10 123

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+15 t^2-24 t+29-24 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6-z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{t^4-3 t^3+3 t^2-3 t+1\right\} }[/math]
Determinant and Signature	{ 121, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+5 q^4-10 q^3+15 q^2-19 q+21-19 q^{-1} +15 q^{-2} -10 q^{-3} +5 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +4 z^6-2 a^2 z^4-2 z^4 a^{-2} +3 z^4+a^2 z^2+z^2 a^{-2} -4 z^2+2 a^2+2 a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 4 a z^9+4 z^9 a^{-1} +10 a^2 z^8+10 z^8 a^{-2} +20 z^8+10 a^3 z^7+14 a z^7+14 z^7 a^{-1} +10 z^7 a^{-3} +5 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +5 z^6 a^{-4} -32 z^6+a^5 z^5-15 a^3 z^5-38 a z^5-38 z^5 a^{-1} -15 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4-3 a^2 z^4-3 z^4 a^{-2} -5 z^4 a^{-4} +4 z^4+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+3 q^{12}-2 q^{10}+3 q^8-3 q^4+4 q^2-5+4 q^{-2} -3 q^{-4} +3 q^{-8} -2 q^{-10} +3 q^{-12} - q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-4 q^{78}+10 q^{76}-20 q^{74}+26 q^{72}-25 q^{70}+10 q^{68}+30 q^{66}-80 q^{64}+140 q^{62}-180 q^{60}+158 q^{58}-71 q^{56}-100 q^{54}+308 q^{52}-473 q^{50}+528 q^{48}-391 q^{46}+69 q^{44}+343 q^{42}-693 q^{40}+822 q^{38}-656 q^{36}+239 q^{34}+267 q^{32}-647 q^{30}+750 q^{28}-495 q^{26}+29 q^{24}+435 q^{22}-675 q^{20}+551 q^{18}-133 q^{16}-414 q^{14}+836 q^{12}-944 q^{10}+710 q^8-173 q^6-472 q^4+970 q^2-1165+970 q^{-2} -472 q^{-4} -173 q^{-6} +710 q^{-8} -944 q^{-10} +836 q^{-12} -414 q^{-14} -133 q^{-16} +551 q^{-18} -675 q^{-20} +435 q^{-22} +29 q^{-24} -495 q^{-26} +750 q^{-28} -647 q^{-30} +267 q^{-32} +239 q^{-34} -656 q^{-36} +822 q^{-38} -693 q^{-40} +343 q^{-42} +69 q^{-44} -391 q^{-46} +528 q^{-48} -473 q^{-50} +308 q^{-52} -100 q^{-54} -71 q^{-56} +158 q^{-58} -180 q^{-60} +140 q^{-62} -80 q^{-64} +30 q^{-66} +10 q^{-68} -25 q^{-70} +26 q^{-72} -20 q^{-74} +10 q^{-76} -4 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_123.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+4 q^9-5 q^7+5 q^5-4 q^3+2 q+2 q^{-1} -4 q^{-3} +5 q^{-5} -5 q^{-7} +4 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-4 q^{30}+q^{28}+15 q^{26}-21 q^{24}-12 q^{22}+53 q^{20}-27 q^{18}-51 q^{16}+75 q^{14}-q^{12}-76 q^{10}+49 q^8+31 q^6-53 q^4-q^2+45- q^{-2} -53 q^{-4} +31 q^{-6} +49 q^{-8} -76 q^{-10} - q^{-12} +75 q^{-14} -51 q^{-16} -27 q^{-18} +53 q^{-20} -12 q^{-22} -21 q^{-24} +15 q^{-26} + q^{-28} -4 