10 54

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+6 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 47, 2 }
Jones polynomial [math]\displaystyle{ -q^6+2 q^5-4 q^4+6 q^3-7 q^2+8 q-6+6 q^{-1} -4 q^{-2} +2 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+4 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} +5 z^2-2 a^2+2 a^{-2} -2 a^{-4} +3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7+3 z^7 a^{-1} +4 z^7 a^{-3} -9 a^2 z^6-5 z^6 a^{-2} +4 z^6 a^{-4} -18 z^6-5 a^3 z^5-13 a z^5-18 z^5 a^{-1} -7 z^5 a^{-3} +3 z^5 a^{-5} +12 a^2 z^4-3 z^4 a^{-2} -6 z^4 a^{-4} +2 z^4 a^{-6} +17 z^4+8 a^3 z^3+20 a z^3+17 z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -6 a^2 z^2+5 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} -7 z^2-4 a^3 z-8 a z-5 z a^{-1} +z a^{-3} +z a^{-5} -z a^{-7} +2 a^2-2 a^{-2} -2 a^{-4} +3 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}-q^8-q^6+q^4+3+2 q^{-2} + q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{60}-q^{58}+4 q^{56}-6 q^{54}+6 q^{52}-5 q^{50}-3 q^{48}+13 q^{46}-22 q^{44}+24 q^{42}-19 q^{40}+2 q^{38}+16 q^{36}-34 q^{34}+38 q^{32}-29 q^{30}+7 q^{28}+12 q^{26}-30 q^{24}+29 q^{22}-18 q^{20}+q^{18}+15 q^{16}-22 q^{14}+17 q^{12}-q^{10}-15 q^8+28 q^6-28 q^4+24 q^2-2-14 q^{-2} +35 q^{-4} -40 q^{-6} +40 q^{-8} -18 q^{-10} -5 q^{-12} +27 q^{-14} -36 q^{-16} +34 q^{-18} -16 q^{-20} - q^{-22} +18 q^{-24} -22 q^{-26} +14 q^{-28} -15 q^{-32} +21 q^{-34} -16 q^{-36} +3 q^{-38} +11 q^{-40} -19 q^{-42} +21 q^{-44} -15 q^{-46} +7 q^{-48} + q^{-50} -11 q^{-52} +13 q^{-54} -14 q^{-56} +12 q^{-58} -8 q^{-60} +3 q^{-62} + q^{-64} -8 q^{-66} +9 q^{-68} -12 q^{-70} +9 q^{-72} -5 q^{-74} + q^{-76} +2 q^{-78} -6 q^{-80} +6 q^{-82} -5 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_54.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^9+q^7-2 q^5+2 q^3+2 q^{-1} + q^{-3} - q^{-5} +2 q^{-7} -2 q^{-9} + q^{-11} - q^{-13} }[/math]
2 [math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+4 q^{22}+q^{20}-7 q^{18}+3 q^{16}+5 q^{14}-8 q^{12}-q^{10}+7 q^8-4 q^6-4 q^4+7 q^2+2-4 q^{-2} +3 q^{-4} +6 q^{-6} -3 q^{-8} -3 q^{-10} +6 q^{-12} + q^{-14} -8 q^{-16} +3 q^{-18} +3 q^{-20} -6 q^{-22} +2 q^{-24} + q^{-26} -3 q^{-28} +2 q^{-30} - q^{-34} + q^{-36} }[/math]
3 [math]\displaystyle{ -q^{57}+q^{55}+2 q^{53}-5 q^{49}-3 q^{47}+7 q^{45}+9 q^{43}-5 q^{41}-15 q^{39}-2 q^{37}+19 q^{35}+11 q^{33}-15 q^{31}-20 q^{29}+6 q^{27}+25 q^{25}+3 q^{23}-22 q^{21}-16 q^{19}+16 q^{17}+20 q^{15}-8 q^{13}-26 q^{11}+q^9+23 q^7+9 q^5-23 q^3-9 q+24 q^{-1} +17 q^{-3} -20 q^{-5} -19 q^{-7} +18 q^{-9} +23 q^{-11} -8 q^{-13} -24 q^{-15} +20 q^{-19} +15 q^{-21} -14 q^{-23} -24 q^{-25} + q^{-27} +27 q^{-29} +6 q^{-31} -23 q^{-33} -16 q^{-35} +17 q^{-37} +13 q^{-39} -9 q^{-41} -10 q^{-43} +3 q^{-45} +6 q^{-47} -2 q^{-51} + q^{-57} + q^{-59} - q^{-61} + q^{-67} - q^{-69} }[/math]
