5 2

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t+2 t^{-1} -3 }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 7, -2 }
Jones polynomial	[math]\displaystyle{ - q^{-6} + q^{-5} - q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^6+a^4 z^2+a^4+a^2 z^2+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^7 z^3-2 a^7 z+a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^3-2 a^5 z+a^4 z^4-a^4 z^2+a^4+a^3 z^3+a^2 z^2-a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{20}-q^{18}+q^{12}+q^{10}+q^8+q^6+q^2 }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2 q^{50}-q^{48}+2 q^{46}+q^{44}+q^{40}+q^{34}+2 q^{24}+q^{20}+q^{14}+q^{10} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 5_2.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+q^7+q^5+q }[/math]
2	[math]\displaystyle{ q^{36}-q^{32}-q^{26}-q^{20}+q^{14}+q^{10}+2 q^8+q^2 }[/math]
3	[math]\displaystyle{ -q^{69}+q^{65}+q^{63}-q^{59}+q^{55}-q^{51}+q^{47}+q^{45}-q^{43}-q^{37}-2 q^{35}-q^{33}-q^{31}+q^{27}+q^{21}+2 q^{19}-q^{15}+q^{13}+2 q^{11}+q^9+q^3 }[/math]
4	[math]\displaystyle{ q^{112}-q^{108}-q^{106}-q^{104}+q^{102}+q^{100}+q^{98}-2 q^{94}+q^{90}+q^{88}-2 q^{84}-q^{82}+2 q^{78}+q^{76}-q^{74}+q^{70}+2 q^{68}+q^{66}-q^{64}-q^{56}-2 q^{54}-q^{52}-q^{50}-q^{48}+q^{44}-q^{42}-q^{40}-q^{38}+2 q^{34}-q^{30}-q^{28}+q^{26}+4 q^{24}+q^{22}-q^{18}+2 q^{14}+q^{12}+q^{10}+q^4 }[/math]
5	[math]\displaystyle{ -q^{165}+q^{161}+q^{159}+q^{157}-q^{153}-2 q^{151}-q^{149}+q^{145}+2 q^{143}+q^{141}-q^{139}-2 q^{137}-q^{135}+q^{133}+2 q^{131}+2 q^{129}-2 q^{125}-3 q^{123}-q^{121}+q^{119}+2 q^{117}+q^{115}-2 q^{113}-2 q^{111}-q^{109}+q^{107}+3 q^{105}+q^{103}-q^{101}-q^{99}+q^{95}+2 q^{93}+q^{91}-q^{89}+q^{85}+q^{83}+q^{81}-q^{77}-q^{73}-q^{71}-q^{69}-q^{63}-2 q^{61}-2 q^{59}-2 q^{57}+2 q^{53}+q^{51}-q^{49}-2 q^{47}-2 q^{45}+q^{43}+3 q^{41}+3 q^{39}-3 q^{35}-2 q^{33}+3 q^{29}+3 q^{27}+2 q^{25}-q^{21}+q^{17}+q^{15}+q^{13}+q^{11}+q^5 }[/math]
6	[math]\displaystyle{ q^{228}-q^{224}-q^{222}-q^{220}+2 q^{214}+2 q^{212}+q^{210}-q^{206}-2 q^{204}-3 q^{202}+q^{198}+2 q^{196}+2 q^{194}+q^{192}-q^{190}-4 q^{188}-2 q^{186}+2 q^{182}+3 q^{180}+4 q^{178}+q^{176}-3 q^{174}-3 q^{172}-3 q^{170}+2 q^{166}+4 q^{164}+2 q^{162}-2 q^{160}-3 q^{158}-4 q^{156}-q^{154}+2 q^{152}+4 q^{150}+2 q^{148}-q^{146}-2 q^{144}-3 q^{142}-q^{140}+q^{138}+3 q^{136}+q^{134}-2 q^{132}-2 q^{130}-2 q^{128}+q^{124}+2 q^{122}+q^{120}-q^{118}+q^{116}+q^{114}+2 q^{112}+2 q^{110}+2 q^{108}+q^{106}-q^{98}-q^{96}-q^{94}+q^{90}-q^{88}-q^{86}-2 q^{84}-2 q^{82}-2 q^{80}+3 q^{76}+q^{74}-2 q^{70}-4 q^{68}-4 q^{66}-q^{64}+4 q^{62}+3 q^{60}+2 q^{58}-2 q^{56}-4 q^{54}-5 q^{52}-q^{50}+5 q^{48}+4 q^{46}+4 q^{44}+q^{42}-2 q^{40}-4 q^{38}-2 q^{36}+2 q^{34}+2 q^{32}+3 q^{30}+2 q^{28}+q^{26}-q^{24}+q^{20}+q^{16}+q^{14}+q^{12}+q^6 }[/math]

