10 18

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -4 t^2+14 t-19+14 t^{-1} -4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -4 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 55, -2 }
Jones polynomial	[math]\displaystyle{ q^3-2 q^2+4 q-6+8 q^{-1} -9 q^{-2} +9 q^{-3} -7 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6-z^4 a^4+a^4-2 z^4 a^2-3 z^2 a^2-a^2-z^4-z^2+z^2 a^{-2} + a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+5 a^2 z^8+2 z^8+4 a^5 z^7+3 a^3 z^7+a z^7+2 z^7 a^{-1} +4 a^6 z^6-5 a^4 z^6-15 a^2 z^6+z^6 a^{-2} -5 z^6+3 a^7 z^5-6 a^5 z^5-12 a^3 z^5-10 a z^5-7 z^5 a^{-1} +a^8 z^4-5 a^6 z^4+6 a^4 z^4+17 a^2 z^4-4 z^4 a^{-2} +z^4-4 a^7 z^3+5 a^5 z^3+14 a^3 z^3+11 a z^3+6 z^3 a^{-1} -a^8 z^2+a^6 z^2-3 a^4 z^2-8 a^2 z^2+4 z^2 a^{-2} +z^2-2 a^5 z-4 a^3 z-4 a z-2 z a^{-1} +a^4+a^2- a^{-2} }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}-q^{20}-q^{18}+2 q^{16}-q^{14}+q^{12}+q^{10}-q^8+q^6-2 q^4+q^2- q^{-2} +2 q^{-4} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-q^{104}-4 q^{102}+12 q^{100}-16 q^{98}+20 q^{96}-18 q^{94}+6 q^{92}+6 q^{90}-21 q^{88}+32 q^{86}-37 q^{84}+33 q^{82}-20 q^{80}+24 q^{76}-41 q^{74}+49 q^{72}-40 q^{70}+20 q^{68}+4 q^{66}-28 q^{64}+39 q^{62}-29 q^{60}+10 q^{58}+17 q^{56}-33 q^{54}+30 q^{52}-8 q^{50}-24 q^{48}+52 q^{46}-63 q^{44}+51 q^{42}-17 q^{40}-25 q^{38}+62 q^{36}-79 q^{34}+72 q^{32}-43 q^{30}+35 q^{26}-58 q^{24}+60 q^{22}-41 q^{20}+9 q^{18}+20 q^{16}-38 q^{14}+34 q^{12}-13 q^{10}-16 q^8+38 q^6-44 q^4+28 q^2+1-33 q^{-2} +58 q^{-4} -57 q^{-6} +40 q^{-8} -10 q^{-10} -21 q^{-12} +42 q^{-14} -47 q^{-16} +39 q^{-18} -21 q^{-20} +2 q^{-22} +13 q^{-24} -20 q^{-26} +20 q^{-28} -14 q^{-30} +9 q^{-32} - q^{-34} -3 q^{-36} +4 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_18.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-2 q^{13}+2 q^{11}-2 q^9+2 q^7-q^3+2 q-2 q^{-1} +2 q^{-3} - q^{-5} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{42}-2 q^{40}-q^{38}+5 q^{36}-4 q^{34}-2 q^{32}+10 q^{30}-7 q^{28}-6 q^{26}+13 q^{24}-6 q^{22}-9 q^{20}+10 q^{18}+q^{16}-6 q^{14}+7 q^{10}-q^8-9 q^6+9 q^4+5 q^2-12+6 q^{-2} +8 q^{-4} -11 q^{-6} +8 q^{-10} -5 q^{-12} -2 q^{-14} +4 q^{-16} - q^{-18} - q^{-20} + q^{-22} }[/math]