4 [math]\displaystyle{ q^{96}-q^{94}-2 q^{92}+q^{88}+7 q^{86}+q^{84}-7 q^{82}-9 q^{80}-9 q^{78}+16 q^{76}+20 q^{74}+8 q^{72}-14 q^{70}-42 q^{68}-10 q^{66}+22 q^{64}+48 q^{62}+33 q^{60}-40 q^{58}-56 q^{56}-41 q^{54}+31 q^{52}+88 q^{50}+38 q^{48}-25 q^{46}-97 q^{44}-64 q^{42}+47 q^{40}+93 q^{38}+80 q^{36}-44 q^{34}-123 q^{32}-66 q^{30}+40 q^{28}+135 q^{26}+66 q^{24}-73 q^{22}-128 q^{20}-60 q^{18}+98 q^{16}+130 q^{14}+13 q^{12}-111 q^{10}-109 q^8+37 q^6+125 q^4+55 q^2-76-107 q^{-2} +9 q^{-4} +112 q^{-6} +60 q^{-8} -69 q^{-10} -104 q^{-12} -8 q^{-14} +111 q^{-16} +85 q^{-18} -41 q^{-20} -110 q^{-22} -70 q^{-24} +62 q^{-26} +120 q^{-28} +53 q^{-30} -55 q^{-32} -141 q^{-34} -67 q^{-36} +81 q^{-38} +149 q^{-40} +75 q^{-42} -118 q^{-44} -169 q^{-46} -34 q^{-48} +132 q^{-50} +164 q^{-52} -19 q^{-54} -149 q^{-56} -98 q^{-58} +44 q^{-60} +132 q^{-62} +35 q^{-64} -64 q^{-66} -70 q^{-68} -8 q^{-70} +62 q^{-72} +26 q^{-74} -15 q^{-76} -25 q^{-78} -12 q^{-80} +21 q^{-82} +6 q^{-84} - q^{-86} -4 q^{-88} -7 q^{-90} +7 q^{-92} - q^{-94} -3 q^{-100} +3 q^{-102} - q^{-104} - q^{-110} + q^{-112} }[/math]
5 [math]\displaystyle{ -q^{145}+q^{143}+2 q^{141}-q^{137}-3 q^{135}-5 q^{133}-q^{131}+9 q^{129}+11 q^{127}+6 q^{125}-4 q^{123}-20 q^{121}-26 q^{119}-9 q^{117}+23 q^{115}+43 q^{113}+40 q^{111}+4 q^{109}-50 q^{107}-80 q^{105}-55 q^{103}+19 q^{101}+93 q^{99}+119 q^{97}+62 q^{95}-54 q^{93}-157 q^{91}-161 q^{89}-48 q^{87}+114 q^{85}+226 q^{83}+195 q^{81}+14 q^{79}-206 q^{77}-304 q^{75}-196 q^{73}+60 q^{71}+314 q^{69}+373 q^{67}+163 q^{65}-192 q^{63}-437 q^{61}-394 q^{59}-59 q^{57}+365 q^{55}+553 q^{53}+338 q^{51}-145 q^{49}-556 q^{47}-585 q^{45}-162 q^{43}+421 q^{41}+704 q^{39}+458 q^{37}-162 q^{35}-692 q^{33}-684 q^{31}-128 q^{29}+544 q^{27}+789 q^{25}+395 q^{23}-337 q^{21}-778 q^{19}-572 q^{17}+105 q^{15}+681 q^{13}+665 q^{11}+74 q^9-545 q^7-650 q^5-188 q^3+407 q+600 q^{-1} +224 q^{-3} -317 q^{-5} -511 q^{-7} -200 q^{-9} +280 q^{-11} +452 q^{-13} +162 q^{-15} -292 q^{-17} -443 q^{-19} -135 q^{-21} +321 q^{-23} +475 q^{-25} +181 q^{-27} -305 q^{-29} -541 q^{-31} -302 q^{-33} +212 q^{-35} +561 q^{-37} +477 q^{-39} +6 q^{-41} -497 q^{-43} -646 q^{-45} -308 q^{-47} +297 q^{-49} +732 q^{-51} +625 q^{-53} +16 q^{-55} -667 q^{-57} -877 q^{-59} -385 q^{-61} +473 q^{-63} +968 q^{-65} +687 q^{-67} -166 q^{-69} -903 q^{-71} -878 q^{-73} -117 q^{-75} +704 q^{-77} +887 q^{-79} +329 q^{-81} -449 q^{-83} -778 q^{-85} -416 q^{-87} +235 q^{-89} +588 q^{-91} +398 q^{-93} -84 q^{-95} -393 q^{-97} -316 q^{-99} +4 q^{-101} +241 q^{-103} +218 q^{-105} +20 q^{-107} -134 q^{-109} -128 q^{-111} -23 q^{-113} +67 q^{-115} +74 q^{-117} +16 