10 25

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+8 t^2-14 t+17-14 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^6-4 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 65, -4 }
Jones polynomial [math]\displaystyle{ 1-2 q^{-1} +5 q^{-2} -7 q^{-3} +10 q^{-4} -11 q^{-5} +10 q^{-6} -9 q^{-7} +6 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^8+2 z^2 a^8+a^8-z^6 a^6-3 z^4 a^6-3 z^2 a^6-2 a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4+z^4 a^2+3 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-3 z^3 a^{11}+z a^{11}+5 z^6 a^{10}-6 z^4 a^{10}+3 z^2 a^{10}+5 z^7 a^9-5 z^5 a^9+2 z^3 a^9+3 z^8 a^8+z^6 a^8-5 z^4 a^8+z^2 a^8+a^8+z^9 a^7+5 z^7 a^7-9 z^5 a^7+3 z^3 a^7-2 z a^7+5 z^8 a^6-8 z^6 a^6+3 z^4 a^6-4 z^2 a^6+2 a^6+z^9 a^5+2 z^7 a^5-7 z^5 a^5+2 z^3 a^5+2 z^8 a^4-3 z^6 a^4-3 z^4 a^4+4 z^2 a^4+2 z^7 a^3-6 z^5 a^3+4 z^3 a^3+z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2 }[/math]
The A2 invariant [math]\displaystyle{ q^{30}-q^{28}+q^{26}+q^{24}-2 q^{22}+q^{20}-3 q^{18}-q^{12}+3 q^{10}-q^8+2 q^6+q^4+1 }[/math]
The G2 invariant [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+5 q^{154}-4 q^{152}-2 q^{150}+12 q^{148}-22 q^{146}+30 q^{144}-32 q^{142}+21 q^{140}-q^{138}-26 q^{136}+59 q^{134}-76 q^{132}+76 q^{130}-52 q^{128}+6 q^{126}+43 q^{124}-86 q^{122}+106 q^{120}-91 q^{118}+48 q^{116}+10 q^{114}-58 q^{112}+79 q^{110}-61 q^{108}+19 q^{106}+31 q^{104}-66 q^{102}+63 q^{100}-22 q^{98}-40 q^{96}+104 q^{94}-132 q^{92}+112 q^{90}-47 q^{88}-43 q^{86}+119 q^{84}-163 q^{82}+151 q^{80}-95 q^{78}+10 q^{76}+67 q^{74}-116 q^{72}+117 q^{70}-77 q^{68}+11 q^{66}+41 q^{64}-71 q^{62}+59 q^{60}-15 q^{58}-38 q^{56}+82 q^{54}-89 q^{52}+58 q^{50}+q^{48}-65 q^{46}+108 q^{44}-111 q^{42}+81 q^{40}-27 q^{38}-31 q^{36}+73 q^{34}-82 q^{32}+71 q^{30}-38 q^{28}+5 q^{26}+20 q^{24}-32 q^{22}+32 q^{20}-22 q^{18}+12 q^{16}-4 q^{12}+6 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_25.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{21}-2 q^{19}+3 q^{17}-3 q^{15}+q^{13}-q^{11}-q^9+3 q^7-2 q^5+3 q^3-q+ q^{-1} }[/math]
2 [math]\displaystyle{ q^{58}-2 q^{56}+6 q^{52}-8 q^{50}-3 q^{48}+16 q^{46}-13 q^{44}-9 q^{42}+23 q^{40}-9 q^{38}-12 q^{36}+16 q^{34}+q^{32}-11 q^{30}+11 q^{26}-3 q^{24}-15 q^{22}+13 q^{20}+7 q^{18}-22 q^{16}+9 q^{14}+14 q^{12}-16 q^{10}+2 q^8+12 q^6-7 q^4-2 q^2+5- q^{-2} - q^{-4} + q^{-6} }[/math]
