9 7

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^2-7 t+9-7 t^{-1} +3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 29, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} - q^{-3} +3 q^{-4} -4 q^{-5} +5 q^{-6} -5 q^{-7} +4 q^{-8} -3 q^{-9} +2 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+2 z^2 a^8+a^8+z^4 a^6+z^2 a^6-a^6+z^4 a^4+3 z^2 a^4+2 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+z a^{13}+2 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+3 z^7 a^9-9 z^5 a^9+11 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-z^5 a^7+2 z^3 a^7-z a^7+z^6 a^6-2 z^2 a^6+a^6+z^5 a^5-z^3 a^5-z a^5+z^4 a^4-3 z^2 a^4+2 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{176}-q^{174}+2 q^{172}-3 q^{170}+2 q^{168}-q^{166}-2 q^{164}+7 q^{162}-8 q^{160}+9 q^{158}-7 q^{156}+6 q^{152}-12 q^{150}+13 q^{148}-11 q^{146}+4 q^{144}+4 q^{142}-10 q^{140}+10 q^{138}-6 q^{136}+5 q^{132}-9 q^{130}+5 q^{128}-7 q^{124}+11 q^{122}-12 q^{120}+8 q^{118}-q^{116}-8 q^{114}+13 q^{112}-16 q^{110}+15 q^{108}-7 q^{106}-q^{104}+9 q^{102}-13 q^{100}+14 q^{98}-7 q^{96}+q^{94}+5 q^{92}-7 q^{90}+5 q^{88}+q^{86}-6 q^{84}+8 q^{82}-7 q^{80}+4 q^{76}-9 q^{74}+10 q^{72}-8 q^{70}+4 q^{68}-q^{66}-4 q^{64}+6 q^{62}-6 q^{60}+7 q^{58}-3 q^{56}+3 q^{54}+q^{52}-q^{50}+4 q^{48}-3 q^{46}+4 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_7.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}+q^{21}-q^{19}+q^{17}-q^{15}+q^{11}-q^9+2 q^7+q^3 }[/math]
2	[math]\displaystyle{ q^{64}-q^{62}-q^{60}+3 q^{58}-q^{56}-4 q^{54}+3 q^{52}+2 q^{50}-4 q^{48}+2 q^{46}+3 q^{44}-4 q^{42}+2 q^{38}-q^{36}-2 q^{34}+3 q^{30}-4 q^{28}-2 q^{26}+5 q^{24}-2 q^{22}-2 q^{20}+4 q^{18}+2 q^{12}+q^6 }[/math]
3	[math]\displaystyle{ -q^{123}+q^{121}+q^{119}-q^{117}-2 q^{115}+q^{113}+5 q^{111}-7 q^{107}-3 q^{105}+6 q^{103}+7 q^{101}-4 q^{99}-10 q^{97}+q^{95}+10 q^{93}+4 q^{91}-10 q^{89}-7 q^{87}+9 q^{85}+9 q^{83}-6 q^{81}-9 q^{79}+5 q^{77}+8 q^{75}-3 q^{73}-8 q^{71}+6 q^{67}+2 q^{65}-3 q^{63}-6 q^{61}+3 q^{59}+9 q^{57}-12 q^{53}-4 q^{51}+10 q^{49}+6 q^{47}-11 q^{45}-7 q^{43}+5 q^{41}+7 q^{39}-2 q^{37}-5 q^{35}+q^{33}+2 q^{31}+q^{29}+q^{25}+q^{21}+q^{19}+2 q^{17}+q^9 }[/math]
4	[math]\displaystyle{ q^{200}-q^{198}-q^{196}+q^{194}+2 q^{190}-3 q^{188}-3 q^{186}+2 q^{184}+2 q^{182}+9 q^{180}-3 q^{178}-11 q^{176}-5 q^{174}-q^{172}+18 q^{170}+9 q^{168}-5 q^{166}-13 q^{164}-20 q^{162}+9 q^{160}+18 q^{158}+16 q^{156}-q^{154}-31 q^{152}-16 q^{150}+5 q^{148}+32 q^{146}+26 q^{144}-21 q^{142}-32 q^{140}-18 q^{138}+26 q^{136}+40 q^{134}-3 q^{132}-30 q^{130}-30 q^{128}+13 q^{126}+36 q^{124}+6 q^{122}-19 q^{120}-25 q^{118}+4 q^{116}+24 q^{114}+10 q^{112}-10 q^{110}-17 q^{108}-3 q^{106}+11 q^{104}+16 q^{102}+q^{100}-11 q^{98}-18 q^{96}-4 q^{94}+27 q^{92}+18 q^{90}-4 q^{88}-33 q^{86}-27 q^{84}+25 q^{82}+35 q^{80}+17 q^{78}-31 q^{76}-44 q^{74}+6 