10 30

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -4 t^2+17 t-25+17 t^{-1} -4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -4 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 67, -2 }
Jones polynomial	[math]\displaystyle{ q-3+6 q^{-1} -8 q^{-2} +11 q^{-3} -11 q^{-4} +10 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8-z^4 a^6-2 z^4 a^4-2 z^2 a^4-a^4-z^4 a^2+z^2 a^2+2 a^2+z^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+9 z^3 a^9-2 z a^9+3 z^8 a^8-7 z^6 a^8+2 z^4 a^8+z^2 a^8+z^9 a^7+5 z^7 a^7-20 z^5 a^7+18 z^3 a^7-6 z a^7+6 z^8 a^6-11 z^6 a^6+2 z^4 a^6+2 z^2 a^6+z^9 a^5+7 z^7 a^5-19 z^5 a^5+16 z^3 a^5-5 z a^5+3 z^8 a^4+2 z^6 a^4-11 z^4 a^4+9 z^2 a^4-a^4+5 z^7 a^3-6 z^5 a^3+4 z^3 a^3-z a^3+5 z^6 a^2-7 z^4 a^2+5 z^2 a^2-2 a^2+3 z^5 a-3 z^3 a+z^4-z^2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{28}-q^{26}-q^{24}+2 q^{22}-2 q^{20}+q^{16}-2 q^{14}+q^{12}-q^{10}+2 q^8+2 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{142}-2 q^{140}+5 q^{138}-9 q^{136}+9 q^{134}-8 q^{132}-2 q^{130}+19 q^{128}-34 q^{126}+46 q^{124}-44 q^{122}+22 q^{120}+15 q^{118}-59 q^{116}+92 q^{114}-96 q^{112}+69 q^{110}-15 q^{108}-49 q^{106}+97 q^{104}-111 q^{102}+89 q^{100}-34 q^{98}-27 q^{96}+70 q^{94}-78 q^{92}+50 q^{90}-49 q^{86}+76 q^{84}-65 q^{82}+16 q^{80}+46 q^{78}-104 q^{76}+131 q^{74}-110 q^{72}+46 q^{70}+34 q^{68}-114 q^{66}+154 q^{64}-147 q^{62}+89 q^{60}-9 q^{58}-66 q^{56}+108 q^{54}-105 q^{52}+64 q^{50}-4 q^{48}-43 q^{46}+60 q^{44}-43 q^{42}+3 q^{40}+48 q^{38}-72 q^{36}+73 q^{34}-39 q^{32}-9 q^{30}+56 q^{28}-87 q^{26}+92 q^{24}-69 q^{22}+33 q^{20}+12 q^{18}-48 q^{16}+66 q^{14}-64 q^{12}+49 q^{10}-24 q^8-q^6+18 q^4-29 q^2+28-19 q^{-2} +12 q^{-4} -2 q^{-6} -3 q^{-8} +5 q^{-10} -6 q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_30.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{19}-2 q^{17}+2 q^{15}-3 q^{13}+2 q^{11}-q^9+3 q^5-2 q^3+3 q-2 q^{-1} + q^{-3} }[/math]
2	[math]\displaystyle{ q^{54}-2 q^{52}-2 q^{50}+7 q^{48}-2 q^{46}-10 q^{44}+12 q^{42}+4 q^{40}-18 q^{38}+11 q^{36}+11 q^{34}-20 q^{32}+3 q^{30}+14 q^{28}-12 q^{26}-5 q^{24}+10 q^{22}+3 q^{20}-11 q^{18}-2 q^{16}+18 q^{14}-10 q^{12}-11 q^{10}+21 q^8-5 q^6-12 q^4+14 q^2-2-7 q^{-2} +6 q^{-4} -2 q^{-8} + q^{-10} }[/math]
