7 1

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1 }[/math]
Conway polynomial	[math]\displaystyle{ z^6+5 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 7, -6 }
Jones polynomial	[math]\displaystyle{ - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^8 \left(-z^4\right)-4 a^8 z^2-3 a^8+a^6 z^6+6 a^6 z^4+10 a^6 z^2+4 a^6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^{13} z+a^{12} z^2+a^{11} z^3-a^{11} z+a^{10} z^4-2 a^{10} z^2+a^9 z^5-3 a^9 z^3+a^9 z+a^8 z^6-5 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^5-4 a^7 z^3+3 a^7 z+a^6 z^6-6 a^6 z^4+10 a^6 z^2-4 a^6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2 q^{84}+2 q^{80}+q^{78}+q^{72}+3 q^{70}+2 q^{68}+q^{64}+2 q^{62}+2 q^{60}+q^{58}+q^{54}+q^{52}+q^{50} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 7_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{21}+q^9+q^7+q^5 }[/math]
2	[math]\displaystyle{ q^{56}-q^{44}-q^{42}-q^{40}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10} }[/math]
3	[math]\displaystyle{ -q^{105}+q^{93}+q^{91}+q^{89}-q^{67}-q^{65}-q^{63}-q^{61}-q^{59}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15} }[/math]
4	[math]\displaystyle{ q^{168}-q^{156}-q^{154}-q^{152}+q^{130}+q^{128}+q^{126}+q^{124}+q^{122}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20} }[/math]
5	[math]\displaystyle{ -q^{245}+q^{233}+q^{231}+q^{229}-q^{207}-q^{205}-q^{203}-q^{201}-q^{199}+q^{167}+q^{165}+q^{163}+q^{161}+q^{159}+q^{157}+q^{155}-q^{113}-q^{111}-q^{109}-q^{107}-q^{105}-q^{103}-q^{101}-q^{99}-q^{97}+q^{45}+q^{43}+q^{41}+q^{39}+q^{37}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25} }[/math]
6	[math]\displaystyle{ q^{336}-q^{324}-q^{322}-q^{320}+q^{298}+q^{296}+q^{294}+q^{292}+q^{290}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}+q^{204}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}-q^{136}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30} }[/math]
8	[math]\displaystyle{ q^{560}-q^{548}-q^{546}-q^{544}+q^{522}+q^{520}+q^{518}+q^{516}+q^{514}-q^{482}-q^{480}-q^{478}-q^{476}-q^{474}-q^{472}-q^{470}+q^{428}+q^{426}+q^{424}+q^{422}+q^{420}+q^{418}+q^{416}+q^{414}+q^{412}-q^{360}-q^{358}-q^{356}-q^{354}-q^{352}-q^{350}-q^{348}-q^{346}-q^{344}-q^{342}-q^{340}+q^{278}+q^{276}+q^{274}+q^{272}+q^{270}+q^{268}+q^{266}+q^{264}+q^{262}+q^{260}+q^{258}+q^{256}+q^{254}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}-q^{172}-q^{170}-q^{168}-q^{166}-q^{164}-q^{162}-q^{160}-q^{158}-q^{156}-q^{154}+q^{72}+q^{70}+q^{68}+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}+q^{40} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10} }[/math]
1,1	[math]\displaystyle{ q^{84}-2 q^{48}-2 q^{46}-4 q^{44}-4 q^{42}-4 q^{40}-2 q^{38}-q^{36}+2 q^{34}+4 q^{32}+4 q^{30}+5 q^{28}+4 q^{26}+4 q^{24}+2 q^{22}+q^{20} }[/math]
2,0	[math]\displaystyle{ q^{74}+q^{72}+2 q^{70}+q^{68}+q^{66}-q^{62}-2 q^{60}-3 q^{58}-3 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{36}+q^{34}+2 q^{32}+2 q^{30}+3 q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{20} }[/math]
3,0	[math]\displaystyle{ -q^{132}-q^{130}-2 q^{128}-2 q^{126}-2 q^{124}-q^{122}+2 q^{118}+4 q^{116}+4 q^{114}+5 q^{112}+4 q^{110}+4 q^{108}+2 q^{106}+q^{104}-q^{94}-2 q^{92}-3 q^{90}-4 q^{88}-5 q^{86}-5 q^{84}-5 q^{82}-4 q^{80}-3 q^{78}-2 q^{76}-q^{74}+q^{54}+q^{52}+2 q^{50}+2 q^{48}+3 q^{46}+3 q^{44}+4 q^{42}+3 q^{40}+3 q^{38}+2 q^{36}+2 q^{34}+q^{32}+q^{30} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{70}-q^{48}-2 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+3 q^{30}+4 q^{28}+3 q^{26}+3 q^{24}+q^{22}+q^{20} }[/math]
1,0,0	[math]\displaystyle{ -q^{39}-q^{37}-2 q^{35}-q^{33}-q^{31}+q^{27}+q^{25}+2 q^{23}+2 q^{21}+2 q^{19}+q^{17}+q^{15} }[/math]
1,0,1	[math]\displaystyle{ q^{112}+q^{78}+q^{76}+3 q^{74}+3 q^{72}+4 q^{70}+3 q^{68}+q^{66}-3 q^{64}-7 q^{62}-10 q^{60}-14 q^{58}-14 q^{56}-13 q^{54}-8 q^{52}-3 q^{50}+3 q^{48}+8 q^{46}+11 q^{44}+12 q^{42}+11 q^{40}+10 q^{38}+7 q^{36}+5 q^{34}+2 q^{32}+q^{30} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{88}+q^{86}+q^{84}+q^{82}+q^{80}-q^{66}-2 q^{64}-4 q^{62}-5 q^{60}-7 q^{58}-7 q^{56}-6 q^{54}-4 q^{52}-q^{50}+2 q^{48}+5 q^{46}+6 q^{44}+8 q^{42}+6 q^{40}+6 q^{38}+4 q^{36}+3 q^{34}+q^{32}+q^{30} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{48}-q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-q^{38}+q^{34}+2 q^{32}+2 q^{30}+3 q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{20} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{70}-q^{48}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+q^{30}+2 q^{28}+q^{26}+q^{24}+q^{22}+q^{20} }[/math]
1,0	[math]\displaystyle{ q^{112}-q^{78}-q^{76}-q^{74}-q^{72}-2 q^{70}-q^{68}-q^{66}-q^{64}-q^{62}+q^{54}+q^{50}+q^{48}+2 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{36}+q^{34}+q^{30} }[/math]

