10 103

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-8 t^2+17 t-21+17 t^{-1} -8 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+4 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{5,t+1\} }[/math]
Determinant and Signature { 75, -2 }
Jones polynomial [math]\displaystyle{ -q^2+3 q-6+10 q^{-1} -11 q^{-2} +13 q^{-3} -12 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^4 a^6-2 z^2 a^6-a^6+z^6 a^4+3 z^4 a^4+3 z^2 a^4+z^6 a^2+3 z^4 a^2+4 z^2 a^2+3 a^2-z^4-2 z^2-1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^9-2 z^3 a^9+3 z^6 a^8-6 z^4 a^8+z^2 a^8+5 z^7 a^7-12 z^5 a^7+10 z^3 a^7-4 z a^7+5 z^8 a^6-12 z^6 a^6+13 z^4 a^6-6 z^2 a^6+a^6+2 z^9 a^5+3 z^7 a^5-16 z^5 a^5+21 z^3 a^5-6 z a^5+9 z^8 a^4-23 z^6 a^4+25 z^4 a^4-8 z^2 a^4+2 z^9 a^3+2 z^7 a^3-9 z^5 a^3+9 z^3 a^3-2 z a^3+4 z^8 a^2-5 z^6 a^2+2 z^2 a^2-3 a^2+4 z^7 a-5 z^5 a-2 z^3 a+z a+3 z^6-6 z^4+3 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}+q^{22}-q^{20}-q^{18}+2 q^{16}-3 q^{14}+q^{12}+q^8+4 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} - q^{-6} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-2 q^{126}+4 q^{124}-7 q^{122}+7 q^{120}-6 q^{118}+12 q^{114}-22 q^{112}+35 q^{110}-42 q^{108}+35 q^{106}-15 q^{104}-25 q^{102}+72 q^{100}-108 q^{98}+118 q^{96}-88 q^{94}+18 q^{92}+74 q^{90}-153 q^{88}+184 q^{86}-149 q^{84}+53 q^{82}+55 q^{80}-142 q^{78}+160 q^{76}-106 q^{74}+11 q^{72}+92 q^{70}-143 q^{68}+115 q^{66}-28 q^{64}-92 q^{62}+180 q^{60}-204 q^{58}+150 q^{56}-38 q^{54}-94 q^{52}+208 q^{50}-252 q^{48}+218 q^{46}-117 q^{44}-25 q^{42}+148 q^{40}-206 q^{38}+189 q^{36}-98 q^{34}-12 q^{32}+112 q^{30}-142 q^{28}+98 q^{26}-2 q^{24}-100 q^{22}+159 q^{20}-140 q^{18}+58 q^{16}+52 q^{14}-136 q^{12}+176 q^{10}-149 q^8+80 q^6+3 q^4-78 q^2+111-107 q^{-2} +78 q^{-4} -34 q^{-6} -3 q^{-8} +28 q^{-10} -42 q^{-12} +39 q^{-14} -29 q^{-16} +15 q^{-18} -3 q^{-20} -6 q^{-22} +8 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_103.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+2 q^{15}-3 q^{13}+3 q^{11}-3 q^9+q^7+2 q^5-q^3+4 q-3 q^{-1} +2 q^{-3} - q^{-5} }[/math]
2 [math]\displaystyle{ q^{48}-2 q^{46}+6 q^{42}-8 q^{40}-4 q^{38}+19 q^{36}-10 q^{34}-19 q^{32}+26 q^{30}-27 q^{26}+20 q^{24}+11 q^{22}-20 q^{20}-q^{18}+13 q^{16}-q^{14}-19 q^{12}+13 q^{10}+19 q^8-26 q^6+2 q^4+27 q^2-19-9 q^{-2} +19 q^{-4} -5 q^{-6} -7 q^{-8} +6 q^{-10} - q^{-12} -2 q^{-14} + q^{-16} }[/math]