3	[math]\displaystyle{ q^{105}-2 q^{103}-2 q^{101}+3 q^{99}+7 q^{97}-2 q^{95}-16 q^{93}-2 q^{91}+24 q^{89}+14 q^{87}-30 q^{85}-32 q^{83}+29 q^{81}+53 q^{79}-20 q^{77}-69 q^{75}-2 q^{73}+80 q^{71}+24 q^{69}-79 q^{67}-46 q^{65}+71 q^{63}+67 q^{61}-56 q^{59}-78 q^{57}+38 q^{55}+85 q^{53}-21 q^{51}-83 q^{49}-2 q^{47}+75 q^{45}+19 q^{43}-59 q^{41}-44 q^{39}+40 q^{37}+59 q^{35}-7 q^{33}-74 q^{31}-22 q^{29}+78 q^{27}+47 q^{25}-70 q^{23}-66 q^{21}+58 q^{19}+72 q^{17}-39 q^{15}-67 q^{13}+27 q^{11}+54 q^9-14 q^7-41 q^5+10 q^3+27 q-5 q^{-1} -18 q^{-3} +4 q^{-5} +12 q^{-7} -3 q^{-9} -6 q^{-11} + q^{-13} +3 q^{-15} -2 q^{-19} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{172}-2 q^{170}-2 q^{168}+3 q^{166}+3 q^{164}+7 q^{162}-9 q^{160}-16 q^{158}-q^{156}+10 q^{154}+42 q^{152}+q^{150}-47 q^{148}-45 q^{146}-16 q^{144}+102 q^{142}+77 q^{140}-29 q^{138}-121 q^{136}-144 q^{134}+95 q^{132}+196 q^{130}+121 q^{128}-103 q^{126}-321 q^{124}-79 q^{122}+191 q^{120}+333 q^{118}+111 q^{116}-357 q^{114}-321 q^{112}-28 q^{110}+396 q^{108}+392 q^{106}-167 q^{104}-425 q^{102}-318 q^{100}+254 q^{98}+536 q^{96}+95 q^{94}-346 q^{92}-478 q^{90}+51 q^{88}+503 q^{86}+268 q^{84}-204 q^{82}-489 q^{80}-96 q^{78}+381 q^{76}+340 q^{74}-67 q^{72}-415 q^{70}-216 q^{68}+203 q^{66}+374 q^{64}+107 q^{62}-262 q^{60}-339 q^{58}-70 q^{56}+333 q^{54}+326 q^{52}+17 q^{50}-387 q^{48}-389 q^{46}+151 q^{44}+440 q^{42}+332 q^{40}-252 q^{38}-557 q^{36}-108 q^{34}+332 q^{32}+486 q^{30}-20 q^{28}-461 q^{26}-234 q^{24}+107 q^{22}+389 q^{20}+112 q^{18}-237 q^{16}-172 q^{14}-34 q^{12}+197 q^{10}+93 q^8-85 q^6-58 q^4-49 q^2+74+36 q^{-2} -33 q^{-4} -4 q^{-6} -23 q^{-8} +28 q^{-10} +7 q^{-12} -17 q^{-14} +5 q^{-16} -8 q^{-18} +11 q^{-20} +2 q^{-22} -7 q^{-24} +2 q^{-26} -2 q^{-28} +3 q^{-30} -2 q^{-34} + q^{-36} }[/math]
5	[math]\displaystyle{ q^{255}-2 q^{253}-2 q^{251}+3 q^{249}+3 q^{247}+3 q^{245}-9 q^{241}-16 q^{239}-q^{237}+20 q^{235}+28 q^{233}+20 q^{231}-17 q^{229}-61 q^{227}-65 q^{225}+4 q^{223}+92 q^{221}+127 q^{219}+66 q^{217}-91 q^{215}-226 q^{213}-195 q^{211}+34 q^{209}+296 q^{207}+375 q^{205}+152 q^{203}-285 q^{201}-584 q^{199}-444 q^{197}+119 q^{195}+701 q^{193}+812 q^{191}+255 q^{189}-634 q^{187}-1146 q^{185}-792 q^{183}+294 q^{181}+1303 q^{179}+1371 q^{177}+324 q^{175}-1140 q^{173}-1857 q^{171}-1119 q^{169}+647 q^{167}+2054 q^{165}+1908 q^{163}+172 q^{161}-1894 q^{159}-2546 q^{157}-1107 q^{155}+1378 q^{153}+2857 q^{151}+2003 q^{149}-600 q^{147}-2824 q^{145}-2704 q^{143}-239 q^{141}+2484 q^{139}+3086 q^{137}+1007 q^{135}-1951 q^{133}-3184 q^{131}-1577 q^{129}+1389 q^{127}+3023 q^{125}+1907 q^{123}-871 q^{121}-2740 q^{119}-2031 q^{117}+479 q^{115}+2403 q^{113}+2034 q^{111}-181 q^{109}-2103 q^{107}-1977 q^{105}-66 q^{103}+1800 q^{101}+1983 q^{99}+363 q^{97}-1521 