D4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{168}-q^{136}-q^{134}-3 q^{132}-3 q^{130}-4 q^{128}-4 q^{126}-q^{124}+3 q^{120}+8 q^{118}+11 q^{116}+12 q^{114}+15 q^{112}+12 q^{110}+12 q^{108}+9 q^{106}+6 q^{104}+2 q^{102}-6 q^{98}-10 q^{96}-16 q^{94}-22 q^{92}-27 q^{90}-32 q^{88}-33 q^{86}-32 q^{84}-27 q^{82}-19 q^{80}-8 q^{78}+12 q^{74}+19 q^{72}+24 q^{70}+26 q^{68}+27 q^{66}+22 q^{64}+20 q^{62}+14 q^{60}+10 q^{58}+6 q^{56}+4 q^{54}+q^{52}+q^{50} }[/math]
1,0,0,0	[math]\displaystyle{ q^{98}-q^{66}-q^{64}-3 q^{62}-3 q^{60}-4 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+2 q^{48}+3 q^{46}+4 q^{44}+5 q^{42}+4 q^{40}+4 q^{38}+3 q^{36}+2 q^{34}+q^{32}+q^{30} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2 q^{84}+2 q^{80}+q^{78}+q^{72}+3 q^{70}+2 q^{68}+q^{64}+2 q^{62}+2 q^{60}+q^{58}+q^{54}+q^{52}+q^{50} }[/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["7 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+5 z^4+6 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]=

{ 7, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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