9 1

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+7 z^6+15 z^4+10 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 9, -8 }
Jones polynomial	[math]\displaystyle{ q^{-4} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} + q^{-12} - q^{-13} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{10}-6 z^4 a^{10}-10 z^2 a^{10}-4 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+20 z^2 a^8+5 a^8 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z a^{17}+z^2 a^{16}+z^3 a^{15}-z a^{15}+z^4 a^{14}-2 z^2 a^{14}+z^5 a^{13}-3 z^3 a^{13}+z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-z a^{11}+z^8 a^{10}-7 z^6 a^{10}+16 z^4 a^{10}-14 z^2 a^{10}+4 a^{10}+z^7 a^9-6 z^5 a^9+10 z^3 a^9-4 z a^9+z^8 a^8-8 z^6 a^8+21 z^4 a^8-20 z^2 a^8+5 a^8 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{38}-q^{36}-q^{34}+q^{22}+q^{20}+2 q^{18}+q^{16}+q^{14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{216}-q^{172}-q^{170}-q^{164}-q^{162}-q^{160}-q^{154}-q^{152}-q^{126}-q^{120}-q^{118}-q^{116}-q^{114}-q^{108}+q^{100}+q^{98}+q^{94}+2 q^{92}+2 q^{90}+2 q^{88}+q^{86}+q^{84}+2 q^{82}+2 q^{80}+q^{78}+q^{74}+q^{72}+q^{70} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{27}+q^{11}+q^9+q^7 }[/math]
2	[math]\displaystyle{ q^{72}-q^{56}-q^{54}-q^{52}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14} }[/math]
3	[math]\displaystyle{ -q^{135}+q^{119}+q^{117}+q^{115}-q^{85}-q^{83}-q^{81}-q^{79}-q^{77}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21} }[/math]
4	[math]\displaystyle{ q^{216}-q^{200}-q^{198}-q^{196}+q^{166}+q^{164}+q^{162}+q^{160}+q^{158}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}-q^{104}-q^{102}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28} }[/math]
5	[math]\displaystyle{ -q^{315}+q^{299}+q^{297}+q^{295}-q^{265}-q^{263}-q^{261}-q^{259}-q^{257}+q^{213}+q^{211}+q^{209}+q^{207}+q^{205}+q^{203}+q^{201}-q^{143}-q^{141}-q^{139}-q^{137}-q^{135}-q^{133}-q^{131}-q^{129}-q^{127}+q^{55}+q^{53}+q^{51}+q^{49}+q^{47}+q^{45}+q^{43}+q^{41}+q^{39}+q^{37}+q^{35} }[/math]
6	[math]\displaystyle{ q^{432}-q^{416}-q^{414}-q^{412}+q^{382}+q^{380}+q^{378}+q^{376}+q^{374}-q^{330}-q^{328}-q^{326}-q^{324}-q^{322}-q^{320}-q^{318}+q^{260}+q^{258}+q^{256}+q^{254}+q^{252}+q^{250}+q^{248}+q^{246}+q^{244}-q^{172}-q^{170}-q^{168}-q^{166}-q^{164}-q^{162}-q^{160}-q^{158}-q^{156}-q^{154}-q^{152}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42} }[/math]
8	[math]\displaystyle{ q^{720}-q^{704}-q^{702}-q^{700}+q^{670}+q^{668}+q^{666}+q^{664}+q^{662}-q^{618}-q^{616}-q^{614}-q^{612}-q^{610}-q^{608}-q^{606}+q^{548}+q^{546}+q^{544}+q^{542}+q^{540}+q^{538}+q^{536}+q^{534}+q^{532}-q^{460}-q^{458}-q^{456}-q^{454}-q^{452}-q^{450}-q^{448}-q^{446}-q^{444}-q^{442}-q^{440}+q^{354}+q^{352}+q^{350}+q^{348}+q^{346}+q^{344}+q^{342}+q^{340}+q^{338}+q^{336}+q^{334}+q^{332}+q^{330}-q^{230}-q^{228}-q^{226}-q^{224}-q^{222}-q^{220}-q^{218}-q^{216}-q^{214}-q^{212}-q^{210}-q^{208}-q^{206}-q^{204}-q^{202}+q^{88}+q^{86}+q^{84}+q^{82}+q^{80}+q^{78}+q^{76}+q^{74}+q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{38}-q^{36}-q^{34}+q^{22}+q^{20}+2 q^{18}+q^{16}+q^{14} }[/math]
1,1	[math]\displaystyle{ q^{108}-2 q^{60}-2 q^{58}-4 q^{56}-4 q^{54}-4 q^{52}-2 q^{50}-2 q^{48}+q^{44}+2 q^{42}+4 q^{40}+4 q^{38}+5 q^{36}+4 q^{34}+4 q^{32}+2 q^{30}+q^{28} }[/math]
2,0	[math]\displaystyle{ q^{94}+q^{92}+2 q^{90}+q^{88}+q^{86}-q^{78}-2 q^{76}-3 q^{74}-3 q^{72}-3 q^{70}-2 q^{68}-q^{66}+q^{44}+q^{42}+2 q^{40}+2 q^{38}+3 q^{36}+2 q^{34}+2 q^{32}+q^{30}+q^{28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{90}-q^{60}-2 q^{58}-3 q^{56}-3 q^{54}-3 q^{52}-2 q^{50}-q^{48}+q^{44}+q^{42}+3 q^{40}+3 q^{38}+4 q^{36}+3 q^{34}+3 q^{32}+q^{30}+q^{28} }[/math]
1,0,0	[math]\displaystyle{ -q^{49}-q^{47}-2 q^{45}-q^{43}-q^{41}+q^{33}+q^{31}+2 q^{29}+2 q^{27}+2 q^{25}+q^{23}+q^{21} }[/math]
1,0,1	[math]\displaystyle{ q^{144}+q^{98}+q^{96}+3 q^{94}+3 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+q^{84}-q^{82}-4 q^{80}-8 q^{78}-10 q^{76}-14 q^{74}-14 q^{72}-14 q^{70}-10 q^{68}-7 q^{66}-2 q^{64}+3 q^{62}+7 q^{60}+10 q^{58}+11 q^{56}+12 q^{54}+11 q^{52}+10 q^{50}+7 q^{48}+5 q^{46}+2 q^{44}+q^{42} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{112}+q^{110}+q^{108}+q^{106}+q^{104}-q^{82}-2 q^{80}-4 q^{78}-5 q^{76}-7 q^{74}-7 q^{72}-7 q^{70}-5 q^{68}-3 q^{66}-q^{64}+2 q^{62}+4 q^{60}+6 q^{58}+6 q^{56}+8 q^{54}+6 q^{52}+6 q^{50}+4 q^{48}+3 q^{46}+q^{44}+q^{42} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{60}-q^{58}-2 q^{56}-2 q^{54}-2 q^{52}-q^{50}-q^{48}+q^{44}+q^{42}+2 q^{40}+2 q^{38}+3 q^{36}+2 q^{34}+2 q^{32}+q^{30}+q^{28} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{90}-q^{60}-q^{56}-q^{54}-q^{52}-q^{48}+q^{44}+q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+q^{32}+q^{30}+q^{28} }[/math]
1,0	[math]\displaystyle{ q^{144}-q^{98}-q^{96}-q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}-q^{84}-q^{82}+q^{66}+q^{62}+q^{60}+2 q^{58}+q^{56}+2 q^{54}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{42} }[/math]