3 [math]\displaystyle{ -q^{93}+2 q^{91}-3 q^{87}+7 q^{83}+q^{81}-18 q^{79}-5 q^{77}+34 q^{75}+22 q^{73}-47 q^{71}-58 q^{69}+48 q^{67}+103 q^{65}-24 q^{63}-143 q^{61}-25 q^{59}+165 q^{57}+82 q^{55}-158 q^{53}-132 q^{51}+125 q^{49}+164 q^{47}-83 q^{45}-170 q^{43}+36 q^{41}+153 q^{39}+16 q^{37}-131 q^{35}-50 q^{33}+94 q^{31}+91 q^{29}-64 q^{27}-128 q^{25}+20 q^{23}+159 q^{21}+25 q^{19}-175 q^{17}-73 q^{15}+168 q^{13}+124 q^{11}-134 q^9-151 q^7+82 q^5+160 q^3-29 q-131 q^{-1} -19 q^{-3} +95 q^{-5} +37 q^{-7} -53 q^{-9} -34 q^{-11} +22 q^{-13} +22 q^{-15} -8 q^{-17} -12 q^{-19} +5 q^{-21} +4 q^{-23} -3 q^{-25} -2 q^{-27} + q^{-29} +2 q^{-31} - q^{-33} }[/math]
4 [math]\displaystyle{ q^{152}-2 q^{150}+3 q^{146}-3 q^{144}+q^{142}-5 q^{140}+7 q^{138}+16 q^{136}-17 q^{134}-17 q^{132}-28 q^{130}+31 q^{128}+93 q^{126}+6 q^{124}-76 q^{122}-177 q^{120}-34 q^{118}+246 q^{116}+248 q^{114}+47 q^{112}-424 q^{110}-459 q^{108}+95 q^{106}+597 q^{104}+677 q^{102}-224 q^{100}-974 q^{98}-687 q^{96}+363 q^{94}+1355 q^{92}+666 q^{90}-793 q^{88}-1447 q^{86}-557 q^{84}+1248 q^{82}+1443 q^{80}+59 q^{78}-1390 q^{76}-1282 q^{74}+494 q^{72}+1410 q^{70}+729 q^{68}-738 q^{66}-1277 q^{64}-178 q^{62}+874 q^{60}+884 q^{58}-132 q^{56}-910 q^{54}-543 q^{52}+378 q^{50}+870 q^{48}+316 q^{46}-600 q^{44}-909 q^{42}-79 q^{40}+938 q^{38}+866 q^{36}-236 q^{34}-1315 q^{32}-732 q^{30}+778 q^{28}+1434 q^{26}+479 q^{24}-1294 q^{22}-1398 q^{20}+57 q^{18}+1430 q^{16}+1246 q^{14}-530 q^{12}-1410 q^{10}-784 q^8+623 q^6+1304 q^4+351 q^2-631-903 q^{-2} -218 q^{-4} +619 q^{-6} +523 q^{-8} +85 q^{-10} -394 q^{-12} -359 q^{-14} +53 q^{-16} +187 q^{-18} +189 q^{-20} -26 q^{-22} -128 q^{-24} -41 q^{-26} -7 q^{-28} +58 q^{-30} +17 q^{-32} -16 q^{-34} +2 q^{-36} -13 q^{-38} +6 q^{-40} - q^{-42} -4 q^{-44} +6 q^{-46} - q^{-48} +2 q^{-50} - q^{-52} -2 q^{-54} + q^{-56} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{24}+q^{22}-q^{20}-q^{18}+2 q^{16}-3 q^{14}+q^{12}+q^8+4 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} - q^{-6} }[/math]