q^{95}-1994 q^{93}-757 q^{91}+1098 q^{89}+2049 q^{87}+1302 q^{85}-565 q^{83}-2017 q^{81}-1890 q^{79}-217 q^{77}+1807 q^{75}+2485 q^{73}+1112 q^{71}-1351 q^{69}-2878 q^{67}-2046 q^{65}+624 q^{63}+2962 q^{61}+2865 q^{59}+251 q^{57}-2687 q^{55}-3356 q^{53}-1131 q^{51}+2064 q^{49}+3459 q^{47}+1838 q^{45}-1278 q^{43}-3138 q^{41}-2230 q^{39}+483 q^{37}+2528 q^{35}+2252 q^{33}+155 q^{31}-1779 q^{29}-1989 q^{27}-540 q^{25}+1099 q^{23}+1530 q^{21}+666 q^{19}-532 q^{17}-1058 q^{15}-623 q^{13}+199 q^{11}+641 q^9+465 q^7-9 q^5-340 q^3-311 q-51 q^{-1} +164 q^{-3} +175 q^{-5} +49 q^{-7} -63 q^{-9} -85 q^{-11} -36 q^{-13} +23 q^{-15} +41 q^{-17} +13 q^{-19} -11 q^{-21} -11 q^{-23} -3 q^{-25} +3 q^{-27} +4 q^{-29} +2 q^{-31} -7 q^{-33} -2 q^{-35} +6 q^{-37} + q^{-39} -2 q^{-41} + q^{-43} - q^{-45} -2 q^{-47} +3 q^{-49} -2 q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{28}-q^{26}-q^{24}+2 q^{22}-2 q^{20}+q^{16}-2 q^{14}+q^{12}-q^{10}+2 q^8+2 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{76}-4 q^{74}+12 q^{72}-30 q^{70}+60 q^{68}-106 q^{66}+168 q^{64}-244 q^{62}+324 q^{60}-390 q^{58}+438 q^{56}-448 q^{54}+407 q^{52}-316 q^{50}+178 q^{48}-4 q^{46}-196 q^{44}+398 q^{42}-578 q^{40}+720 q^{38}-811 q^{36}+834 q^{34}-792 q^{32}+686 q^{30}-534 q^{28}+354 q^{26}-166 q^{24}-8 q^{22}+155 q^{20}-268 q^{18}+336 q^{16}-362 q^{14}+366 q^{12}-338 q^{10}+298 q^8-242 q^6+191 q^4-142 q^2+98-62 q^{-2} +38 q^{-4} -20 q^{-6} +10 q^{-8} -4 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{72}-q^{70}-2 q^{68}+4 q^{64}+2 q^{62}-7 q^{60}-2 q^{58}+8 q^{56}+5 q^{54}-9 q^{52}-6 q^{50}+11 q^{48}+7 q^{46}-12 q^{44}-8 q^{42}+9 q^{40}+4 q^{38}-8 q^{36}-3 q^{34}+7 q^{32}+q^{30}-3 q^{28}+3 q^{26}-4 q^{24}-7 q^{22}+9 q^{20}+6 q^{18}-10 q^{16}-4 q^{14}+15 q^{12}+7 q^{10}-13 q^8-4 q^6+13 q^4+2 q^2-9- q^{-2} +5 q^{-4} +2 q^{-6} -2 q^{-8} - q^{-10} + q^{-12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{60}-2 q^{58}+q^{56}+2 q^{54}-7 q^{52}+5 q^{50}+3 q^{48}-11 q^{46}+10 q^{44}+6 q^{42}-14 q^{40}+11 q^{38}+8 q^{36}-16 q^{34}+2 q^{32}+5 q^{30}-9 q^{28}-5 q^{26}+3 q^{24}+7 q^{22}-7 q^{20}-2 q^{18}+17 q^{16}-9 q^{14}-7 q^{12}+18 q^{10}-5 q^8-9 q^6+13 q^4-q^2-6+5 q^{-2} -2 q^{-6} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{37}-q^{35}-q^{31}+2 q^{29}-2 q^{27}+q^{25}-q^{23}+q^{21}-2 q^{19}-q^{13}+2 q^{11}+q^9+3 q^7-q^5+3 q^3-q- q^{-3} + q^{-5} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{60}-2 q^{58}+5 q^{56}-8 q^{54}+11 q^{52}-15 q^{50}+17 q^{48}-19 q^{46}+18 q^{44}-16 q^{42}+10 