9 45

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^2+6 t-9+6 t^{-1} - t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 23, -2 }
Jones polynomial [math]\displaystyle{ 2 q^{-1} -3 q^{-2} +4 q^{-3} -4 q^{-4} +4 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^8+2 z^2 a^6+2 a^6-z^4 a^4-2 z^2 a^4-2 a^4+2 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^9-3 z^3 a^9+2 z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+z^7 a^7-5 z^3 a^7+2 z a^7+4 z^6 a^6-10 z^4 a^6+7 z^2 a^6-2 a^6+z^7 a^5-z^3 a^5+2 z^6 a^4-4 z^4 a^4+6 z^2 a^4-2 a^4+z^5 a^3+z^3 a^3+3 z^2 a^2-2 a^2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}-q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{10}+q^8+q^6+2 q^2 }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+2 q^{120}-5 q^{116}+9 q^{114}-10 q^{112}+8 q^{110}-3 q^{108}-7 q^{106}+10 q^{104}-12 q^{102}+8 q^{100}-3 q^{98}-6 q^{96}+9 q^{94}-7 q^{92}+2 q^{90}+6 q^{88}-9 q^{86}+10 q^{84}-4 q^{82}-2 q^{80}+9 q^{78}-12 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+7 q^{68}-12 q^{66}+14 q^{64}-12 q^{62}+5 q^{60}+2 q^{58}-9 q^{56}+9 q^{54}-8 q^{52}+q^{50}+6 q^{48}-10 q^{46}+6 q^{44}-q^{42}-7 q^{40}+11 q^{38}-11 q^{36}+8 q^{34}-5 q^{30}+9 q^{28}-8 q^{26}+8 q^{24}-q^{22}-q^{20}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_45.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+q^{15}-q^{13}+q^{11}+q^5-q^3+2 q }[/math]
2 [math]\displaystyle{ q^{48}-q^{46}-2 q^{44}+3 q^{42}+q^{40}-4 q^{38}+2 q^{36}+3 q^{34}-4 q^{32}-q^{30}+3 q^{28}-2 q^{26}-2 q^{24}+2 q^{22}+q^{20}-2 q^{18}-q^{16}+5 q^{14}-q^{12}-3 q^{10}+5 q^8-3 q^4+2 q^2+1 }[/math]
3 [math]\displaystyle{ -q^{93}+q^{91}+2 q^{89}-4 q^{85}-3 q^{83}+5 q^{81}+6 q^{79}-3 q^{77}-9 q^{75}-q^{73}+11 q^{71}+6 q^{69}-9 q^{67}-10 q^{65}+5 q^{63}+13 q^{61}-3 q^{59}-13 q^{57}-q^{55}+13 q^{53}+4 q^{51}-11 q^{49}-4 q^{47}+9 q^{45}+4 q^{43}-7 q^{41}-7 q^{39}+2 q^{37}+7 q^{35}+q^{33}-9 q^{31}-6 q^{29}+9 q^{27}+12 q^{25}-6 q^{23}-14 q^{21}+4 q^{19}+15 q^{17}-10 q^{13}-3 q^{11}+8 q^9+4 q^7-4 q^5-q^3+2 q^{-1} }[/math]
4 [math]\displaystyle{ q^{152}-q^{150}-2 q^{148}+q^{144}+6 q^{142}+q^{140}-5 q^{138}-7 q^{136}-7 q^{134}+11 q^{132}+13 q^{130}+5 q^{128}-9 q^{126}-25 q^{124}-5 q^{122}+14 q^{120}+28 q^{118}+16 q^{116}-26 q^{114}-32 q^{112}-15 q^{110}+27 q^{108}+45 q^{106}+4 q^{104}-33 q^{102}-46 q^{100}+49 q^{96}+34 q^{94}-12 q^{92}-51 q^{90}-23 q^{88}+33 q^{86}+42 q^{84}+4 q^{82}-40 q^{80}-26 q^{78}+18 q^{76}+34 q^{74}+7 q^{72}-27 q^{70}-23 q^{68}+6 q^{66}+27 q^{64}+11 q^{62}-13 q^{60}-23 q^{58}-12 q^{56}+17 q^{54}+25 q^{52}+15 q^{50}-23 q^{48}-42 q^{46}-5 q^{44}+33 q^{42}+48 q^{40}-4 q^{38}-57 q^{36}-35 q^{34}+17 q^{32}+61 q^{30}+25 q^{28}-37 q^{26}-40 q^{24}-11 q^{22}+36 q^{20}+31 q^{18}-6 q^{16}-19 q^{14}-16 q^{12}+7 q^{10}+12 q^8+3 q^6-q^4-5 q^2-1+2 q^{-2} + q^{-4} }[/math]
5 [math]\displaystyle{ -q^{225}+q^{223}+2 q^{221}-q^{217}-3 q^{215}-4 q^{213}-q^{211}+7 q^{209}+9 q^{207}+5 q^{205}-3 q^{203}-14 q^{201}-18 q^{199}-7 q^{197}+16 q^{195}+28 q^{193}+24 q^{191}+q^{189}-32 q^{187}-48 q^{185}-30 q^{183}+17 q^{181}+59 q^{179}+67 q^{177}+25 q^{175}-47 q^{173}-97 q^{171}-77 q^{169}+8 q^{167}+96 q^{165}+125 q^{163}+56 q^{161}-67 q^{159}-151 q^{157}-122 q^{155}+12 q^{153}+144 q^{151}+170 q^{149}+58 q^{147}-109 q^{145}-196 q^{143}-116 q^{141}+61 q^{139}+187 q^{137}+159 q^{135}-9 q^{133}-165 q^{131}-175 q^{129}-31 q^{127}+133 q^{125}+168 q^{123}+56 q^{121}-102 q^{119}-153 q^{117}-61 q^{115}+76 q^{113}+131 q^{111}+57 q^{109}-59 q^{107}-108 q^{105}-49 q^{103}+50 