10 38

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -4 t^2+15 t-21+15 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 { 59, -2 }
Jones polynomial [math]\displaystyle{ q-2+5 q^{-1} -7 q^{-2} +9 q^{-3} -10 q^{-4} +9 q^{-5} -7 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+a^6-2 z^4 a^4-3 z^2 a^4-2 a^4-z^4 a^2+a^2+z^2+1 }[/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+8 z^3 a^9-z a^9+3 z^8 a^8-8 z^6 a^8+4 z^4 a^8+z^9 a^7+3 z^7 a^7-13 z^5 a^7+8 z^3 a^7-z a^7+5 z^8 a^6-10 z^6 a^6+3 z^4 a^6+2 z^2 a^6-a^6+z^9 a^5+3 z^7 a^5-7 z^5 a^5+3 z^3 a^5+2 z^8 a^4+2 z^6 a^4-8 z^4 a^4+8 z^2 a^4-2 a^4+3 z^7 a^3-2 z^5 a^3+z^3 a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ q^{28}-q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18}+q^{16}-2 q^{14}-2 q^{10}+q^8+q^6-q^4+3 q^2+ q^{-4} }[/math]
The G2 invariant [math]\displaystyle{ q^{142}-2 q^{140}+5 q^{138}-9 q^{136}+9 q^{134}-7 q^{132}-3 q^{130}+20 q^{128}-33 q^{126}+41 q^{124}-36 q^{122}+12 q^{120}+20 q^{118}-55 q^{116}+77 q^{114}-70 q^{112}+40 q^{110}+6 q^{108}-49 q^{106}+73 q^{104}-70 q^{102}+40 q^{100}-35 q^{96}+51 q^{94}-39 q^{92}+8 q^{90}+32 q^{88}-53 q^{86}+54 q^{84}-32 q^{82}-12 q^{80}+55 q^{78}-87 q^{76}+94 q^{74}-64 q^{72}+15 q^{70}+44 q^{68}-89 q^{66}+101 q^{64}-81 q^{62}+35 q^{60}+13 q^{58}-54 q^{56}+65 q^{54}-46 q^{52}+12 q^{50}+21 q^{48}-40 q^{46}+30 q^{44}-7 q^{42}-26 q^{40}+46 q^{38}-50 q^{36}+40 q^{34}-12 q^{32}-19 q^{30}+43 q^{28}-54 q^{26}+52 q^{24}-35 q^{22}+14 q^{20}+8 q^{18}-26 q^{16}+37 q^{14}-35 q^{12}+29 q^{10}-13 q^8+q^6+9 q^4-15 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_38.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{19}-2 q^{17}+2 q^{15}-2 q^{13}+2 q^{11}-q^9-q^7+2 q^5-2 q^3+3 q- q^{-1} + q^{-3} }[/math]
2 [math]\displaystyle{ q^{54}-2 q^{52}-2 q^{50}+7 q^{48}-q^{46}-10 q^{44}+9 q^{42}+5 q^{40}-15 q^{38}+6 q^{36}+11 q^{34}-13 q^{32}+11 q^{28}-7 q^{26}-6 q^{24}+6 q^{22}+5 q^{20}-8 q^{18}-4 q^{16}+15 q^{14}-5 q^{12}-11 q^{10}+14 q^8-2 q^6-9 q^4+9 q^2-1-4 q^{-2} +4 q^{-4} - q^{-8} + q^{-10} }[/math]
3 [math]\displaystyle{ q^{105}-2 q^{103}-2 q^{101}+3 q^{99}+7 q^{97}-q^{95}-16 q^{93}-5 q^{91}+21 q^{89}+18 q^{87}-20 q^{85}-34 q^{83}+13 q^{81}+47 q^{79}+2 q^{77}-52 q^{75}-23 q^{73}+49 q^{71}+40 q^{69}-40 q^{67}-52 q^{65}+25 q^{63}+60 q^{61}-9 q^{59}-60 q^{57}-2 q^{55}+57 q^{53}+13 q^{51}-52 q^{49}-23 q^{47}+40 q^{45}+33 q^{43}-26 q^{41}-42 q^{39}+7 q^{37}+45 q^{35}+19 q^{33}-46 q^{31}-40 q^{29}+36 q^{27}+55 q^{25}-23 q^{23}-58 q^{21}+8 q^{19}+55 q^{17}-40 q^{13}-4 q^{11}+28 q^9+4 q^7-18 q^5+q^3+10 q- q^{-1} -7 q^{-3} +3 q^{-5} +4 q^{-7} -2 q^{-9} -3 q^{-11} +2 q^{-13} + q^{-15} - 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}-8 q^{160}-16 q^{158}-4 q^{156}+7 q^{154}+42 q^{152}+11 q^{150}-35 q^{148}-48 q^{146}-37 q^{144}+71 q^{142}+82 q^{140}+21 q^{138}-72 q^{136}-150 q^{134}+q^{132}+118 q^{130}+152 q^{128}+31 q^{126}-202 q^{124}-155 q^{122}-3 q^{120}+214 q^{118}+222 