3	[math]\displaystyle{ q^{81}-2 q^{79}-q^{77}+2 q^{75}+3 q^{73}-q^{71}-5 q^{69}+3 q^{67}+3 q^{65}-5 q^{63}-4 q^{61}+10 q^{59}+6 q^{57}-17 q^{55}-11 q^{53}+25 q^{51}+21 q^{49}-30 q^{47}-31 q^{45}+27 q^{43}+41 q^{41}-20 q^{39}-43 q^{37}+6 q^{35}+41 q^{33}+5 q^{31}-31 q^{29}-17 q^{27}+18 q^{25}+28 q^{23}-8 q^{21}-32 q^{19}-3 q^{17}+37 q^{15}+13 q^{13}-39 q^{11}-22 q^9+37 q^7+31 q^5-31 q^3-38 q+22 q^{-1} +44 q^{-3} -7 q^{-5} -43 q^{-7} -7 q^{-9} +37 q^{-11} +17 q^{-13} -25 q^{-15} -23 q^{-17} +14 q^{-19} +22 q^{-21} -5 q^{-23} -16 q^{-25} - q^{-27} +11 q^{-29} +2 q^{-31} -6 q^{-33} -2 q^{-35} +3 q^{-37} + q^{-39} - q^{-41} - q^{-43} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{132}-2 q^{130}-q^{128}+2 q^{126}+6 q^{122}-4 q^{120}-4 q^{118}-2 q^{116}-8 q^{114}+17 q^{112}+2 q^{110}+3 q^{108}-2 q^{106}-30 q^{104}+8 q^{102}+3 q^{100}+33 q^{98}+28 q^{96}-48 q^{94}-35 q^{92}-41 q^{90}+57 q^{88}+109 q^{86}-4 q^{84}-79 q^{82}-149 q^{80}+12 q^{78}+182 q^{76}+112 q^{74}-39 q^{72}-235 q^{70}-104 q^{68}+153 q^{66}+195 q^{64}+72 q^{62}-197 q^{60}-180 q^{58}+29 q^{56}+157 q^{54}+150 q^{52}-66 q^{50}-150 q^{48}-77 q^{46}+50 q^{44}+139 q^{42}+55 q^{40}-70 q^{38}-130 q^{36}-37 q^{34}+104 q^{32}+132 q^{30}-14 q^{28}-162 q^{26}-89 q^{24}+81 q^{22}+195 q^{20}+36 q^{18}-184 q^{16}-147 q^{14}+33 q^{12}+231 q^{10}+111 q^8-143 q^6-190 q^4-72 q^2+187+180 q^{-2} -21 q^{-4} -150 q^{-6} -168 q^{-8} +55 q^{-10} +156 q^{-12} +96 q^{-14} -24 q^{-16} -164 q^{-18} -64 q^{-20} +47 q^{-22} +107 q^{-24} +78 q^{-26} -71 q^{-28} -79 q^{-30} -41 q^{-32} +40 q^{-34} +83 q^{-36} +2 q^{-38} -29 q^{-40} -46 q^{-42} -8 q^{-44} +38 q^{-46} +13 q^{-48} +2 q^{-50} -19 q^{-52} -11 q^{-54} +11 q^{-56} +3 q^{-58} +4 q^{-60} -4 q^{-62} -4 q^{-64} +3 q^{-66} + q^{-70} - q^{-72} - q^{-74} + q^{-76} }[/math]
5	[math]\displaystyle{ q^{195}-2 q^{193}-q^{191}+2 q^{189}+3 q^{185}+3 q^{183}-3 q^{181}-9 q^{179}-5 q^{177}-q^{175}+9 q^{173}+21 q^{171}+12 q^{169}-13 q^{167}-35 q^{165}-28 q^{163}+q^{161}+44 q^{159}+67 q^{157}+26 q^{155}-56 q^{153}-105 q^{151}-75 q^{149}+30 q^{147}+153 q^{145}+171 q^{143}+15 q^{141}-192 q^{139}-280 q^{137}-132 q^{135}+190 q^{133}+425 q^{131}+316 q^{129}-128 q^{127}-554 q^{125}-555 q^{123}-37 q^{121}+620 q^{119}+833 q^{117}+299 q^{115}-593 q^{113}-1063 q^{111}-626 q^{109}+424 q^{107}+1194 q^{105}+955 q^{103}-148 q^{101}-1168 q^{99}-1200 q^{97}-190 q^{95}+985 q^{93}+1288 q^{91}+502 q^{89}-664 q^{87}-1218 q^{85}-723 q^{83}+321 q^{81}+994 q^{79}+802 q^{77}+17 q^{75}-698 q^{73}-778 q^{71}-254 q^{69}+394 q^{67}+663 q^{65}+401 q^{63}-128 q^{61}-532 q^{59}-497 q^{57}-49 q^{55}+431 q^{53}+547 q^{51}+181 q^{49}-374 q^{47}-625 q^{45}-281 q^{43}+375 q^{41}+718 q^{39}+383 q^{37}-372 