9 1

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_1.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

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

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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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+7z^{6}+15z^{4}+10z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, -8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-4}+q^{-6}-q^{-7}+q^{-8}-q^{-9}+q^{-10}-q^{-11}+q^{-12}-q^{-13}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{10}-6z^{4}a^{10}-10z^{2}a^{10}-4a^{10}+z^{8}a^{8}+8z^{6}a^{8}+21z^{4}a^{8}+20z^{2}a^{8}+5a^{8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$za^{17}+z^{2}a^{16}+z^{3}a^{15}-za^{15}+z^{4}a^{14}-2z^{2}a^{14}+z^{5}a^{13}-3z^{3}a^{13}+za^{13}+z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+z^{7}a^{11}-5z^{5}a^{11}+6z^{3}a^{11}-za^{11}+z^{8}a^{10}-7z^{6}a^{10}+16z^{4}a^{10}-14z^{2}a^{10}+4a^{10}+z^{7}a^{9}-6z^{5}a^{9}+10z^{3}a^{9}-4za^{9}+z^{8}a^{8}-8z^{6}a^{8}+21z^{4}a^{8}-20z^{2}a^{8}+5a^{8}$