^{-132} -28523 q^{-134} +7750 q^{-136} +30246 q^{-138} +31650 q^{-140} +16041 q^{-142} -2320 q^{-144} -14323 q^{-146} -16186 q^{-148} -8833 q^{-150} +532 q^{-152} +6743 q^{-154} +7066 q^{-156} +4246 q^{-158} +192 q^{-160} -2846 q^{-162} -3109 q^{-164} -1724 q^{-166} +84 q^{-168} +903 q^{-170} +1160 q^{-172} +715 q^{-174} -57 q^{-176} -394 q^{-178} -370 q^{-180} -129 q^{-182} -35 q^{-184} +115 q^{-186} +135 q^{-188} +26 q^{-190} -23 q^{-192} -31 q^{-194} +5 q^{-196} -21 q^{-198} - q^{-200} +16 q^{-202} -2 q^{-204} -3 q^{-206} -3 q^{-208} +6 q^{-210} -3 q^{-214} + q^{-216} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{14}+2 q^{12}-2 q^{10}+2 q^8-2 q^4+4 q^2-3+4 q^{-2} -2 q^{-4} +2 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} }[/math]
1,1 [math]\displaystyle{ q^{44}-6 q^{42}+20 q^{40}-50 q^{38}+105 q^{36}-196 q^{34}+336 q^{32}-542 q^{30}+804 q^{28}-1110 q^{26}+1446 q^{24}-1736 q^{22}+1906 q^{20}-1892 q^{18}+1646 q^{16}-1140 q^{14}+379 q^{12}+542 q^{10}-1518 q^8+2440 q^6-3194 q^4+3692 q^2-3858+3692 q^{-2} -3194 q^{-4} +2440 q^{-6} -1518 q^{-8} +542 q^{-10} +379 q^{-12} -1140 q^{-14} +1646 q^{-16} -1892 q^{-18} +1906 q^{-20} -1736 q^{-22} +1446 q^{-24} -1110 q^{-26} +804 q^{-28} -542 q^{-30} +336 q^{-32} -196 q^{-34} +105 q^{-36} -50 q^{-38} +20 q^{-40} -6 q^{-42} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{38}-2 q^{36}-q^{34}+5 q^{32}-4 q^{30}-6 q^{28}+8 q^{26}+6 q^{24}-8 q^{22}-6 q^{20}+13 q^{18}+7 q^{16}-21 q^{14}+2 q^{12}+16 q^{10}-12 q^8-9 q^6+13 q^4+6 q^2-10+6 q^{-2} +13 q^{-4} -9 q^{-6} -12 q^{-8} +16 q^{-10} +2 q^{-12} -21 q^{-14} +7 q^{-16} +13 q^{-18} -6 q^{-20} -8 q^{-22} +6 q^{-24} +8 q^{-26} -6 q^{-28} -4 q^{-30} +5 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{34}-3 q^{32}+q^{30}+7 q^{28}-12 q^{26}+3 q^{24}+18 q^{22}-27 q^{20}+4 q^{18}+30 q^{16}-35 q^{14}+2 q^{12}+31 q^{10}-25 q^8-5 q^6+19 q^4-2 q^2-8-2 q^{-2} +19 q^{-4} -5 q^{-6} -25 q^{-8} +31 q^{-10} +2 q^{-12} -35 q^{-14} +30 q^{-16} +4 q^{-18} -27 q^{-20} +18 q^{-22} +3 q^{-24} -12 q^{-26} +7 q^{-28} + q^{-30} -3 q^{-32} + q^{-34} }[/math]
1,0,0 [math]\displaystyle{ -q^{17}+2 q^{15}-3 q^{13}+4 q^{11}-3 q^9+3 q^7-2 q^5+2 q^3+2 q^{-3} -2 q^{-5} +3 q^{-7} -3 q^{-9} +4 q^{-11} -3 q^{-13} +2 q^{-15} - q^{-17} }[/math]
1,0,1 [math]\displaystyle{ q^{56}-6 q^{54}+17 q^{52}-27 q^{50}+16 q^{48}+36 q^{46}-120 q^{44}+182 q^{42}-140 q^{40}-66 q^{38}+374 q^{36}-615 q^{34}+575 q^{32}-129 q^{30}-594 q^{28}+1268 q^{26}-1479 q^{24}+995 q^{22}+75 q^{20}-1276 q^{18}+2014 q^{16}-1920 q^{14}+1030 q^{12}+162 q^{10}-1046 q^8+1230 q^6-780 q^4+170 q^2+121+170 q^{-2} -780 q^{-4} +1230 q^{-6} -1046 q^{-8} +162 q^{-10} +1030 q^{-12} -1920 q^{-14} +2014 q^{-16} -1276 q^{-18} +75 q^{-20} +995 q^{-22} -1479 q^{-24} +1268 q^{-26} -594 q^{-28} -129 q^{-30} +575 q^{-32} -615 q^{-34} +374 q^{-36} -66 q^{-38} -140 q^{-40} +182 q^{-42} -120 q^{-44} +36 q^{-46} +16 q^{-48} -27 q^{-50} +17 q^{-52} -6 q^{-54} + q^{-56} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{40}-2 q^{38}-q^{36}+6 q^{34}-5 q^{32}-6 q^{30}+15 q^{28}-3 q^{26}-18 q^{24}+14 q^{22}+16 q^{20}-23 q^{18}-8 q^{16}+25 q^{14}-30 q^{10}+11 q^8+30 q^6-29 q^4-8 q^2+40-8 q^{-2} -29 q^{-4} +30 q^{-6} +11 q^{-8} -30 q^{-10} +25 q^{-14} -8 q^{-16} -23 q^{-18} +16 q^{-20} +14 q^{-22} -18 q^{-24} -3 q^{-26} +15 q^{-28} -6 q^{-30} -5 q^{-32} +6 q^{-34} - q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{20}+2 q^{18}-3 q^{16}+3 q^{14}-q^{12}+2 q^8-2 q^6+3 q^4-2 q^2+3-2 q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-8} - q^{-12} +3 q^{-14} -3 q^{-16} +2 q^{-18} - q^{-20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+13 q^{28}-20 q^{26}+29 q^{24}-38 q^{22}+43 q^{20}-44 q^{18}+40 q^{16}-29 q^{14}+14 q^{12}+7 q^{10}-29 q^8+51 q^6-69 q^4+82 q^2-86+82 q^{-2} -69 q^{-4} +51 q^{-6} -29 q^{-8} +7 q^{-10} +14 q^{-12} -29 q^{-14} +40 q^{-16} -44 q^{-18} +43 q^{-20} -38 q^{-22} +29 q^{-24} -20 q^{-26} +13 q^{-28} -7 q^{-30} +3 q^{-32} - q^{-34} }[/math]