q^{35}-842 q^{33}-524 q^{31}+349 q^{29}+963 q^{27}+693 q^{25}-260 q^{23}-1030 q^{21}-884 q^{19}+85 q^{17}+1016 q^{15}+1055 q^{13}+154 q^{11}-882 q^9-1144 q^7-430 q^5+627 q^3+1126 q+676 q^{-1} -294 q^{-3} -965 q^{-5} -818 q^{-7} -61 q^{-9} +675 q^{-11} +833 q^{-13} +357 q^{-15} -326 q^{-17} -692 q^{-19} -521 q^{-21} -20 q^{-23} +438 q^{-25} +549 q^{-27} +267 q^{-29} -160 q^{-31} -431 q^{-33} -382 q^{-35} -87 q^{-37} +243 q^{-39} +375 q^{-41} +229 q^{-43} -56 q^{-45} -269 q^{-47} -265 q^{-49} -82 q^{-51} +139 q^{-53} +229 q^{-55} +136 q^{-57} -33 q^{-59} -147 q^{-61} -135 q^{-63} -30 q^{-65} +74 q^{-67} +100 q^{-69} +48 q^{-71} -25 q^{-73} -60 q^{-75} -38 q^{-77} +27 q^{-81} +28 q^{-83} +5 q^{-85} -13 q^{-87} -12 q^{-89} -3 q^{-91} + q^{-93} +6 q^{-95} +4 q^{-97} -3 q^{-99} -2 q^{-101} + q^{-103} + q^{-109} - q^{-111} - q^{-113} + q^{-115} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}-q^{20}-q^{18}+2 q^{16}-q^{14}+q^{12}+q^{10}-q^8+q^6-2 q^4+q^2- q^{-2} +2 q^{-4} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+32 q^{52}-48 q^{50}+70 q^{48}-90 q^{46}+108 q^{44}-126 q^{42}+144 q^{40}-156 q^{38}+156 q^{36}-148 q^{34}+128 q^{32}-88 q^{30}+28 q^{28}+48 q^{26}-128 q^{24}+212 q^{22}-288 q^{20}+340 q^{18}-376 q^{16}+378 q^{14}-351 q^{12}+302 q^{10}-226 q^8+144 q^6-44 q^4-44 q^2+118-172 q^{-2} +204 q^{-4} -210 q^{-6} +192 q^{-8} -164 q^{-10} +130 q^{-12} -96 q^{-14} +64 q^{-16} -40 q^{-18} +25 q^{-20} -12 q^{-22} +6 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{56}-q^{54}-2 q^{52}+q^{50}+3 q^{48}-5 q^{44}+2 q^{42}+7 q^{40}-3 q^{38}-7 q^{36}+3 q^{34}+8 q^{32}-4 q^{30}-8 q^{28}+4 q^{26}+4 q^{24}-5 q^{22}-q^{20}+3 q^{18}-2 q^{16}+4 q^{12}-q^{10}-4 q^8+4 q^6+9 q^4-3 q^2-5+6 q^{-2} +4 q^{-4} -6 q^{-6} -6 q^{-8} +2 q^{-10} +4 q^{-12} -2 q^{-14} -3 q^{-16} +2 q^{-18} +3 q^{-20} - q^{-24} + q^{-28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-2 q^{46}+4 q^{42}-5 q^{40}+8 q^{36}-8 q^{34}-4 q^{32}+11 q^{30}-6 q^{28}-5 q^{26}+11 q^{24}-q^{22}-3 q^{20}+3 q^{18}+3 q^{16}-2 q^{14}-6 q^{12}+5 q^{10}+q^8-11 q^6+5 q^4+6 q^2-9+5 q^{-2} +6 q^{-4} -6 q^{-6} +3 q^{-8} +2 q^{-10} -3 q^{-12} +2 q^{-14} + q^{-16} - q^{-18} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ q^{29}-q^{27}-q^{23}+2 q^{21}-q^{19}+2 q^{17}+q^{13}-q^{11}-2 q^5+q^3-q+ q^{-1} - q^{-3} +2 q^{-5} + q^{-9} + q^{-13} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-2 q^{46}+4 q^{44}-6 q^{42}+7 q^{40}-10 q^{38}+12 q^{36}-12 q^{34}+12 q^{32}-9 q^{30}+6 q^{28}-q^{26}-5 q^{24}+11 q^{22}-17 q^{20}+21 q^{18}-23 q^{16}+24 q^{14}-22 q^{12}+19 q^{10}-13 q^8+7 q^6-q^4-4 q^2+7-11 q^{-2} +12 q^{-4} -12 q^{-6} +11 q^{-8} -8 q^{-10} +7 q^{-12} -4 q^{-14} +3 q^{-16} - q^{-18} + q^{-20} }[/math]