3 [math]\displaystyle{ q^{111}-2 q^{109}+3 q^{105}+q^{103}-7 q^{101}-4 q^{99}+15 q^{97}+6 q^{95}-24 q^{93}-12 q^{91}+36 q^{89}+25 q^{87}-51 q^{85}-41 q^{83}+63 q^{81}+59 q^{79}-63 q^{77}-81 q^{75}+59 q^{73}+92 q^{71}-41 q^{69}-98 q^{67}+17 q^{65}+88 q^{63}+11 q^{61}-69 q^{59}-39 q^{57}+47 q^{55}+58 q^{53}-17 q^{51}-74 q^{49}-7 q^{47}+80 q^{45}+37 q^{43}-84 q^{41}-55 q^{39}+73 q^{37}+76 q^{35}-61 q^{33}-92 q^{31}+39 q^{29}+96 q^{27}-15 q^{25}-92 q^{23}-7 q^{21}+78 q^{19}+25 q^{17}-57 q^{15}-32 q^{13}+35 q^{11}+34 q^9-17 q^7-24 q^5+4 q^3+17 q+ q^{-1} -8 q^{-3} -2 q^{-5} +4 q^{-7} + q^{-9} - q^{-11} - q^{-13} + q^{-15} }[/math]
4 [math]\displaystyle{ q^{180}-2 q^{178}+3 q^{174}-2 q^{172}+2 q^{170}-8 q^{168}+q^{166}+14 q^{164}-5 q^{162}+2 q^{160}-27 q^{158}+4 q^{156}+48 q^{154}-5 q^{152}-16 q^{150}-80 q^{148}+12 q^{146}+132 q^{144}+31 q^{142}-58 q^{140}-214 q^{138}-19 q^{136}+277 q^{134}+169 q^{132}-74 q^{130}-425 q^{128}-161 q^{126}+378 q^{124}+396 q^{122}+45 q^{120}-563 q^{118}-398 q^{116}+284 q^{114}+543 q^{112}+275 q^{110}-455 q^{108}-536 q^{106}+8 q^{104}+438 q^{102}+440 q^{100}-141 q^{98}-444 q^{96}-258 q^{94}+152 q^{92}+424 q^{90}+181 q^{88}-209 q^{86}-402 q^{84}-128 q^{82}+303 q^{80}+401 q^{78}+15 q^{76}-444 q^{74}-324 q^{72}+154 q^{70}+535 q^{68}+210 q^{66}-420 q^{64}-465 q^{62}-30 q^{60}+569 q^{58}+398 q^{56}-274 q^{54}-525 q^{52}-274 q^{50}+437 q^{48}+524 q^{46}-6 q^{44}-420 q^{42}-462 q^{40}+155 q^{38}+461 q^{36}+234 q^{34}-158 q^{32}-444 q^{30}-101 q^{28}+226 q^{26}+273 q^{24}+77 q^{22}-242 q^{20}-167 q^{18}+8 q^{16}+143 q^{14}+133 q^{12}-56 q^{10}-85 q^8-55 q^6+24 q^4+70 q^2+7-13 q^{-2} -29 q^{-4} -8 q^{-6} +19 q^{-8} +4 q^{-10} +3 q^{-12} -6 q^{-14} -4 q^{-16} +4 q^{-18} + q^{-22} - q^{-24} - q^{-26} + q^{-28} }[/math]
5 [math]\displaystyle{ q^{265}-2 q^{263}+3 q^{259}-2 q^{257}-q^{255}+q^{253}-3 q^{251}+9 q^{247}+q^{245}-9 q^{243}-5 q^{241}-2 q^{239}+10 q^{237}+17 q^{235}+3 q^{233}-31 q^{231}-40 q^{229}+14 q^{227}+69 q^{225}+71 q^{223}-18 q^{221}-142 q^{219}-164 q^{217}+19 q^{215}+280 q^{213}+320 q^{211}+11 q^{209}-450 q^{207}-597 q^{205}-139 q^{203}+653 q^{201}+1023 q^{199}+400 q^{197}-825 q^{195}-1544 q^{193}-867 q^{191}+858 q^{189}+2109 q^{187}+1539 q^{185}-673 q^{183}-2601 q^{181}-2306 q^{179}+210 q^{177}+2815 q^{175}+3087 q^{173}+517 q^{171}-2721 q^{169}-3641 q^{167}-1339 q^{165}+2193 q^{163}+3855 q^{161}+2144 q^{159}-1401 q^{157}-3640 q^{155}-2706 q^{153}+442 q^{151}+3028 q^{149}+2958 q^{147}+489 q^{145}-2171 q^{143}-2875 q^{141}-1244 q^{139}+1217 q^{137}+2528 q^{135}+1785 q^{133}-315 q^{131}-2073 q^{129}-2095 q^{127}-440 q^{125}+1587 q^{123}+2299 q^{121}+1016 q^{119}-1182 q^{117}-2406 q^{115}-1502 q^{113}+859 q^{111}+2547 q^{109}+1892 q^{107}-592 q^{105}-2641 