8 11

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+7 t-9+7 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 27, -2 }
Jones polynomial	[math]\displaystyle{ q-2+4 q^{-1} -4 q^{-2} +5 q^{-3} -5 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-2 a^4-z^4 a^2-z^2 a^2+a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8-2 z^2 a^8+2 z^5 a^7-4 z^3 a^7+2 z a^7+2 z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z a^5+4 z^6 a^4-7 z^4 a^4+6 z^2 a^4-2 a^4+z^7 a^3+z^5 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-2 z^4 a^2-a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}+q^{16}-2 q^{14}-q^{12}-q^{10}+2 q^6+2 q^2+ q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-q^{104}-3 q^{102}+7 q^{100}-8 q^{98}+8 q^{96}-6 q^{94}+2 q^{92}+7 q^{90}-13 q^{88}+16 q^{86}-13 q^{84}+7 q^{82}+3 q^{80}-10 q^{78}+13 q^{76}-9 q^{74}+8 q^{72}+4 q^{70}-10 q^{68}+7 q^{66}-2 q^{64}-8 q^{62}+14 q^{60}-18 q^{58}+10 q^{56}-3 q^{54}-9 q^{52}+18 q^{50}-25 q^{48}+19 q^{46}-14 q^{44}-q^{42}+11 q^{40}-17 q^{38}+16 q^{36}-9 q^{34}+4 q^{32}+6 q^{30}-9 q^{28}+7 q^{26}-7 q^{22}+14 q^{20}-11 q^{18}+5 q^{16}+6 q^{14}-11 q^{12}+17 q^{10}-14 q^8+9 q^6-2 q^4-7 q^2+10-9 q^{-2} +8 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_11.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-q^{13}+q^{11}-2 q^9+q^5+2 q- q^{-1} + q^{-3} }[/math]
2	[math]\displaystyle{ q^{42}-q^{40}-q^{38}+3 q^{36}-2 q^{34}-4 q^{32}+5 q^{30}-5 q^{26}+6 q^{24}+2 q^{22}-4 q^{20}+q^{16}-4 q^{12}+2 q^{10}+4 q^8-5 q^6+q^4+6 q^2-4- q^{-2} +4 q^{-4} - q^{-6} - q^{-8} + q^{-10} }[/math]
3	[math]\displaystyle{ q^{81}-q^{79}-q^{77}+q^{75}+2 q^{73}-2 q^{71}-5 q^{69}+2 q^{67}+8 q^{65}-11 q^{61}-2 q^{59}+14 q^{57}+8 q^{55}-15 q^{53}-11 q^{51}+12 q^{49}+14 q^{47}-13 q^{45}-15 q^{43}+8 q^{41}+13 q^{39}-q^{37}-11 q^{35}+6 q^{31}+6 q^{29}-4 q^{27}-9 q^{25}-q^{23}+14 q^{21}+3 q^{19}-16 q^{17}-6 q^{15}+16 q^{13}+10 q^{11}-14 q^9-12 q^7+11 q^5+14 q^3-7 q-11 q^{-1} +2 q^{-3} +10 q^{-5} + q^{-7} -6 q^{-9} - q^{-11} +3 q^{-13} + q^{-15} - q^{-17} - q^{-19} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{132}-q^{130}-q^{128}+q^{126}+2 q^{122}-4 q^{120}-3 q^{118}+4 q^{116}+3 q^{114}+8 q^{112}-8 q^{110}-13 q^{108}+q^{106}+9 q^{104}+25 q^{102}-5 q^{100}-28 q^{98}-20 q^{96}+3 q^{94}+49 q^{92}+19 q^{90}-28 q^{88}-47 q^{86}-23 q^{84}+56 q^{82}+47 q^{80}-7 q^{78}-54 q^{76}-47 q^{74}+43 q^{72}+53 q^{70}+12 q^{68}-39 q^{66}-49 q^{64}+15 q^{62}+36 q^{60}+22 q^{58}-16 q^{56}-33 q^{54}-7 q^{52}+17 q^{50}+24 q^{48}+6 q^{46}-15 q^{44}-31 q^{42}-5 q^{40}+30 q^{38}+29 q^{36}-2 q^{34}-52 q^{32}-25 q^{30}+32 q^{28}+51 q^{26}+18 q^{24}-60 q^{22}-43 q^{20}+16 q^{18}+54 q^{16}+40 q^{14}-43 q^{12}-48 q^{10}-11 q^8+35 q^6+49 q^4-13 q^2-29-23 q^{-2} +6 q^{-4} +32 q^{-6} +5 q^{-8} -6 q^{-10} -16 q^{-12} -6 q^{-14} +12 q^{-16} +3 q^{-18} +2 q^{-20} -4 q^{-22} -3 q^{-24} +3 q^{-26} + q^{-30} - q^{-32} - q^{-34} + q^{-36} }[/math]