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+4 q^{61}-q^{59}-11 q^{57}+q^{55}+27 q^{53}+12 q^{51}-73 q^{49}-38 q^{47}+131 q^{45}+126 q^{43}-184 q^{41}-289 q^{39}+185 q^{37}+493 q^{35}-82 q^{33}-682 q^{31}-127 q^{29}+783 q^{27}+387 q^{25}-754 q^{23}-619 q^{21}+601 q^{19}+772 q^{17}-376 q^{15}-810 q^{13}+130 q^{11}+751 q^9+102 q^7-636 q^5-298 q^3+478 q+478 q^{-1} -298 q^{-3} -636 q^{-5} +102 q^{-7} +751 q^{-9} +130 q^{-11} -810 q^{-13} -376 q^{-15} +772 q^{-17} +601 q^{-19} -619 q^{-21} -754 q^{-23} +387 q^{-25} +783 q^{-27} -127 q^{-29} -682 q^{-31} -82 q^{-33} +493 q^{-35} +185 q^{-37} -289 q^{-39} -184 q^{-41} +126 q^{-43} +131 q^{-45} -38 q^{-47} -73 q^{-49} +12 q^{-51} +27 q^{-53} + q^{-55} -11 q^{-57} - q^{-59} +4 q^{-61} - q^{-63} }[/math]
5	[math]\displaystyle{ -q^{155}+4 q^{153}-q^{151}-11 q^{149}+5 q^{147}+11 q^{145}+7 q^{143}-3 q^{141}-28 q^{139}-43 q^{137}+23 q^{135}+137 q^{133}+116 q^{131}-95 q^{129}-381 q^{127}-416 q^{125}+53 q^{123}+958 q^{121}+1448 q^{119}+427 q^{117}-1828 q^{115}-3593 q^{113}-2582 q^{111}+1979 q^{109}+7323 q^{107}+7942 q^{105}+557 q^{103}-11096 q^{101}-17370 q^{99}-9355 q^{97}+11333 q^{95}+29603 q^{93}+26624 q^{91}-2590 q^{89}-39209 q^{87}-51313 q^{85}-19986 q^{83}+38304 q^{81}+77229 q^{79}+56261 q^{77}-19581 q^{75}-93770 q^{73}-99666 q^{71}-19217 q^{69}+90984 q^{67}+138225 q^{65}+72047 q^{63}-64007 q^{61}-159134 q^{59}-126635 q^{57}+16445 q^{55}+154719 q^{53}+168882 q^{51}+40575 q^{49}-125523 q^{47}-188457 q^{45}-93249 q^{43}+79592 q^{41}+182862 q^{39}+130390 q^{37}-29132 q^{35}-156911 q^{33}-146829 q^{31}-15052 q^{29}+119894 q^{27}+144524 q^{25}+46592 q^{23}-81740 q^{21}-129908 q^{19}-64506 q^{17}+49163 q^{15}+110867 q^{13}+72745 q^{11}-24867 q^9-94167 q^7-76937 q^5+7459 q^3+82883 q+82883 q^{-1} +7459 q^{-3} -76937 q^{-5} -94167 q^{-7} -24867 q^{-9} +72745 q^{-11} +110867 q^{-13} +49163 q^{-15} -64506 q^{-17} -129908 q^{-19} -81740 q^{-21} +46592 q^{-23} +144524 q^{-25} +119894 q^{-27} -15052 q^{-29} -146829 q^{-31} -156911 q^{-33} -29132 q^{-35} +130390 q^{-37} +182862 q^{-39} +79592 q^{-41} -93249 q^{-43} -188457 q^{-45} -125523 q^{-47} +40575 q^{-49} +168882 q^{-51} +154719 q^{-53} +16445 q^{-55} -126635 q^{-57} -159134 q^{-59} -64007 q^{-61} +72047 q^{-63} +138225 q^{-65} +90984 q^{-67} -19217 q^{-69} -99666 q^{-71} -93770 q^{-73} -19581 q^{-75} +56261 q^{-77} +77229 q^{-79} +38304 q^{-81} -19986 q^{-83} -51313 q^{-85} -39209 q^{-87} -2590 q^{-89} +26624 q^{-91} +29603 q^{-93} +11333 q^{-95} -9355 q^{-97} -17370 q^{-99} -11096 q^{-101} +557 q^{-103} +7942 q^{-105} +7323 q^{-107} +1979 q^{-109} -2582 q^{-111} -3593 q^{-113} -1828 q^{-115} +427 q^{-117} +1448 q^{-119} +958 q^{-121} +53 q^{-123} -416 q^{-125} -381 q^{-127} -95 q^{-129} +116 q^{-131} +137 q^{-133} +23 q^{-135} -43 q^{-137} -28 q^{-139} -3 q^{-141} +7 q^{-143} +11 q^{-145} +5 q^{-147} -11 q^{-149} - q^{-151} +4 q^{-153} - q^{-155} }[/math]