q^{116}-86 q^{114}-233 q^{112}-210 q^{110}+106 q^{108}+327 q^{106}+120 q^{104}-153 q^{102}-338 q^{100}-74 q^{98}+285 q^{96}+254 q^{94}-14 q^{92}-329 q^{90}-190 q^{88}+177 q^{86}+277 q^{84}+73 q^{82}-257 q^{80}-219 q^{78}+82 q^{76}+253 q^{74}+123 q^{72}-165 q^{70}-228 q^{68}-31 q^{66}+206 q^{64}+188 q^{62}-15 q^{60}-212 q^{58}-202 q^{56}+68 q^{54}+234 q^{52}+209 q^{50}-100 q^{48}-335 q^{46}-142 q^{44}+155 q^{42}+364 q^{40}+99 q^{38}-292 q^{36}-277 q^{34}-23 q^{32}+324 q^{30}+218 q^{28}-130 q^{26}-226 q^{24}-132 q^{22}+163 q^{20}+177 q^{18}-11 q^{16}-90 q^{14}-114 q^{12}+43 q^{10}+76 q^8+12 q^6-4 q^4-52 q^2+6+16 q^{-2} - q^{-4} +14 q^{-6} -14 q^{-8} +4 q^{-10} - q^{-12} -7 q^{-14} +9 q^{-16} -3 q^{-18} +3 q^{-20} - q^{-22} -4 q^{-24} +3 q^{-26} - q^{-28} + q^{-30} - 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}-8 q^{241}-16 q^{239}-4 q^{237}+17 q^{235}+28 q^{233}+26 q^{231}-5 q^{229}-51 q^{227}-74 q^{225}-26 q^{223}+57 q^{221}+117 q^{219}+108 q^{217}-5 q^{215}-154 q^{213}-217 q^{211}-104 q^{209}+116 q^{207}+296 q^{205}+289 q^{203}+33 q^{201}-308 q^{199}-469 q^{197}-276 q^{195}+162 q^{193}+552 q^{191}+582 q^{189}+143 q^{187}-476 q^{185}-817 q^{183}-554 q^{181}+180 q^{179}+877 q^{177}+978 q^{175}+299 q^{173}-709 q^{171}-1270 q^{169}-853 q^{167}+299 q^{165}+1332 q^{163}+1373 q^{161}+265 q^{159}-1159 q^{157}-1723 q^{155}-863 q^{153}+779 q^{151}+1843 q^{149}+1388 q^{147}-283 q^{145}-1759 q^{143}-1739 q^{141}-202 q^{139}+1507 q^{137}+1886 q^{135}+612 q^{133}-1182 q^{131}-1876 q^{129}-874 q^{127}+867 q^{125}+1733 q^{123}+996 q^{121}-594 q^{119}-1551 q^{117}-1025 q^{115}+413 q^{113}+1370 q^{111}+999 q^{109}-265 q^{107}-1214 q^{105}-1004 q^{103}+117 q^{101}+1092 q^{99}+1046 q^{97}+92 q^{95}-921 q^{93}-1154 q^{91}-423 q^{89}+681 q^{87}+1269 q^{85}+837 q^{83}-297 q^{81}-1287 q^{79}-1325 q^{77}-232 q^{75}+1175 q^{73}+1733 q^{71}+856 q^{69}-846 q^{67}-1973 q^{65}-1474 q^{63}+348 q^{61}+1961 q^{59}+1952 q^{57}+233 q^{55}-1707 q^{53}-2162 q^{51}-770 q^{49}+1240 q^{47}+2114 q^{45}+1144 q^{43}-734 q^{41}-1817 q^{39}-1272 q^{37}+249 q^{35}+1384 q^{33}+1218 q^{31}+78 q^{29}-936 q^{27}-1005 q^{25}-248 q^{23}+543 q^{21}+740 q^{19}+309 q^{17}-271 q^{15}-495 q^{13}-265 q^{11}+103 q^9+288 q^7+207 q^5-16 q^3-159 q-132 q^{-1} -18 q^{-3} +73 q^{-5} +80 q^{-7} +23 q^{-9} -28 q^{-11} -38 q^{-13} -23 q^{-15} +6 q^{-17} +22 q^{-19} +11 q^{-21} + q^{-23} -3 q^{-25} -9 q^{-27} -6 q^{-29} +6 q^{-31} +2 q^{-33} +3 q^{-37} -2 q^{-39} -3 q^{-41} +2 q^{-43} - q^{-47} + q^{-49} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{28}-q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18}+q^{16}-2 q^{14}-2 q^{10}+q^8+q^6-q^4+3 q^2+ q^{-4} }[/math]