1,0 [math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+10 q^{46}-16 q^{42}-12 q^{40}+16 q^{38}+28 q^{36}-5 q^{34}-39 q^{32}-17 q^{30}+36 q^{28}+38 q^{26}-19 q^{24}-47 q^{22}-3 q^{20}+44 q^{18}+20 q^{16}-32 q^{14}-28 q^{12}+20 q^{10}+30 q^8-10 q^6-30 q^4+4 q^2+31+4 q^{-2} -30 q^{-4} -10 q^{-6} +30 q^{-8} +20 q^{-10} -28 q^{-12} -32 q^{-14} +20 q^{-16} +44 q^{-18} -3 q^{-20} -47 q^{-22} -19 q^{-24} +38 q^{-26} +36 q^{-28} -17 q^{-30} -39 q^{-32} -5 q^{-34} +28 q^{-36} +16 q^{-38} -12 q^{-40} -16 q^{-42} +10 q^{-46} +4 q^{-48} -3 q^{-50} -3 q^{-52} + q^{-56} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{46}-3 q^{44}+4 q^{42}-6 q^{40}+11 q^{38}-16 q^{36}+19 q^{34}-23 q^{32}+31 q^{30}-35 q^{28}+33 q^{26}-33 q^{24}+34 q^{22}-27 q^{20}+14 q^{18}-8 q^{16}+18 q^{12}-33 q^{10}+39 q^8-48 q^6+64 q^4-64 q^2+64-64 q^{-2} +64 q^{-4} -48 q^{-6} +39 q^{-8} -33 q^{-10} +18 q^{-12} -8 q^{-16} +14 q^{-18} -27 q^{-20} +34 q^{-22} -33 q^{-24} +33 q^{-26} -35 q^{-28} +31 q^{-30} -23 q^{-32} +19 q^{-34} -16 q^{-36} +11 q^{-38} -6 q^{-40} +4 q^{-42} -3 q^{-44} + q^{-46} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-13 q^{70}+2 q^{68}+22 q^{66}-48 q^{64}+77 q^{62}-93 q^{60}+75 q^{58}-27 q^{56}-60 q^{54}+162 q^{52}-237 q^{50}+259 q^{48}-188 q^{46}+27 q^{44}+173 q^{42}-342 q^{40}+401 q^{38}-319 q^{36}+115 q^{34}+129 q^{32}-313 q^{30}+362 q^{28}-237 q^{26}+12 q^{24}+212 q^{22}-326 q^{20}+260 q^{18}-55 q^{16}-209 q^{14}+412 q^{12}-458 q^{10}+343 q^8-79 q^6-234 q^4+477 q^2-571+477 q^{-2} -234 q^{-4} -79 q^{-6} +343 q^{-8} -458 q^{-10} +412 q^{-12} -209 q^{-14} -55 q^{-16} +260 q^{-18} -326 q^{-20} +212 q^{-22} +12 q^{-24} -237 q^{-26} +362 q^{-28} -313 q^{-30} +129 q^{-32} +115 q^{-34} -319 q^{-36} +401 q^{-38} -342 q^{-40} +173 q^{-42} +27 q^{-44} -188 q^{-46} +259 q^{-48} -237 q^{-50} +162 q^{-52} -60 q^{-54} -27 q^{-56} +75 q^{-58} -93 q^{-60} +77 q^{-62} -48 q^{-64} +22 q^{-66} +2 q^{-68} -13 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/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["10 118"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-5 t^3+12 t^2-19 t+23-19 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+3 z^6+2 z^4+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]= { 97, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+4 q^4-8 q^3+12 q^2-15 q+17-15 q^{-1} +12 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +5 z^6-3 a^2 z^4-3 z^4 a^{-2} +8 z^4-2 a^2 z^2-2 z^2 a^{-2} +4 z^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 a z^9+3 z^9 a^{-1} +7 a^2 z^8+7 z^8 a^{-2} +14 z^8+7 a^3 z^7+6 a z^7+6 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +4 z^6 a^{-4} -30 z^6+a^5 z^5-12 a^3 z^5-20 a z^5-20 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+6 a^2 z^4+6 z^4 a^{-2} -6 z^4 a^{-4} +24 z^4-a^5 z^3+5 a^3 z^3+15 a z^3+15 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -6 z^2-a^3 z-3 a z-3 z a^{-1} -z a^{-3} +1 }[/math]
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