6	[math]\displaystyle{ q^{216}-4 q^{214}+q^{212}+11 q^{210}-5 q^{208}-11 q^{206}-11 q^{204}+23 q^{202}+13 q^{200}-12 q^{198}+32 q^{196}-63 q^{194}-102 q^{192}-35 q^{190}+216 q^{188}+331 q^{186}+151 q^{184}-63 q^{182}-828 q^{180}-1302 q^{178}-822 q^{176}+1158 q^{174}+3193 q^{172}+3762 q^{170}+2167 q^{168}-3467 q^{166}-9707 q^{164}-11888 q^{162}-4605 q^{160}+9930 q^{158}+24650 q^{156}+29895 q^{154}+12734 q^{152}-22818 q^{150}-59258 q^{148}-66630 q^{146}-29644 q^{144}+45161 q^{142}+122100 q^{140}+139844 q^{138}+65184 q^{136}-85492 q^{134}-227185 q^{132}-263331 q^{130}-129523 q^{128}+140894 q^{126}+394984 q^{124}+456871 q^{122}+224564 q^{120}-215686 q^{118}-631328 q^{116}-729568 q^{114}-362254 q^{112}+323275 q^{110}+944757 q^{108}+1069701 q^{106}+528876 q^{104}-463516 q^{102}-1326402 q^{100}-1467670 q^{98}-686860 q^{96}+650850 q^{94}+1746741 q^{92}+1872976 q^{90}+805959 q^{88}-889284 q^{86}-2184458 q^{84}-2211519 q^{82}-835611 q^{80}+1167684 q^{78}+2577899 q^{76}+2429580 q^{74}+743587 q^{72}-1482269 q^{70}-2853591 q^{68}-2467337 q^{66}-527285 q^{64}+1780716 q^{62}+2971580 q^{60}+2302095 q^{58}+195777 q^{56}-1998465 q^{54}-2893466 q^{52}-1955828 q^{50}+181758 q^{48}+2103658 q^{46}+2622082 q^{44}+1470680 q^{42}-527658 q^{40}-2065674 q^{38}-2201421 q^{36}-938315 q^{34}+801680 q^{32}+1892462 q^{30}+1687640 q^{28}+437651 q^{26}-973947 q^{24}-1624896 q^{22}-1162065 q^{20}+1047 q^{18}+1054281 q^{16}+1308526 q^{14}+672400 q^{12}-368280 q^{10}-1077408 q^8-993482 q^6-218554 q^4+688126 q^2+1077985+688126 q^{-2} -218554 q^{-4} -993482 q^{-6} -1077408 q^{-8} -368280 q^{-10} +672400 q^{-12} +1308526 q^{-14} +1054281 q^{-16} +1047 q^{-18} -1162065 q^{-20} -1624896 q^{-22} -973947 q^{-24} +437651 q^{-26} +1687640 q^{-28} +1892462 q^{-30} +801680 q^{-32} -938315 q^{-34} -2201421 q^{-36} -2065674 q^{-38} -527658 q^{-40} +1470680 q^{-42} +2622082 q^{-44} +2103658 q^{-46} +181758 q^{-48} -1955828 q^{-50} -2893466 q^{-52} -1998465 q^{-54} +195777 q^{-56} +2302095 q^{-58} +2971580 q^{-60} +1780716 q^{-62} -527285 q^{-64} -2467337 q^{-66} -2853591 q^{-68} -1482269 q^{-70} +743587 q^{-72} +2429580 q^{-74} +2577899 q^{-76} +1167684 q^{-78} -835611 