1,1 [math]\displaystyle{ q^{76}-4 q^{74}+12 q^{72}-30 q^{70}+58 q^{68}-98 q^{66}+150 q^{64}-204 q^{62}+256 q^{60}-290 q^{58}+294 q^{56}-270 q^{54}+206 q^{52}-112 q^{50}-4 q^{48}+138 q^{46}-262 q^{44}+374 q^{42}-460 q^{40}+508 q^{38}-519 q^{36}+480 q^{34}-410 q^{32}+310 q^{30}-197 q^{28}+86 q^{26}+20 q^{24}-96 q^{22}+156 q^{20}-186 q^{18}+200 q^{16}-202 q^{14}+186 q^{12}-170 q^{10}+144 q^8-120 q^6+95 q^4-68 q^2+50-30 q^{-2} +21 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{72}-q^{70}-2 q^{68}+4 q^{64}+3 q^{62}-6 q^{60}-3 q^{58}+5 q^{56}+4 q^{54}-6 q^{52}-6 q^{50}+8 q^{48}+6 q^{46}-6 q^{44}-5 q^{42}+6 q^{40}+2 q^{38}-7 q^{36}-3 q^{34}+3 q^{32}+q^{30}-q^{28}+6 q^{26}-q^{24}-4 q^{22}+8 q^{20}+5 q^{18}-9 q^{16}-7 q^{14}+8 q^{12}+4 q^{10}-10 q^8-4 q^6+9 q^4+2 q^2-4+ q^{-2} +4 q^{-4} + q^{-6} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{60}-2 q^{58}+5 q^{56}-8 q^{54}+10 q^{52}-13 q^{50}+14 q^{48}-15 q^{46}+14 q^{44}-10 q^{42}+5 q^{40}+2 q^{38}-8 q^{36}+16 q^{34}-22 q^{32}+26 q^{30}-28 q^{28}+27 q^{26}-25 q^{24}+19 q^{22}-13 q^{20}+5 q^{18}+q^{16}-7 q^{14}+11 q^{12}-13 q^{10}+14 q^8-12 q^6+12 q^4-8 q^2+7-4 q^{-2} +3 q^{-4} - 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}-8 q^{84}-q^{82}+11 q^{80}+7 q^{78}-10 q^{76}-13 q^{74}+4 q^{72}+16 q^{70}+4 q^{68}-14 q^{66}-11 q^{64}+8 q^{62}+14 q^{60}-12 q^{56}-3 q^{54}+9 q^{52}+5 q^{50}-7 q^{48}-6 q^{46}+7 q^{44}+7 q^{42}-6 q^{40}-10 q^{38}+4 q^{36}+11 q^{34}-q^{32}-13 q^{30}-4 q^{28}+12 q^{26}+8 q^{24}-9 q^{22}-13 q^{20}+2 q^{18}+14 q^{16}+6 q^{14}-9 q^{12}-10 q^{10}+2 q^8+11 q^6+4 q^4-4 q^2-5+ q^{-2} +4 q^{-4} +2 q^{-6} - q^{-8} - 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}-7 q^{132}-3 q^{130}+20 q^{128}-33 q^{126}+41 q^{124}-36 q^{122}+12 q^{120}+20 q^{118}-55 q^{116}+77 q^{114}-70 q^{112}+40 q^{110}+6 q^{108}-49 q^{106}+73 q^{104}-70 q^{102}+40 q^{100}-35 q^{96}+51 q^{94}-39 q^{92}+8 q^{90}+32 q^{88}-53 q^{86}+54 q^{84}-32 q^{82}-12 q^{80}+55 q^{78}-87 q^{76}+94 q^{74}-64 q^{72}+15 q^{70}+44 q^{68}-89 q^{66}+101 q^{64}-81 q^{62}+35 q^{60}+13 q^{58}-54 q^{56}+65 q^{54}-46 q^{52}+12 q^{50}+21 q^{48}-40 q^{46}+30 q^{44}-7 q^{42}-26 q^{40}+46 q^{38}-50 q^{36}+40 q^{34}-12 q^{32}-19 q^{30}+43 q^{28}-54 q^{26}+52 q^{24}-35 q^{22}+14 q^{20}+8 q^{18}-26 q^{16}+37 q^{14}-35 q^{12}+29 q^{10}-13 q^8+q^6+9 q^4-15 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-14} - 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 38"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -4 t^2+15 t-21+15 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]= { 59, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q-2+5 q^{-1} -7 q^{-2} +9 q^{-3} -10 q^{-4} +9 q^{-5} -7 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+a^6-2 z^4 a^4-3 z^2 a^4-2 a^4-z^4 a^2+a^2+z^2+1 }[/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+8 z^3 a^9-z a^9+3 z^8 a^8-8 z^6 a^8+4 z^4 a^8+z^9 a^7+3 z^7 a^7-13 z^5 a^7+8 z^3 a^7-z a^7+5 z^8 a^6-10 z^6 a^6+3 z^4 a^6+2 z^2 a^6-a^6+z^9 a^5+3 z^7 a^5-7 z^5 a^5+3 z^3 a^5+2 z^8 a^4+2 z^6 a^4-8 z^4 a^4+8 z^2 a^4-2 a^4+3 z^7 a^3-2 z^5 a^3+z^3 a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1 }[/math]
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