q^{40}-3 q^{38}-6 q^{36}+16 q^{34}-24 q^{32}+31 q^{30}-35 q^{28}+37 q^{26}-35 q^{24}+29 q^{22}-21 q^{20}+12 q^{18}-3 q^{16}-5 q^{14}+13 q^{12}-16 q^{10}+19 q^8-17 q^6+17 q^4-13 q^2+10-7 q^{-2} +4 q^{-4} -2 q^{-6} + q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{98}-2 q^{94}-2 q^{92}+3 q^{90}+5 q^{88}-3 q^{86}-9 q^{84}-2 q^{82}+12 q^{80}+9 q^{78}-10 q^{76}-16 q^{74}+3 q^{72}+20 q^{70}+9 q^{68}-16 q^{66}-16 q^{64}+8 q^{62}+20 q^{60}+2 q^{58}-17 q^{56}-8 q^{54}+10 q^{52}+8 q^{50}-9 q^{48}-11 q^{46}+5 q^{44}+10 q^{42}-5 q^{40}-13 q^{38}+2 q^{36}+15 q^{34}+3 q^{32}-14 q^{30}-7 q^{28}+15 q^{26}+14 q^{24}-8 q^{22}-18 q^{20}+q^{18}+19 q^{16}+10 q^{14}-12 q^{12}-15 q^{10}+2 q^8+15 q^6+6 q^4-7 q^2-8+ q^{-2} +6 q^{-4} +2 q^{-6} -2 q^{-8} -2 q^{-10} + q^{-14} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{142}-2 q^{140}+5 q^{138}-9 q^{136}+9 q^{134}-8 q^{132}-2 q^{130}+19 q^{128}-34 q^{126}+46 q^{124}-44 q^{122}+22 q^{120}+15 q^{118}-59 q^{116}+92 q^{114}-96 q^{112}+69 q^{110}-15 q^{108}-49 q^{106}+97 q^{104}-111 q^{102}+89 q^{100}-34 q^{98}-27 q^{96}+70 q^{94}-78 q^{92}+50 q^{90}-49 q^{86}+76 q^{84}-65 q^{82}+16 q^{80}+46 q^{78}-104 q^{76}+131 q^{74}-110 q^{72}+46 q^{70}+34 q^{68}-114 q^{66}+154 q^{64}-147 q^{62}+89 q^{60}-9 q^{58}-66 q^{56}+108 q^{54}-105 q^{52}+64 q^{50}-4 q^{48}-43 q^{46}+60 q^{44}-43 q^{42}+3 q^{40}+48 q^{38}-72 q^{36}+73 q^{34}-39 q^{32}-9 q^{30}+56 q^{28}-87 q^{26}+92 q^{24}-69 q^{22}+33 q^{20}+12 q^{18}-48 q^{16}+66 q^{14}-64 q^{12}+49 q^{10}-24 q^8-q^6+18 q^4-29 q^2+28-19 q^{-2} +12 q^{-4} -2 q^{-6} -3 q^{-8} +5 q^{-10} -6 q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18} }[/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 30"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -4 t^2+17 t-25+17 t^{-1} -4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -4 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]=

{ 67, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q-3+6 q^{-1} -8 q^{-2} +11 q^{-3} -11 q^{-4} +10 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^2 a^8-z^4 a^6-2 z^4 a^4-2 z^2 a^4-a^4-z^4 a^2+z^2 a^2+2 a^2+z^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+9 z^3 a^9-2 z a^9+3 z^8 a^8-7 z^6 a^8+2 z^4 a^8+z^2 a^8+z^9 a^7+5 z^7 a^7-20 z^5 a^7+18 z^3 a^7-6 z a^7+6 z^8 a^6-11 z^6 a^6+2 z^4 a^6+2 z^2 a^6+z^9 a^5+7 z^7 a^5-19 z^5 a^5+16 z^3 a^5-5 z a^5+3 z^8 a^4+2 z^6 a^4-11 z^4 a^4+9 z^2 a^4-a^4+5 z^7 a^3-6 z^5 a^3+4 z^3 a^3-z a^3+5 z^6 a^2-7 z^4 a^2+5 z^2 a^2-2 a^2+3 z^5 a-3 z^3 a+z^4-z^2 }[/math]