q^{-80} -2211519 q^{-82} -2184458 q^{-84} -889284 q^{-86} +805959 q^{-88} +1872976 q^{-90} +1746741 q^{-92} +650850 q^{-94} -686860 q^{-96} -1467670 q^{-98} -1326402 q^{-100} -463516 q^{-102} +528876 q^{-104} +1069701 q^{-106} +944757 q^{-108} +323275 q^{-110} -362254 q^{-112} -729568 q^{-114} -631328 q^{-116} -215686 q^{-118} +224564 q^{-120} +456871 q^{-122} +394984 q^{-124} +140894 q^{-126} -129523 q^{-128} -263331 q^{-130} -227185 q^{-132} -85492 q^{-134} +65184 q^{-136} +139844 q^{-138} +122100 q^{-140} +45161 q^{-142} -29644 q^{-144} -66630 q^{-146} -59258 q^{-148} -22818 q^{-150} +12734 q^{-152} +29895 q^{-154} +24650 q^{-156} +9930 q^{-158} -4605 q^{-160} -11888 q^{-162} -9707 q^{-164} -3467 q^{-166} +2167 q^{-168} +3762 q^{-170} +3193 q^{-172} +1158 q^{-174} -822 q^{-176} -1302 q^{-178} -828 q^{-180} -63 q^{-182} +151 q^{-184} +331 q^{-186} +216 q^{-188} -35 q^{-190} -102 q^{-192} -63 q^{-194} +32 q^{-196} -12 q^{-198} +13 q^{-200} +23 q^{-202} -11 q^{-204} -11 q^{-206} -5 q^{-208} +11 q^{-210} + q^{-212} -4 q^{-214} + q^{-216} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{14}+3 q^{12}-2 q^{10}+3 q^8-3 q^4+4 q^2-5+4 q^{-2} -3 q^{-4} +3 q^{-8} -2 q^{-10} +3 q^{-12} - q^{-14} }[/math]
1,1	[math]\displaystyle{ q^{44}-8 q^{42}+32 q^{40}-88 q^{38}+196 q^{36}-392 q^{34}+716 q^{32}-1194 q^{30}+1831 q^{28}-2600 q^{26}+3434 q^{24}-4172 q^{22}+4631 q^{20}-4638 q^{18}+4072 q^{16}-2860 q^{14}+1036 q^{12}+1192 q^{10}-3588 q^8+5840 q^6-7694 q^4+8922 q^2-9330+8922 q^{-2} -7694 q^{-4} +5840 q^{-6} -3588 q^{-8} +1192 q^{-10} +1036 q^{-12} -2860 q^{-14} +4072 q^{-16} -4638 q^{-18} +4631 q^{-20} -4172 q^{-22} +3434 q^{-24} -2600 q^{-26} +1831 q^{-28} -1194 q^{-30} +716 q^{-32} -392 q^{-34} +196 q^{-36} -88 q^{-38} +32 q^{-40} -8 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{38}-3 q^{36}-q^{34}+9 q^{32}-6 q^{30}-9 q^{28}+14 q^{26}+9 q^{24}-13 q^{22}-10 q^{20}+20 q^{18}+10 q^{16}-35 q^{14}+2 q^{12}+24 q^{10}-20 q^8-14 q^6+21 q^4+11 q^2-14+11 q^{-2} +21 q^{-4} -14 q^{-6} -20 q^{-8} +24 q^{-10} +2 q^{-12} -35 q^{-14} +10 q^{-16} +20 q^{-18} -10 q^{-20} -13 q^{-22} +9 q^{-24} +14 q^{-26} -9 q^{-28} -6 q^{-30} +9 q^{-32} - q^{-34} -3 q^{-36} + q^{-38} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-4 q^{32}+2 q^{30}+11 q^{28}-19 q^{26}+4 q^{24}+29 q^{22}-43 q^{20}+6 q^{18}+49 q^{16}-54 q^{14}+5 q^{12}+50 q^{10}-40 q^8-8 q^6+28 q^4-6 q^2-16-6 q^{-2} +28 q^{-4} -8 q^{-6} -40 q^{-8} +50 q^{-10} +5 q^{-12} -54 q^{-14} +49 q^{-16} +6 q^{-18} -43 q^{-20} +29 q^{-22} +4 q^{-24} -19 q^{-26} +11 q^{-28} +2 q^{-30} -4 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{17}+3 q^{15}-3 q^{13}+6 q^{11}-3 q^9+4 q^7-3 q^5+q^3-2 q-2 q^{-1} + q^{-3} -3 q^{-5} +4 q^{-7} -3 q^{-9} +6 q^{-11} -3 q^{-13} +3 q^{-15} - q^{-17} }[/math]