q^{72}+28 q^{70}+31 q^{68}-8 q^{66}-35 q^{64}-12 q^{62}+5 q^{60}+24 q^{58}+9 q^{56}-13 q^{54}-9 q^{52}-8 q^{50}+8 q^{48}+9 q^{46}-q^{42}-7 q^{40}+q^{38}+4 q^{36}+2 q^{34}+3 q^{32}-3 q^{30}+q^{26}+q^{24}+3 q^{22}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{295}+q^{293}+q^{291}-q^{289}+q^{281}+2 q^{279}-2 q^{277}-5 q^{275}-2 q^{273}+q^{271}+6 q^{269}+9 q^{267}+5 q^{265}-9 q^{263}-17 q^{261}-11 q^{259}+q^{257}+19 q^{255}+24 q^{253}+13 q^{251}-12 q^{249}-30 q^{247}-27 q^{245}-9 q^{243}+20 q^{241}+40 q^{239}+35 q^{237}+2 q^{235}-35 q^{233}-54 q^{231}-39 q^{229}+11 q^{227}+61 q^{225}+75 q^{223}+27 q^{221}-51 q^{219}-96 q^{217}-73 q^{215}+18 q^{213}+105 q^{211}+112 q^{209}+21 q^{207}-92 q^{205}-132 q^{203}-62 q^{201}+67 q^{199}+140 q^{197}+92 q^{195}-40 q^{193}-132 q^{191}-105 q^{189}+11 q^{187}+112 q^{185}+107 q^{183}+6 q^{181}-91 q^{179}-95 q^{177}-15 q^{175}+67 q^{173}+81 q^{171}+19 q^{169}-50 q^{167}-63 q^{165}-18 q^{163}+35 q^{161}+48 q^{159}+20 q^{157}-20 q^{155}-44 q^{153}-26 q^{151}+8 q^{149}+35 q^{147}+43 q^{145}+12 q^{143}-36 q^{141}-62 q^{139}-36 q^{137}+31 q^{135}+86 q^{133}+72 q^{131}-15 q^{129}-102 q^{127}-111 q^{125}-14 q^{123}+111 q^{121}+142 q^{119}+50 q^{117}-93 q^{115}-160 q^{113}-92 q^{111}+65 q^{109}+157 q^{107}+110 q^{105}-18 q^{103}-127 q^{101}-124 q^{99}-19 q^{97}+87 q^{95}+109 q^{93}+40 q^{91}-44 q^{89}-84 q^{87}-53 q^{85}+11 q^{83}+55 q^{81}+45 q^{79}+5 q^{77}-27 q^{75}-35 q^{73}-14 q^{71}+13 q^{69}+22 q^{67}+13 q^{65}-3 q^{63}-15 q^{61}-11 q^{59}+q^{57}+7 q^{55}+8 q^{53}+4 q^{51}-6 q^{49}-5 q^{47}+2 q^{43}+4 q^{41}+5 q^{39}-q^{37}-2 q^{35}+q^{29}+3 q^{27}+q^{25}+q^{15} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6 }[/math]
1,1	[math]\displaystyle{ q^{92}-2 q^{90}+4 q^{88}-8 q^{86}+15 q^{84}-20 q^{82}+26 q^{80}-34 q^{78}+35 q^{76}-34 q^{74}+28 q^{72}-16 q^{70}+3 q^{68}+14 q^{66}-30 q^{64}+42 q^{62}-53 q^{60}+58 q^{58}-58 q^{56}+56 q^{54}-46 q^{52}+36 q^{50}-22 q^{48}+6 q^{46}+q^{44}-18 q^{42}+18 q^{40}-24 q^{38}+21 q^{36}-20 q^{34}+20 q^{32}-14 q^{30}+16 q^{28}-10 q^{26}+12 q^{24}-6 q^{22}+8 q^{20}-2 q^{18}+4 q^{16}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{86}+q^{78}-3 q^{74}-q^{72}+q^{70}+q^{68}-q^{66}+3 q^{62}+q^{60}-2 q^{58}-q^{56}+q^{54}-q^{52}-q^{50}-q^{46}-3 q^{44}-2 q^{42}-3 q^{38}-q^{36}+4 q^{34}+2 q^{32}-q^{30}+2 q^{28}+4 q^{26}+2 q^{24}-q^{22}+2 q^{20}+2 q^{18}+q^{12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{74}-q^{72}+q^{68}-2 q^{66}+2 q^{64}+2 q^{62}-3 q^{60}+2 q^{58}+q^{56}-5 q^{54}-q^{52}+q^{50}-3 q^{48}-q^{46}+q^{44}+q^{42}+4 q^{36}-2 q^{34}-3 q^{32}+3 q^{30}-2 q^{28}-3 q^{26}+4 q^{24}+2 q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ -q^{45}-q^{41}-q^{37}+q^{35}+q^{31}-q^{25}-q^{23}+q^{21}+2 q^{17}+q^{15}+2 q^{13}+q^9 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{96}-q^{92}+q^{90}+q^{88}-2 q^{86}+4 