q^{-119} -34 q^{-121} -38 q^{-123} -7 q^{-125} +13 q^{-127} +15 q^{-129} +6 q^{-131} -5 q^{-133} -9 q^{-135} -2 q^{-137} +4 q^{-139} +2 q^{-145} -2 q^{-147} +2 q^{-151} - q^{-153} - q^{-155} + q^{-157} + q^{-163} - q^{-165} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{12}-q^8-q^6+q^4+3+2 q^{-2} + q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{36}-2 q^{34}+8 q^{32}-18 q^{30}+33 q^{28}-54 q^{26}+76 q^{24}-96 q^{22}+108 q^{20}-108 q^{18}+90 q^{16}-62 q^{14}+18 q^{12}+24 q^{10}-78 q^8+116 q^6-150 q^4+170 q^2-172+172 q^{-2} -134 q^{-4} +112 q^{-6} -56 q^{-8} +28 q^{-10} +15 q^{-12} -40 q^{-14} +50 q^{-16} -62 q^{-18} +49 q^{-20} -48 q^{-22} +38 q^{-24} -36 q^{-26} +30 q^{-28} -28 q^{-30} +26 q^{-32} -26 q^{-34} +22 q^{-36} -18 q^{-38} +16 q^{-40} -12 q^{-42} +9 q^{-44} -6 q^{-46} +4 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{34}-q^{30}+2 q^{26}+q^{24}-2 q^{22}-2 q^{20}+2 q^{18}-4 q^{14}-3 q^{12}-q^{10}-q^8-5 q^6-q^4+4 q^2+3+4 q^{-2} +7 q^{-4} +4 q^{-6} +3 q^{-8} +5 q^{-10} +5 q^{-12} - q^{-14} - q^{-16} +2 q^{-18} - q^{-20} -9 q^{-22} -4 q^{-24} -3 q^{-28} -2 q^{-30} +2 q^{-34} + q^{-40} + q^{-46} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{26}-q^{24}+2 q^{22}+q^{20}-3 q^{18}+2 q^{16}-3 q^{14}-7 q^{12}-4 q^8-4 q^6+7 q^4+3 q^2+2+9 q^{-2} +5 q^{-4} +3 q^{-8} +3 q^{-10} +2 q^{-12} -4 q^{-14} + q^{-16} + q^{-18} -8 q^{-20} - q^{-22} +3 q^{-24} -6 q^{-26} - q^{-28} +5 q^{-30} -3 q^{-32} -2 q^{-34} +3 q^{-36} - q^{-40} + q^{-42} }[/math]
1,0,0 [math]\displaystyle{ -q^{15}-2 q^{11}-2 q^7+q^5+3 q+2 q^{-1} +3 q^{-3} +2 q^{-5} + q^{-7} +2 q^{-9} - q^{-11} + q^{-13} -2 q^{-15} -2 q^{-19} - q^{-23} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{32}+q^{28}+3 q^{26}+q^{24}+2 q^{20}-3 q^{18}-8 q^{16}-6 q^{14}-8 q^{12}-12 q^{10}-8 q^8+3 q^6+4 q^4+2 q^2+13+19 q^{-2} +7 q^{-4} +6 q^{-6} +12 q^{-8} +4 q^{-10} -2 q^{-12} +3 q^{-14} +2 q^{-16} -3 q^{-18} -4 q^{-20} + q^{-22} -4 q^{-24} -9 q^{-26} -3 q^{-28} -6 q^{-32} -4 q^{-34} +2 q^{-36} + q^{-38} - q^{-40} - q^{-42} +2 q^{-44} +2 q^{-46} + q^{-52} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{18}-2 q^{14}-q^{12}-q^{10}-2 q^8+q^6+3 q^2+2+3 q^{-2} +3 q^{-4} +2 q^{-6} +2 q^{-8} + q^{-10} +2 q^{-12} - q^{-14} + q^{-16} -2 q^{-18} - q^{-20} - q^{-22} -2 q^{-24} - q^{-28} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{26}+q^{24}-4 q^{22}+5 q^{20}-7 q^{18}+8 q^{16}-9 q^{14}+9 q^{12}-8 q^{10}+6 q^8-2 q^6-q^4+7 q^2-10+15 q^{-2} -15 q^{-4} +18 q^{-6} -15 q^{-8} +15 q^{-10} -10 q^{-12} +8 q^{-14} -3 q^{-16} - q^{-18} +4 q^{-20} -7 q^{-22} +7 q^{-24} -8 q^{-26} +7 q^{-28} -7 q^{-30} +5 q^{-32} -4 q^{-34} +3 q^{-36} -2 q^{-38} + q^{-40} - q^{-42} }[/math]