1,0,1	[math]\displaystyle{ q^{56}-8 q^{54}+28 q^{52}-52 q^{50}+38 q^{48}+65 q^{46}-253 q^{44}+409 q^{42}-326 q^{40}-151 q^{38}+912 q^{36}-1524 q^{34}+1436 q^{32}-334 q^{30}-1493 q^{28}+3199 q^{26}-3741 q^{24}+2539 q^{22}+148 q^{20}-3147 q^{18}+5022 q^{16}-4816 q^{14}+2595 q^{12}+368 q^{10}-2626 q^8+3081 q^6-1935 q^4+375 q^2+395+375 q^{-2} -1935 q^{-4} +3081 q^{-6} -2626 q^{-8} +368 q^{-10} +2595 q^{-12} -4816 q^{-14} +5022 q^{-16} -3147 q^{-18} +148 q^{-20} +2539 q^{-22} -3741 q^{-24} +3199 q^{-26} -1493 q^{-28} -334 q^{-30} +1436 q^{-32} -1524 q^{-34} +912 q^{-36} -151 q^{-38} -326 q^{-40} +409 q^{-42} -253 q^{-44} +65 q^{-46} +38 q^{-48} -52 q^{-50} +28 q^{-52} -8 q^{-54} + q^{-56} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{40}-3 q^{38}-q^{36}+10 q^{34}-8 q^{32}-9 q^{30}+25 q^{28}-5 q^{26}-30 q^{24}+23 q^{22}+26 q^{20}-36 q^{18}-11 q^{16}+43 q^{14}+q^{12}-48 q^{10}+18 q^8+46 q^6-50 q^4-18 q^2+62-18 q^{-2} -50 q^{-4} +46 q^{-6} +18 q^{-8} -48 q^{-10} + q^{-12} +43 q^{-14} -11 q^{-16} -36 q^{-18} +26 q^{-20} +23 q^{-22} -30 q^{-24} -5 q^{-26} +25 q^{-28} -9 q^{-30} -8 q^{-32} +10 q^{-34} - q^{-36} -3 q^{-38} + q^{-40} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{20}+3 q^{18}-3 q^{16}+5 q^{14}+q^{10}+3 q^8-3 q^6+2 q^4-5 q^2+1-5 q^{-2} +2 q^{-4} -3 q^{-6} +3 q^{-8} + q^{-10} +5 q^{-14} -3 q^{-16} +3 q^{-18} - q^{-20} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+4 q^{32}-10 q^{30}+19 q^{28}-31 q^{26}+46 q^{24}-59 q^{22}+69 q^{20}-70 q^{18}+65 q^{16}-48 q^{14}+23 q^{12}+10 q^{10}-46 q^8+80 q^6-110 q^4+130 q^2-138+130 q^{-2} -110 q^{-4} +80 q^{-6} -46 q^{-8} +10 q^{-10} +23 q^{-12} -48 q^{-14} +65 q^{-16} -70 q^{-18} +69 q^{-20} -59 q^{-22} +46 q^{-24} -31 q^{-26} +19 q^{-28} -10 q^{-30} +4 q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-4 q^{52}-4 q^{50}+6 q^{48}+15 q^{46}+q^{44}-25 q^{42}-20 q^{40}+25 q^{38}+44 q^{36}-6 q^{34}-62 q^{32}-28 q^{30}+57 q^{28}+61 q^{26}-29 q^{24}-75 q^{22}-4 q^{20}+71 q^{18}+32 q^{16}-52 q^{14}-45 q^{12}+32 q^{10}+48 q^8-17 q^6-49 q^4+5 q^2+49+5 q^{-2} -49 q^{-4} -17 q^{-6} +48 q^{-8} +32 q^{-10} -45 q^{-12} -52 q^{-14} +32 q^{-16} +71 q^{-18} -4 q^{-20} -75 q^{-22} -29 q^{-24} +61 q^{-26} +57 q^{-28} -28 q^{-30} -62 q^{-32} -6 q^{-34} +44 q^{-36} +25 q^{-38} -20 q^{-40} -25 