q^{82}+2 q^{80}-2 q^{78}+2 q^{74}-3 q^{72}-6 q^{70}-q^{68}-2 q^{66}-5 q^{64}-q^{62}+q^{60}-q^{58}+4 q^{54}+3 q^{52}+q^{48}+3 q^{46}-2 q^{44}-5 q^{42}-q^{40}-q^{36}+3 q^{32}+5 q^{30}+3 q^{28}+3 q^{26}+3 q^{24}+2 q^{22}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{56}-q^{52}-q^{50}-q^{46}+q^{44}+q^{40}+q^{38}-q^{32}-q^{30}-q^{28}+q^{26}+2 q^{22}+2 q^{20}+q^{18}+2 q^{16}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{74}+q^{72}-2 q^{70}+3 q^{68}-4 q^{66}+4 q^{64}-4 q^{62}+3 q^{60}-2 q^{58}+q^{56}+q^{54}-3 q^{52}+5 q^{50}-7 q^{48}+7 q^{46}-7 q^{44}+7 q^{42}-6 q^{40}+4 q^{38}-2 q^{36}+q^{32}-3 q^{30}+4 q^{28}-3 q^{26}+4 q^{24}-2 q^{22}+3 q^{20}-q^{18}+2 q^{16}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{120}-q^{116}-q^{114}+q^{112}+2 q^{110}-q^{108}-3 q^{106}+4 q^{102}+3 q^{100}-3 q^{98}-4 q^{96}+q^{94}+4 q^{92}+q^{90}-4 q^{88}-3 q^{86}+q^{84}+2 q^{82}-q^{80}-3 q^{78}+2 q^{74}-3 q^{70}-q^{68}+3 q^{66}+2 q^{64}-2 q^{62}-2 q^{60}+2 q^{58}+3 q^{56}-q^{54}-4 q^{52}-q^{50}+4 q^{48}+2 q^{46}-3 q^{44}-3 q^{42}+4 q^{38}+2 q^{36}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+q^{18} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{102}-q^{100}+q^{98}-2 q^{96}+3 q^{94}-3 q^{92}+3 q^{90}-3 q^{88}+4 q^{86}-2 q^{84}+2 q^{82}-q^{80}-q^{76}-4 q^{74}+q^{72}-5 q^{70}+3 q^{68}-7 q^{66}+5 q^{64}-5 q^{62}+7 q^{60}-4 q^{58}+5 q^{56}-2 q^{54}+4 q^{52}-q^{46}-3 q^{44}+2 q^{42}-4 q^{40}+q^{38}-3 q^{36}+4 q^{34}+4 q^{30}+q^{28}+4 q^{26}+q^{24}+2 q^{22}+q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{176}-q^{174}+2 q^{172}-3 q^{170}+2 q^{168}-q^{166}-2 q^{164}+7 q^{162}-8 q^{160}+9 q^{158}-7 q^{156}+6 q^{152}-12 q^{150}+13 q^{148}-11 q^{146}+4 q^{144}+4 q^{142}-10 q^{140}+10 q^{138}-6 q^{136}+5 q^{132}-9 q^{130}+5 q^{128}-7 q^{124}+11 q^{122}-12 q^{120}+8 q^{118}-q^{116}-8 q^{114}+13 q^{112}-16 q^{110}+15 q^{108}-7 q^{106}-q^{104}+9 q^{102}-13 q^{100}+14 q^{98}-7 q^{96}+q^{94}+5 q^{92}-7 q^{90}+5 q^{88}+q^{86}-6 q^{84}+8 q^{82}-7 q^{80}+4 q^{76}-9 q^{74}+10 q^{72}-8 q^{70}+4 q^{68}-q^{66}-4 q^{64}+6 q^{62}-6 q^{60}+7 q^{58}-3 q^{56}+3 q^{54}+q^{52}-q^{50}+4 q^{48}-3 q^{46}+4 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30} }[/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 7"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 3 t^2-7 t+9-7 t^{-1} +3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 3 z^4+5 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]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+2 z^2 a^8+a^8+z^4 a^6+z^2 a^6-a^6+z^4 a^4+3 z^2 a^4+2 a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+z a^{13}+2 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+3 z^7 a^9-9 z^5 a^9+11 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-z^5 a^7+2 z^3 a^7-z a^7+z^6 a^6-2 z^2 a^6+a^6+z^5 a^5-z^3 a^5-z a^5+z^4 a^4-3 z^2 a^4+2 a^4 }[/math]