1,1 [math]\displaystyle{ q^{68}-4 q^{66}+10 q^{64}-22 q^{62}+46 q^{60}-78 q^{58}+124 q^{56}-202 q^{54}+301 q^{52}-406 q^{50}+528 q^{48}-644 q^{46}+711 q^{44}-702 q^{42}+606 q^{40}-428 q^{38}+145 q^{36}+198 q^{34}-556 q^{32}+896 q^{30}-1167 q^{28}+1354 q^{26}-1420 q^{24}+1360 q^{22}-1198 q^{20}+912 q^{18}-578 q^{16}+228 q^{14}+115 q^{12}-388 q^{10}+596 q^8-672 q^6+682 q^4-632 q^2+534-418 q^{-2} +311 q^{-4} -224 q^{-6} +146 q^{-8} -92 q^{-10} +54 q^{-12} -28 q^{-14} +12 q^{-16} -4 q^{-18} + q^{-20} }[/math]
2,0 [math]\displaystyle{ q^{62}-q^{60}+2 q^{56}-q^{54}-2 q^{52}-q^{50}+7 q^{48}+q^{46}-11 q^{44}+9 q^{40}-q^{38}-13 q^{36}+2 q^{34}+12 q^{32}-2 q^{30}-10 q^{28}+5 q^{26}+3 q^{24}-11 q^{22}+3 q^{20}+5 q^{18}-3 q^{16}+12 q^{12}+3 q^{10}-10 q^8+4 q^6+14 q^4-6 q^2-11+7 q^{-2} +7 q^{-4} -5 q^{-6} -6 q^{-8} +2 q^{-10} +3 q^{-12} -2 q^{-14} - q^{-16} + q^{-18} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{68}-q^{66}-2 q^{64}+4 q^{62}+q^{60}-8 q^{58}+4 q^{56}+11 q^{54}-6 q^{52}-8 q^{50}+12 q^{48}+9 q^{46}-18 q^{44}-10 q^{42}+14 q^{40}-6 q^{38}-22 q^{36}+8 q^{34}+10 q^{32}-18 q^{30}-2 q^{28}+15 q^{26}-3 q^{24}-9 q^{22}+17 q^{20}+19 q^{18}-9 q^{16}+2 q^{14}+22 q^{12}+q^{10}-16 q^8+5 q^6+7 q^4-8 q^2-8+2 q^{-2} +2 q^{-4} -4 q^{-6} - q^{-8} +3 q^{-10} - q^{-14} + q^{-16} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{38}+q^{36}-2 q^{34}-2 q^{28}+2 q^{26}-3 q^{24}+q^{22}-q^{20}+q^{16}+2 q^{14}+3 q^{12}+2 q^{10}+4 q^8-q^6+3 q^4-2 q^2-2 q^{-4} + q^{-6} - q^{-8} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{54}+2 q^{52}-4 q^{50}+8 q^{48}-12 q^{46}+16 q^{44}-23 q^{42}+25 q^{40}-26 q^{38}+23 q^{36}-16 q^{34}+9 q^{32}+3 q^{30}-16 q^{28}+30 q^{26}-42 q^{24}+47 q^{22}-50 q^{20}+47 q^{18}-42 q^{16}+32 q^{14}-16 q^{12}+6 q^{10}+9 q^8-14 q^6+24 q^4-26 q^2+25-23 q^{-2} +17 q^{-4} -12 q^{-6} +8 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-14} }[/math]
1,0 [math]\displaystyle{ q^{88}-2 q^{84}-2 q^{82}+2 q^{80}+6 q^{78}-9 q^{74}-6 q^{72}+9 q^{70}+17 q^{68}-3 q^{66}-23 q^{64}-10 q^{62}+21 q^{60}+22 q^{58}-12 q^{56}-28 q^{54}-3 q^{52}+25 q^{50}+12 q^{48}-19 q^{46}-17 q^{44}+11 q^{42}+17 q^{40}-7 q^{38}-18 q^{36}+q^{34}+17 q^{32}+2 q^{30}-17 q^{28}-6 q^{26}+18 q^{24}+14 q^{22}-14 q^{20}-16 q^{18}+13 q^{16}+28 q^{14}-q^{12}-26 q^{10}-12 q^8+23 q^6+21 q^4-10 q^2-23-4 q^{-2} +16 q^{-4} +9 q^{-6} -7 q^{-8} -10 q^{-10} - q^{-12} +6 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{74}-2 q^{72}+2 q^{70}-4 q^{68}+7 q^{66}-9 q^{64}+11 q^{62}-13 q^{60}+19 q^{58}-20 q^{56}+20 