q^{-42} + q^{-44} +15 q^{-46} +6 q^{-48} -4 q^{-50} -4 q^{-52} + q^{-56} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{46}-4 q^{44}+6 q^{42}-8 q^{40}+16 q^{38}-25 q^{36}+30 q^{34}-36 q^{32}+48 q^{30}-56 q^{28}+53 q^{26}-53 q^{24}+56 q^{22}-42 q^{20}+26 q^{18}-12 q^{16}+2 q^{14}+30 q^{12}-51 q^{10}+61 q^8-79 q^6+98 q^4-106 q^2+98-106 q^{-2} +98 q^{-4} -79 q^{-6} +61 q^{-8} -51 q^{-10} +30 q^{-12} +2 q^{-14} -12 q^{-16} +26 q^{-18} -42 q^{-20} +56 q^{-22} -53 q^{-24} +53 q^{-26} -56 q^{-28} +48 q^{-30} -36 q^{-32} +30 q^{-34} -25 q^{-36} +16 q^{-38} -8 q^{-40} +6 q^{-42} -4 q^{-44} + q^{-46} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-4 q^{78}+10 q^{76}-20 q^{74}+26 q^{72}-25 q^{70}+10 q^{68}+30 q^{66}-80 q^{64}+140 q^{62}-180 q^{60}+158 q^{58}-71 q^{56}-100 q^{54}+308 q^{52}-473 q^{50}+528 q^{48}-391 q^{46}+69 q^{44}+343 q^{42}-693 q^{40}+822 q^{38}-656 q^{36}+239 q^{34}+267 q^{32}-647 q^{30}+750 q^{28}-495 q^{26}+29 q^{24}+435 q^{22}-675 q^{20}+551 q^{18}-133 q^{16}-414 q^{14}+836 q^{12}-944 q^{10}+710 q^8-173 q^6-472 q^4+970 q^2-1165+970 q^{-2} -472 q^{-4} -173 q^{-6} +710 q^{-8} -944 q^{-10} +836 q^{-12} -414 q^{-14} -133 q^{-16} +551 q^{-18} -675 q^{-20} +435 q^{-22} +29 q^{-24} -495 q^{-26} +750 q^{-28} -647 q^{-30} +267 q^{-32} +239 q^{-34} -656 q^{-36} +822 q^{-38} -693 q^{-40} +343 q^{-42} +69 q^{-44} -391 q^{-46} +528 q^{-48} -473 q^{-50} +308 q^{-52} -100 q^{-54} -71 q^{-56} +158 q^{-58} -180 q^{-60} +140 q^{-62} -80 q^{-64} +30 q^{-66} +10 q^{-68} -25 q^{-70} +26 q^{-72} -20 q^{-74} +10 q^{-76} -4 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["10 123"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+15 t^2-24 t+29-24 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6-z^4-2 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{ \left\{t^4-3 t^3+3 t^2-3 t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+5 q^4-10 q^3+15 q^2-19 q+21-19 q^{-1} +15 q^{-2} -10 q^{-3} +5 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} +4 z^6-2 a^2 z^4-2 z^4 a^{-2} +3 z^4+a^2 z^2+z^2 a^{-2} -4 z^2+2 a^2+2 a^{-2} -3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 4 a z^9+4 z^9 a^{-1} +10 a^2 z^8+10 z^8 a^{-2} +20 z^8+10 a^3 z^7+14 a z^7+14 z^7 a^{-1} +10 z^7 a^{-3} +5 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +5 z^6 a^{-4} -32 z^6+a^5 z^5-15 a^3 z^5-38 a z^5-38 z^5 a^{-1} -15 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4-3 a^2 z^4-3 z^4 a^{-2} -5 z^4 a^{-4} +4 z^4+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3 }[/math]