q^{103}-2317 q^{101}+274 q^{99}+2763 q^{97}+2727 q^{95}+111 q^{93}-2719 q^{91}-3143 q^{89}-670 q^{87}+2525 q^{85}+3474 q^{83}+1302 q^{81}-2047 q^{79}-3605 q^{77}-1997 q^{75}+1329 q^{73}+3462 q^{71}+2597 q^{69}-442 q^{67}-2983 q^{65}-2948 q^{63}-500 q^{61}+2212 q^{59}+2971 q^{57}+1292 q^{55}-1276 q^{53}-2621 q^{51}-1795 q^{49}+335 q^{47}+1982 q^{45}+1931 q^{43}+421 q^{41}-1216 q^{39}-1722 q^{37}-864 q^{35}+491 q^{33}+1269 q^{31}+1004 q^{29}+48 q^{27}-772 q^{25}-865 q^{23}-327 q^{21}+319 q^{19}+611 q^{17}+408 q^{15}-44 q^{13}-341 q^{11}-327 q^9-96 q^7+139 q^5+214 q^3+117 q-26 q^{-1} -106 q^{-3} -91 q^{-5} -14 q^{-7} +41 q^{-9} +47 q^{-11} +25 q^{-13} -10 q^{-15} -25 q^{-17} -12 q^{-19} +3 q^{-21} +4 q^{-23} +7 q^{-25} +3 q^{-27} -5 q^{-29} -2 q^{-31} +2 q^{-33} + q^{-39} - q^{-41} - q^{-43} + q^{-45} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{30}-q^{28}+q^{26}+q^{24}-2 q^{22}+q^{20}-3 q^{18}-q^{12}+3 q^{10}-q^8+2 q^6+q^4+1 }[/math]
1,1 [math]\displaystyle{ q^{84}-4 q^{82}+10 q^{80}-20 q^{78}+40 q^{76}-72 q^{74}+112 q^{72}-164 q^{70}+233 q^{68}-306 q^{66}+364 q^{64}-412 q^{62}+439 q^{60}-418 q^{58}+342 q^{56}-218 q^{54}+53 q^{52}+150 q^{50}-372 q^{48}+578 q^{46}-749 q^{44}+866 q^{42}-906 q^{40}+876 q^{38}-769 q^{36}+610 q^{34}-402 q^{32}+176 q^{30}+25 q^{28}-210 q^{26}+346 q^{24}-436 q^{22}+456 q^{20}-436 q^{18}+390 q^{16}-314 q^{14}+237 q^{12}-162 q^{10}+110 q^8-62 q^6+35 q^4-16 q^2+8-2 q^{-2} + q^{-4} }[/math]
2,0 [math]\displaystyle{ q^{76}-q^{74}+2 q^{70}-3 q^{66}-q^{64}+4 q^{62}-2 q^{60}-7 q^{58}+3 q^{56}+7 q^{54}-6 q^{52}-4 q^{50}+9 q^{48}+6 q^{46}-7 q^{44}-q^{42}+8 q^{40}-3 q^{38}-6 q^{36}+6 q^{34}+q^{32}-9 q^{30}+q^{28}+4 q^{26}-7 q^{24}-7 q^{22}+7 q^{20}+5 q^{18}-6 q^{16}+9 q^{12}+q^{10}-4 q^8+q^6+4 q^4+q^2-1+ q^{-4} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{68}-2 q^{66}+5 q^{62}-7 q^{60}-2 q^{58}+13 q^{56}-11 q^{54}-6 q^{52}+18 q^{50}-10 q^{48}-8 q^{46}+16 q^{44}-2 q^{42}-6 q^{40}+7 q^{38}+4 q^{36}-4 q^{34}-10 q^{32}+6 q^{30}+2 q^{28}-18 q^{26}+8 q^{24}+9 q^{22}-16 q^{20}+6 q^{18}+9 q^{16}-10 q^{14}+6 q^{12}+6 q^{10}-3 q^8+3 q^6+2 q^4-q^2+1 }[/math]
1,0,0 [math]\displaystyle{ q^{39}-q^{37}+2 q^{35}-q^{33}+2 q^{31}-2 q^{29}+q^{27}-3 q^{25}-q^{23}-q^{21}-q^{19}+q^{17}-q^{15}+3 q^{13}-q^{11}+3 q^9+2 q^5+q }[/math]

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{68}-2 q^{66}+4 q^{64}-7 q^{62}+11 q^{60}-14 q^{58}+17 q^{56}-19 q^{54}+18 q^{52}-16 q^{50}+10 q^{48}-2 q^{46}-6 q^{44}+16 q^{42}-24 q^{40}+31 q^{38}-36 q^{36}+36 q^{34}-34 q^{32}+26 q^{30}-20 q^{28}+10 q^{26}-2 q^{24}-7 q^{22}+14 q^{20}-16 q^{18}+19 q^{16}-16 q^{14}+16 q^{12}-12 q^{10}+9 q^8-5 q^6+4 q^4-q^2+1 }[/math]