q^{54}-19 q^{52}+19 q^{50}-17 q^{48}+6 q^{46}-7 q^{44}-q^{42}+8 q^{40}-21 q^{38}+22 q^{36}-29 q^{34}+37 q^{32}-38 q^{30}+36 q^{28}-38 q^{26}+38 q^{24}-27 q^{22}+26 q^{20}-17 q^{18}+15 q^{16}+4 q^{14}+10 q^{10}-14 q^8+20 q^6-20 q^4+18 q^2-22+17 q^{-2} -15 q^{-4} +10 q^{-6} -10 q^{-8} +7 q^{-10} -4 q^{-12} +3 q^{-14} -2 q^{-16} + q^{-18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}-2 q^{126}+4 q^{124}-7 q^{122}+7 q^{120}-6 q^{118}+12 q^{114}-22 q^{112}+35 q^{110}-42 q^{108}+35 q^{106}-15 q^{104}-25 q^{102}+72 q^{100}-108 q^{98}+118 q^{96}-88 q^{94}+18 q^{92}+74 q^{90}-153 q^{88}+184 q^{86}-149 q^{84}+53 q^{82}+55 q^{80}-142 q^{78}+160 q^{76}-106 q^{74}+11 q^{72}+92 q^{70}-143 q^{68}+115 q^{66}-28 q^{64}-92 q^{62}+180 q^{60}-204 q^{58}+150 q^{56}-38 q^{54}-94 q^{52}+208 q^{50}-252 q^{48}+218 q^{46}-117 q^{44}-25 q^{42}+148 q^{40}-206 q^{38}+189 q^{36}-98 q^{34}-12 q^{32}+112 q^{30}-142 q^{28}+98 q^{26}-2 q^{24}-100 q^{22}+159 q^{20}-140 q^{18}+58 q^{16}+52 q^{14}-136 q^{12}+176 q^{10}-149 q^8+80 q^6+3 q^4-78 q^2+111-107 q^{-2} +78 q^{-4} -34 q^{-6} -3 q^{-8} +28 q^{-10} -42 q^{-12} +39 q^{-14} -29 q^{-16} +15 q^{-18} -3 q^{-20} -6 q^{-22} +8 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }[/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["10 103"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-8 t^2+17 t-21+17 t^{-1} -8 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+4 z^4+3 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{ \{5,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 75, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^2+3 q-6+10 q^{-1} -11 q^{-2} +13 q^{-3} -12 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^4 a^6-2 z^2 a^6-a^6+z^6 a^4+3 z^4 a^4+3 z^2 a^4+z^6 a^2+3 z^4 a^2+4 z^2 a^2+3 a^2-z^4-2 z^2-1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^5 a^9-2 z^3 a^9+3 z^6 a^8-6 z^4 a^8+z^2 a^8+5 z^7 a^7-12 z^5 a^7+10 z^3 a^7-4 z a^7+5 z^8 a^6-12 z^6 a^6+13 z^4 a^6-6 z^2 a^6+a^6+2 z^9 a^5+3 z^7 a^5-16 z^5 a^5+21 z^3 a^5-6 z a^5+9 z^8 a^4-23 z^6 a^4+25 z^4 a^4-8 z^2 a^4+2 z^9 a^3+2 z^7 a^3-9 z^5 a^3+9 z^3 a^3-2 z a^3+4 z^8 a^2-5 z^6 a^2+2 z^2 a^2-3 a^2+4 z^7 a-5 z^5 a-2 z^3 a+z a+3 z^6-6 z^4+3 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }[/math]
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