1,0 [math]\displaystyle{ q^{44}-q^{40}-q^{38}+3 q^{36}+3 q^{34}-3 q^{32}-5 q^{30}+q^{28}+6 q^{26}+q^{24}-9 q^{22}-7 q^{20}+4 q^{18}+7 q^{16}-3 q^{14}-11 q^{12}-3 q^{10}+8 q^8+8 q^6-3 q^4-5 q^2+3+9 q^{-2} +3 q^{-4} -2 q^{-6} +7 q^{-10} +3 q^{-12} -4 q^{-14} -3 q^{-16} +6 q^{-18} +6 q^{-20} -3 q^{-22} -8 q^{-24} +7 q^{-28} + q^{-30} -8 q^{-32} -6 q^{-34} +3 q^{-36} +6 q^{-38} - q^{-40} -7 q^{-42} -4 q^{-44} +3 q^{-46} +6 q^{-48} -4 q^{-52} -3 q^{-54} + q^{-56} +3 q^{-58} + q^{-60} - q^{-62} - q^{-64} + q^{-68} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{34}-q^{32}+3 q^{30}-3 q^{28}+6 q^{26}-6 q^{24}+5 q^{22}-9 q^{20}+5 q^{18}-12 q^{16}+q^{14}-9 q^{12}+2 q^{10}-2 q^8+7 q^4+15-5 q^{-2} +16 q^{-4} -9 q^{-6} +16 q^{-8} -11 q^{-10} +14 q^{-12} -9 q^{-14} +10 q^{-16} -6 q^{-18} +5 q^{-20} -3 q^{-22} -2 q^{-24} -7 q^{-28} +2 q^{-30} -7 q^{-32} +5 q^{-34} -7 q^{-36} +4 q^{-38} -5 q^{-40} +6 q^{-42} -4 q^{-44} +2 q^{-46} -3 q^{-48} +3 q^{-50} - q^{-52} + q^{-54} - q^{-56} + q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{60}-q^{58}+4 q^{56}-6 q^{54}+6 q^{52}-5 q^{50}-3 q^{48}+13 q^{46}-22 q^{44}+24 q^{42}-19 q^{40}+2 q^{38}+16 q^{36}-34 q^{34}+38 q^{32}-29 q^{30}+7 q^{28}+12 q^{26}-30 q^{24}+29 q^{22}-18 q^{20}+q^{18}+15 q^{16}-22 q^{14}+17 q^{12}-q^{10}-15 q^8+28 q^6-28 q^4+24 q^2-2-14 q^{-2} +35 q^{-4} -40 q^{-6} +40 q^{-8} -18 q^{-10} -5 q^{-12} +27 q^{-14} -36 q^{-16} +34 q^{-18} -16 q^{-20} - q^{-22} +18 q^{-24} -22 q^{-26} +14 q^{-28} -15 q^{-32} +21 q^{-34} -16 q^{-36} +3 q^{-38} +11 q^{-40} -19 q^{-42} +21 q^{-44} -15 q^{-46} +7 q^{-48} + q^{-50} -11 q^{-52} +13 q^{-54} -14 q^{-56} +12 q^{-58} -8 q^{-60} +3 q^{-62} + q^{-64} -8 q^{-66} +9 q^{-68} -12 q^{-70} +9 q^{-72} -5 q^{-74} + q^{-76} +2 q^{-78} -6 q^{-80} +6 q^{-82} -5 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/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 54"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+6 z^4+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]= { 47, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^6+2 q^5-4 q^4+6 q^3-7 q^2+8 q-6+6 q^{-1} -4 q^{-2} +2 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+4 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} +5 z^2-2 a^2+2 a^{-2} -2 a^{-4} +3 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7+3 z^7 a^{-1} +4 z^7 a^{-3} -9 a^2 z^6-5 z^6 a^{-2} +4 z^6 a^{-4} -18 z^6-5 a^3 z^5-13 a z^5-18 z^5 a^{-1} -7 z^5 a^{-3} +3 z^5 a^{-5} +12 a^2 z^4-3 z^4 a^{-2} -6 z^4 a^{-4} +2 z^4 a^{-6} +17 z^4+8 a^3 z^3+20 a z^3+17 z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -6 a^2 z^2+5 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} -7 z^2-4 a^3 z-8 a z-5 z a^{-1} +z a^{-3} +z a^{-5} -z a^{-7} +2 a^2-2 a^{-2} -2 a^{-4} +3 }[/math]
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