5	[math]\displaystyle{ q^{195}-q^{193}-q^{191}+q^{189}-2 q^{181}-2 q^{179}+4 q^{177}+6 q^{175}+q^{173}-4 q^{171}-9 q^{169}-8 q^{167}+4 q^{165}+18 q^{163}+18 q^{161}-q^{159}-24 q^{157}-35 q^{155}-16 q^{153}+25 q^{151}+58 q^{149}+43 q^{147}-15 q^{145}-74 q^{143}-84 q^{141}-17 q^{139}+82 q^{137}+126 q^{135}+65 q^{133}-65 q^{131}-161 q^{129}-120 q^{127}+33 q^{125}+172 q^{123}+175 q^{121}+18 q^{119}-170 q^{117}-209 q^{115}-64 q^{113}+138 q^{111}+220 q^{109}+109 q^{107}-107 q^{105}-212 q^{103}-124 q^{101}+62 q^{99}+182 q^{97}+130 q^{95}-23 q^{93}-146 q^{91}-124 q^{89}+3 q^{87}+103 q^{85}+101 q^{83}+25 q^{81}-67 q^{79}-93 q^{77}-33 q^{75}+37 q^{73}+71 q^{71}+61 q^{69}-5 q^{67}-69 q^{65}-76 q^{63}-24 q^{61}+61 q^{59}+107 q^{57}+56 q^{55}-63 q^{53}-136 q^{51}-92 q^{49}+49 q^{47}+164 q^{45}+130 q^{43}-34 q^{41}-179 q^{39}-170 q^{37}+5 q^{35}+181 q^{33}+199 q^{31}+37 q^{29}-161 q^{27}-211 q^{25}-79 q^{23}+118 q^{21}+208 q^{19}+115 q^{17}-67 q^{15}-176 q^{13}-131 q^{11}+9 q^9+131 q^7+134 q^5+28 q^3-77 q-106 q^{-1} -52 q^{-3} +28 q^{-5} +76 q^{-7} +55 q^{-9} - q^{-11} -40 q^{-13} -43 q^{-15} -13 q^{-17} +16 q^{-19} +27 q^{-21} +16 q^{-23} -5 q^{-25} -13 q^{-27} -9 q^{-29} +3 q^{-33} +5 q^{-35} +2 q^{-37} -3 q^{-39} - q^{-41} + q^{-43} + q^{-49} - q^{-51} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}+q^{16}-2 q^{14}-q^{12}-q^{10}+2 q^6+2 q^2+ q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{60}-2 q^{58}+4 q^{56}-8 q^{54}+15 q^{52}-20 q^{50}+26 q^{48}-38 q^{46}+43 q^{44}-44 q^{42}+40 q^{40}-34 q^{38}+20 q^{36}+4 q^{34}-26 q^{32}+50 q^{30}-66 q^{28}+84 q^{26}-88 q^{24}+86 q^{22}-79 q^{20}+56 q^{18}-38 q^{16}+14 q^{14}+6 q^{12}-26 q^{10}+42 q^8-44 q^6+44 q^4-40 q^2+36-22 q^{-2} +17 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{56}+q^{48}-q^{46}-4 q^{44}-q^{42}+2 q^{40}-3 q^{36}+2 q^{34}+5 q^{32}+3 q^{30}-q^{28}+2 q^{26}-4 q^{22}-q^{20}-q^{18}-2 q^{16}-q^{14}+3 q^{12}-2 q^8+q^6+4 q^4-q^2-2+3 q^{-2} +3 q^{-4} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-q^{46}+q^{42}-4 q^{40}+4 q^{36}-3 q^{34}+2 q^{32}+7 q^{30}-2 q^{28}+q^{24}-3 q^{22}-4 q^{20}-3 q^{18}+q^{16}-q^{14}-2 q^{12}+5 q^{10}+2 q^8-4 q^6+5 q^4+2 q^2-2+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{29}+q^{25}+q^{21}-2 q^{19}-q^{17}-2 q^{15}-q^{13}+q^9+2 q^7+2 q^3+ q^{-1} + q^{-5} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{62}-q^{58}+q^{56}+q^{54}-3 q^{52}-3 q^{50}+q^{48}+q^{46}-3 q^{44}+7 q^{40}+4 q^{38}+q^{36}+5 q^{34}+4 q^{32}-4 q^{30}-4 q^{28}-3 q^{26}-6 q^{24}-7 q^{22}-q^{20}+2 q^{18}-3 q^{16}+q^{14}+6 q^{12}+2 q^{10}-2 q^8+3 q^6+4 q^4+q^2+2 q^{-2} +2 q^{-4} + q^{-10} }[/math]