D4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{216}-q^{172}-q^{170}-3 q^{168}-3 q^{166}-4 q^{164}-4 q^{162}-4 q^{160}-3 q^{158}+2 q^{154}+5 q^{152}+9 q^{150}+12 q^{148}+12 q^{146}+15 q^{144}+12 q^{142}+12 q^{140}+9 q^{138}+6 q^{136}+3 q^{134}+3 q^{132}-q^{126}-3 q^{124}-6 q^{122}-10 q^{120}-16 q^{118}-22 q^{116}-28 q^{114}-33 q^{112}-36 q^{110}-37 q^{108}-33 q^{106}-27 q^{104}-18 q^{102}-8 q^{100}+4 q^{98}+12 q^{96}+22 q^{94}+26 q^{92}+29 q^{90}+29 q^{88}+28 q^{86}+22 q^{84}+20 q^{82}+14 q^{80}+10 q^{78}+6 q^{76}+4 q^{74}+q^{72}+q^{70} }[/math]
1,0,0,0	[math]\displaystyle{ q^{126}-q^{82}-q^{80}-3 q^{78}-3 q^{76}-4 q^{74}-4 q^{72}-4 q^{70}-3 q^{68}-2 q^{66}+q^{62}+3 q^{60}+4 q^{58}+4 q^{56}+5 q^{54}+4 q^{52}+4 q^{50}+3 q^{48}+2 q^{46}+q^{44}+q^{42} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{216}-q^{172}-q^{170}-q^{164}-q^{162}-q^{160}-q^{154}-q^{152}-q^{126}-q^{120}-q^{118}-q^{116}-q^{114}-q^{108}+q^{100}+q^{98}+q^{94}+2 q^{92}+2 q^{90}+2 q^{88}+q^{86}+q^{84}+2 q^{82}+2 q^{80}+q^{78}+q^{74}+q^{72}+q^{70} }[/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["9 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+7 z^6+15 z^4+10 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]=

{ 9, -8 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-4} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} + q^{-12} - q^{-13} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^6 a^{10}-6 z^4 a^{10}-10 z^2 a^{10}-4 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+20 z^2 a^8+5 a^8 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z a^{17}+z^2 a^{16}+z^3 a^{15}-z a^{15}+z^4 a^{14}-2 z^2 a^{14}+z^5 a^{13}-3 z^3 a^{13}+z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-z a^{11}+z^8 a^{10}-7 z^6 a^{10}+16 z^4 a^{10}-14 z^2 a^{10}+4 a^{10}+z^7 a^9-6 z^5 a^9+10 z^3 a^9-4 z a^9+z^8 a^8-8 z^6 a^8+21 z^4 a^8-20 z^2 a^8+5 a^8 }[/math]