1,0 [math]\displaystyle{ q^{110}-2 q^{106}-2 q^{104}+2 q^{102}+6 q^{100}+q^{98}-9 q^{96}-8 q^{94}+6 q^{92}+15 q^{90}+2 q^{88}-17 q^{86}-12 q^{84}+11 q^{82}+19 q^{80}-2 q^{78}-19 q^{76}-6 q^{74}+15 q^{72}+12 q^{70}-9 q^{68}-12 q^{66}+6 q^{64}+14 q^{62}-13 q^{58}-2 q^{56}+11 q^{54}+3 q^{52}-13 q^{50}-7 q^{48}+10 q^{46}+9 q^{44}-11 q^{42}-17 q^{40}+4 q^{38}+20 q^{36}+5 q^{34}-18 q^{32}-14 q^{30}+10 q^{28}+19 q^{26}-14 q^{22}-7 q^{20}+10 q^{18}+10 q^{16}-q^{14}-6 q^{12}-q^{10}+4 q^8+3 q^6-q^4-q^2+ q^{-2} }[/math]

D4 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+5 q^{154}-4 q^{152}-2 q^{150}+12 q^{148}-22 q^{146}+30 q^{144}-32 q^{142}+21 q^{140}-q^{138}-26 q^{136}+59 q^{134}-76 q^{132}+76 q^{130}-52 q^{128}+6 q^{126}+43 q^{124}-86 q^{122}+106 q^{120}-91 q^{118}+48 q^{116}+10 q^{114}-58 q^{112}+79 q^{110}-61 q^{108}+19 q^{106}+31 q^{104}-66 q^{102}+63 q^{100}-22 q^{98}-40 q^{96}+104 q^{94}-132 q^{92}+112 q^{90}-47 q^{88}-43 q^{86}+119 q^{84}-163 q^{82}+151 q^{80}-95 q^{78}+10 q^{76}+67 q^{74}-116 q^{72}+117 q^{70}-77 q^{68}+11 q^{66}+41 q^{64}-71 q^{62}+59 q^{60}-15 q^{58}-38 q^{56}+82 q^{54}-89 q^{52}+58 q^{50}+q^{48}-65 q^{46}+108 q^{44}-111 q^{42}+81 q^{40}-27 q^{38}-31 q^{36}+73 q^{34}-82 q^{32}+71 q^{30}-38 q^{28}+5 q^{26}+20 q^{24}-32 q^{22}+32 q^{20}-22 q^{18}+12 q^{16}-4 q^{12}+6 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/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 25"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+8 t^2-14 t+17-14 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^6-4 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]= { 65, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 1-2 q^{-1} +5 q^{-2} -7 q^{-3} +10 q^{-4} -11 q^{-5} +10 q^{-6} -9 q^{-7} +6 q^{-8} -3 q^{-9} + q^{-10} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^8+2 z^2 a^8+a^8-z^6 a^6-3 z^4 a^6-3 z^2 a^6-2 a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4+z^4 a^2+3 z^2 a^2+2 a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-3 z^3 a^{11}+z a^{11}+5 z^6 a^{10}-6 z^4 a^{10}+3 z^2 a^{10}+5 z^7 a^9-5 z^5 a^9+2 z^3 a^9+3 z^8 a^8+z^6 a^8-5 z^4 a^8+z^2 a^8+a^8+z^9 a^7+5 z^7 a^7-9 z^5 a^7+3 z^3 a^7-2 z a^7+5 z^8 a^6-8 z^6 a^6+3 z^4 a^6-4 z^2 a^6+2 a^6+z^9 a^5+2 z^7 a^5-7 z^5 a^5+2 z^3 a^5+2 z^8 a^4-3 z^6 a^4-3 z^4 a^4+4 z^2 a^4+2 z^7 a^3-6 z^5 a^3+4 z^3 a^3+z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2 }[/math]
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