A2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{20}-q^{18}+q^{12}+q^{10}+q^8+q^6+q^2 }[/math]
0,2	[math]\displaystyle{ q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}+q^4 }[/math]
1,0	[math]\displaystyle{ -q^{20}-q^{18}+q^{12}+q^{10}+q^8+q^6+q^2 }[/math]
1,1	[math]\displaystyle{ q^{52}+2 q^{48}-2 q^{46}-2 q^{42}+2 q^{34}-2 q^{32}-5 q^{28}-4 q^{24}+2 q^{22}+q^{20}+2 q^{18}+4 q^{16}+2 q^{14}+4 q^{12}+2 q^8+q^4 }[/math]
2,0	[math]\displaystyle{ q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}+q^4 }[/math]
3,0	[math]\displaystyle{ -q^{90}-q^{88}-q^{86}+2 q^{82}+2 q^{80}+2 q^{78}-q^{66}+q^{62}+2 q^{60}+q^{58}-q^{54}-q^{52}-3 q^{50}-4 q^{48}-5 q^{46}-4 q^{44}-3 q^{42}-2 q^{40}+2 q^{36}+3 q^{34}+2 q^{32}+3 q^{30}+2 q^{28}+3 q^{26}+2 q^{24}+q^{22}+q^{20}+2 q^{18}+3 q^{16}+2 q^{14}+q^6 }[/math]

A3 Invariants.

Weight	Invariant
0,0,1	[math]\displaystyle{ -q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^7+q^3 }[/math]
0,1,0	[math]\displaystyle{ q^{42}+q^{38}-2 q^{34}-2 q^{32}-2 q^{30}-2 q^{28}-q^{26}+q^{24}+q^{22}+2 q^{20}+q^{18}+2 q^{16}+q^{14}+q^{12}+2 q^{10}+q^8+q^4 }[/math]
1,0,0	[math]\displaystyle{ -q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^7+q^3 }[/math]
1,0,1	[math]\displaystyle{ q^{68}+2 q^{64}-q^{60}-3 q^{56}+q^{54}-q^{52}+q^{50}+3 q^{48}-q^{46}+q^{44}-2 q^{42}-4 q^{40}-4 q^{38}-4 q^{36}-5 q^{34}-3 q^{32}-q^{30}+5 q^{26}+3 q^{24}+7 q^{22}+6 q^{20}+3 q^{18}+5 q^{16}+q^{14}+2 q^{12}+2 q^{10}+q^6 }[/math]

A4 Invariants.

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

A5 Invariants.

Weight	Invariant
0,0,0,0,1	[math]\displaystyle{ -q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^5 }[/math]
1,0,0,0,0	[math]\displaystyle{ -q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^5 }[/math]

A6 Invariants.

Weight	Invariant
0,0,0,0,0,1	[math]\displaystyle{ -q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^6 }[/math]
1,0,0,0,0,0	[math]\displaystyle{ -q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^6 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}-q^{38}+q^{26}-q^{24}+q^{22}+q^{18}+q^{14}+q^{12}+q^8+q^4 }[/math]
1,0	[math]\displaystyle{ q^{68}+q^{60}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}+q^{34}+q^{32}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{16}+q^{14}+q^6 }[/math]

B3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ q^{100}+q^{92}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}-q^{64}+q^{56}+q^{54}+q^{50}+q^{48}+q^{46}+q^{42}+q^{40}+q^{38}+q^{34}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{10} }[/math]

B4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{132}+q^{124}-q^{110}-q^{106}-q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}+q^{74}+q^{72}+q^{70}+q^{66}+q^{64}+q^{62}+q^{58}+q^{56}+q^{54}+q^{50}+q^{46}+q^{42}+q^{38}+q^{34}+q^{32}+q^{30}+q^{26}+q^{22}+q^{14} }[/math]

C3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ -q^{58}-q^{54}-q^{48}+q^{30}+q^{26}+q^{22}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^6 }[/math]

C4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ -q^{74}-q^{70}-q^{62}-q^{60}-q^{48}+q^{42}+q^{38}+q^{34}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{14}+q^{12}+q^8 }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{58}+q^{54}-q^{48}-2 q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-2 q^{38}+2 q^{32}+q^{30}+2 q^{28}+q^{26}+2 q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^6 }[/math]

G2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{204}+q^{192}+q^{190}+q^{188}-q^{186}-q^{184}-q^{176}+q^{172}-2 q^{168}-q^{166}-q^{156}+q^{154}+q^{152}+q^{142}+q^{140}+q^{138}+2 q^{136}+q^{134}-q^{132}-2 q^{130}-q^{122}-q^{120}-q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}-q^{108}-2 q^{106}-2 q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-2 q^{94}-2 q^{92}+q^{88}+q^{86}+2 q^{84}+q^{82}+q^{78}+2 q^{74}+2 q^{72}+2 q^{70}+2 q^{68}+3 q^{66}+2 q^{64}+q^{62}+q^{60}+2 q^{58}+2 q^{56}+q^{54}+q^{50}+3 q^{48}+q^{46}+q^{36}+q^{34}+q^{32}+q^{30}+q^{18} }[/math]
1,0	[math]\displaystyle{ q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2 q^{50}-q^{48}+2 q^{46}+q^{44}+q^{40}+q^{34}+2 q^{24}+q^{20}+q^{14}+q^{10} }[/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["5 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 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]=

{ 7, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ - q^{-6} + q^{-5} - q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^7 z^3-2 a^7 z+a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^3-2 a^5 z+a^4 z^4-a^4 z^2+a^4+a^3 z^3+a^2 z^2-a^2 }[/math]