1,0,0,0	[math]\displaystyle{ q^{36}+q^{32}+q^{30}+q^{26}-2 q^{24}-q^{22}-2 q^{20}-2 q^{18}-q^{16}+q^{12}+q^{10}+2 q^8+2 q^4+1+ q^{-2} + q^{-6} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-q^{46}+2 q^{44}-3 q^{42}+4 q^{40}-4 q^{38}+4 q^{36}-3 q^{34}+2 q^{32}-q^{30}-2 q^{28}+4 q^{26}-7 q^{24}+7 q^{22}-8 q^{20}+7 q^{18}-7 q^{16}+5 q^{14}-2 q^{12}+q^{10}+2 q^8-2 q^6+5 q^4-4 q^2+4-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{78}-q^{74}-q^{72}+q^{70}+2 q^{68}-q^{66}-4 q^{64}-2 q^{62}+3 q^{60}+4 q^{58}-q^{56}-4 q^{54}+5 q^{50}+4 q^{48}-2 q^{46}-2 q^{44}+2 q^{42}+3 q^{40}-2 q^{38}-4 q^{36}-q^{34}+2 q^{32}-q^{30}-4 q^{28}-2 q^{26}+2 q^{24}+2 q^{22}-2 q^{20}-2 q^{18}+2 q^{16}+5 q^{14}-4 q^{10}-2 q^8+5 q^6+4 q^4-q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{66}-q^{64}+q^{62}-2 q^{60}+2 q^{58}-4 q^{56}+2 q^{54}-3 q^{52}+4 q^{50}-2 q^{48}+3 q^{46}+q^{44}+4 q^{42}+3 q^{40}-2 q^{38}+3 q^{36}-5 q^{34}+4 q^{32}-9 q^{30}+2 q^{28}-9 q^{26}+4 q^{24}-5 q^{22}+3 q^{20}-3 q^{18}+3 q^{16}+2 q^{14}+q^{12}+2 q^{10}-2 q^8+5 q^6-2 q^4+4 q^2-2+4 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} + q^{-10} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-q^{104}-3 q^{102}+7 q^{100}-8 q^{98}+8 q^{96}-6 q^{94}+2 q^{92}+7 q^{90}-13 q^{88}+16 q^{86}-13 q^{84}+7 q^{82}+3 q^{80}-10 q^{78}+13 q^{76}-9 q^{74}+8 q^{72}+4 q^{70}-10 q^{68}+7 q^{66}-2 q^{64}-8 q^{62}+14 q^{60}-18 q^{58}+10 q^{56}-3 q^{54}-9 q^{52}+18 q^{50}-25 q^{48}+19 q^{46}-14 q^{44}-q^{42}+11 q^{40}-17 q^{38}+16 q^{36}-9 q^{34}+4 q^{32}+6 q^{30}-9 q^{28}+7 q^{26}-7 q^{22}+14 q^{20}-11 q^{18}+5 q^{16}+6 q^{14}-11 q^{12}+17 q^{10}-14 q^8+9 q^6-2 q^4-7 q^2+10-9 q^{-2} +8 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/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["8 11"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^4-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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q-2+4 q^{-1} -4 q^{-2} +5 q^{-3} -5 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-2 a^4-z^4 a^2-z^2 a^2+a^2+z^2+1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^4 a^8-2 z^2 a^8+2 z^5 a^7-4 z^3 a^7+2 z a^7+2 z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z a^5+4 z^6 a^4-7 z^4 a^4+6 z^2 a^4-2 a^4+z^7 a^3+z^5 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-2 z^4 a^2-a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1 }[/math]