q^{101}+92 q^{99}+44 q^{97}-39 q^{95}-78 q^{93}-47 q^{91}+22 q^{89}+72 q^{87}+63 q^{85}+4 q^{83}-63 q^{81}-87 q^{79}-47 q^{77}+41 q^{75}+112 q^{73}+98 q^{71}-9 q^{69}-128 q^{67}-155 q^{65}-44 q^{63}+121 q^{61}+202 q^{59}+108 q^{57}-94 q^{55}-222 q^{53}-164 q^{51}+41 q^{49}+209 q^{47}+200 q^{45}+22 q^{43}-170 q^{41}-199 q^{39}-67 q^{37}+101 q^{35}+170 q^{33}+94 q^{31}-42 q^{29}-119 q^{27}-85 q^{25}+q^{23}+64 q^{21}+64 q^{19}+21 q^{17}-25 q^{15}-38 q^{13}-13 q^{11}+4 q^9+13 q^7+13 q^5-6 q-3 q^{-1} +2 q^{-7} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{26}-q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{10}+q^8+q^6+2 q^2 }[/math]
1,1 [math]\displaystyle{ q^{68}-2 q^{66}+6 q^{64}-12 q^{62}+17 q^{60}-24 q^{58}+30 q^{56}-30 q^{54}+25 q^{52}-18 q^{50}+6 q^{48}+12 q^{46}-27 q^{44}+38 q^{42}-48 q^{40}+52 q^{38}-54 q^{36}+44 q^{34}-36 q^{32}+22 q^{30}-6 q^{28}-6 q^{26}+20 q^{24}-26 q^{22}+31 q^{20}-28 q^{18}+22 q^{16}-16 q^{14}+13 q^{12}-4 q^{10}+2 q^8+2 q^4+2 q^2 }[/math]
2,0 [math]\displaystyle{ q^{66}+q^{64}-3 q^{60}-2 q^{58}+q^{56}+2 q^{54}+3 q^{48}+2 q^{46}-2 q^{44}-4 q^{42}-q^{40}-2 q^{38}-2 q^{36}-q^{34}+2 q^{32}+2 q^{30}+q^{28}+2 q^{26}-q^{24}-q^{22}+2 q^{20}+2 q^{18}-2 q^{16}-q^{14}+3 q^{12}+2 q^{10}-q^8-q^6+3 q^4+q^2 }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{54}-q^{52}+q^{50}+q^{48}-3 q^{46}+q^{44}-q^{42}-4 q^{40}+2 q^{38}+q^{36}-2 q^{34}+2 q^{32}+2 q^{30}+q^{22}-2 q^{20}-2 q^{18}+3 q^{16}-2 q^{14}+5 q^{10}+3 q^4 }[/math]
1,0,0 [math]\displaystyle{ -q^{35}-q^{33}-q^{31}+q^{29}+2 q^{25}+q^{23}+q^{21}-q^{19}-q^{17}-q^{15}-q^{13}+q^{11}+q^9+2 q^7+2 q^3 }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{72}+q^{70}+q^{64}-q^{62}-4 q^{60}-2 q^{58}+q^{56}-2 q^{54}-3 q^{52}+3 q^{50}+4 q^{48}-q^{46}-q^{44}+2 q^{42}-q^{40}-3 q^{38}+q^{36}+3 q^{34}-q^{32}+q^{30}+4 q^{28}-2 q^{26}-4 q^{24}+q^{20}-2 q^{18}+5 q^{14}+4 q^{12}+q^{10}+q^8+3 q^6 }[/math]
1,0,0,0 [math]\displaystyle{ -q^{44}-q^{42}-q^{40}-q^{38}+q^{36}+2 q^{32}+2 q^{30}+q^{28}+q^{26}-q^{24}-q^{22}-2 q^{20}-q^{18}-q^{16}+q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^4 }[/math]

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+2 q^{120}-5 q^{116}+9 q^{114}-10 q^{112}+8 q^{110}-3 q^{108}-7 q^{106}+10 q^{104}-12 q^{102}+8 q^{100}-3 q^{98}-6 q^{96}+9 q^{94}-7 q^{92}+2 q^{90}+6 q^{88}-9 q^{86}+10 q^{84}-4 q^{82}-2 q^{80}+9 q^{78}-12 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+7 q^{68}-12 q^{66}+14 q^{64}-12 q^{62}+5 q^{60}+2 q^{58}-9 q^{56}+9 q^{54}-8 q^{52}+q^{50}+6 q^{48}-10 q^{46}+6 q^{44}-q^{42}-7 q^{40}+11 q^{38}-11 q^{36}+8 q^{34}-5 q^{30}+9 q^{28}-8 q^{26}+8 q^{24}-q^{22}-q^{20}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/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["9 45"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^2+6 t-9+6 t^{-1} - t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -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]= { 23, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 2 q^{-1} -3 q^{-2} +4 q^{-3} -4 q^{-4} +4 q^{-5} -3 q^{-6} +2 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{ -a^8+2 z^2 a^6+2 a^6-z^4 a^4-2 z^2 a^4-2 a^4+2 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^5 a^9-3 z^3 a^9+2 z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+z^7 a^7-5 z^3 a^7+2 z a^7+4 z^6 a^6-10 z^4 a^6+7 z^2 a^6-2 a^6+z^7 a^5-z^3 a^5+2 z^6 a^4-4 z^4 a^4+6 z^2 a^4-2 a^4+z^5 a^3+z^3 a^3+3 z^2 a^2-2 a^2 }[/math]
Retrieved from "https://katlas.org/index.php?title=9_45&oldid=4604"