1,0	[math]\displaystyle{ q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+5 q^{68}-6 q^{64}-5 q^{62}+5 q^{60}+10 q^{58}-11 q^{54}-8 q^{52}+7 q^{50}+12 q^{48}-q^{46}-12 q^{44}-4 q^{42}+10 q^{40}+8 q^{38}-6 q^{36}-8 q^{34}+4 q^{32}+9 q^{30}-q^{28}-8 q^{26}-q^{24}+7 q^{22}+2 q^{20}-7 q^{18}-4 q^{16}+7 q^{14}+6 q^{12}-7 q^{10}-11 q^8+3 q^6+13 q^4+4 q^2-11-10 q^{-2} +7 q^{-4} +13 q^{-6} + q^{-8} -10 q^{-10} -5 q^{-12} +6 q^{-14} +6 q^{-16} - q^{-18} -5 q^{-20} - q^{-22} +3 q^{-24} +2 q^{-26} - q^{-28} - q^{-30} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-q^{104}-4 q^{102}+12 q^{100}-16 q^{98}+20 q^{96}-18 q^{94}+6 q^{92}+6 q^{90}-21 q^{88}+32 q^{86}-37 q^{84}+33 q^{82}-20 q^{80}+24 q^{76}-41 q^{74}+49 q^{72}-40 q^{70}+20 q^{68}+4 q^{66}-28 q^{64}+39 q^{62}-29 q^{60}+10 q^{58}+17 q^{56}-33 q^{54}+30 q^{52}-8 q^{50}-24 q^{48}+52 q^{46}-63 q^{44}+51 q^{42}-17 q^{40}-25 q^{38}+62 q^{36}-79 q^{34}+72 q^{32}-43 q^{30}+35 q^{26}-58 q^{24}+60 q^{22}-41 q^{20}+9 q^{18}+20 q^{16}-38 q^{14}+34 q^{12}-13 q^{10}-16 q^8+38 q^6-44 q^4+28 q^2+1-33 q^{-2} +58 q^{-4} -57 q^{-6} +40 q^{-8} -10 q^{-10} -21 q^{-12} +42 q^{-14} -47 q^{-16} +39 q^{-18} -21 q^{-20} +2 q^{-22} +13 q^{-24} -20 q^{-26} +20 q^{-28} -14 q^{-30} +9 q^{-32} - q^{-34} -3 q^{-36} +4 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} }[/math]

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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 18"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -4 t^2+14 t-19+14 t^{-1} -4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -4 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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 55, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-2 q^2+4 q-6+8 q^{-1} -9 q^{-2} +9 q^{-3} -7 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+5 a^2 z^8+2 z^8+4 a^5 z^7+3 a^3 z^7+a z^7+2 z^7 a^{-1} +4 a^6 z^6-5 a^4 z^6-15 a^2 z^6+z^6 a^{-2} -5 z^6+3 a^7 z^5-6 a^5 z^5-12 a^3 z^5-10 a z^5-7 z^5 a^{-1} +a^8 z^4-5 a^6 z^4+6 a^4 z^4+17 a^2 z^4-4 z^4 a^{-2} +z^4-4 a^7 z^3+5 a^5 z^3+14 a^3 z^3+11 a z^3+6 z^3 a^{-1} -a^8 z^2+a^6 z^2-3 a^4 z^2-8 a^2 z^2+4 z^2 a^{-2} +z^2-2 a^5 z-4 a^3 z-4 a z-2 z a^{-1} +a^4+a^2- a^{-2} }[/math]