8 14

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 1-2 z^4 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 31, -2 }
Jones polynomial	[math]\displaystyle{ q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -5 q^{-4} +4 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6-z^4 a^4-z^2 a^4-z^4 a^2-z^2 a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8-z^2 a^8+3 z^5 a^7-5 z^3 a^7+z a^7+3 z^6 a^6-4 z^4 a^6+z^2 a^6+z^7 a^5+4 z^5 a^5-8 z^3 a^5+3 z a^5+5 z^6 a^4-7 z^4 a^4+3 z^2 a^4+z^7 a^3+3 z^5 a^3-6 z^3 a^3+3 z a^3+2 z^6 a^2-z^4 a^2-z^2 a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^6-q^4+2 q^2+ q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+3 q^{106}-6 q^{102}+14 q^{100}-16 q^{98}+17 q^{96}-11 q^{94}-4 q^{92}+17 q^{90}-25 q^{88}+25 q^{86}-17 q^{84}+4 q^{82}+11 q^{80}-17 q^{78}+17 q^{76}-9 q^{74}-3 q^{72}+13 q^{70}-16 q^{68}+5 q^{66}+7 q^{64}-19 q^{62}+28 q^{60}-26 q^{58}+15 q^{56}+q^{54}-21 q^{52}+33 q^{50}-37 q^{48}+28 q^{46}-11 q^{44}-7 q^{42}+21 q^{40}-23 q^{38}+19 q^{36}-7 q^{34}-6 q^{32}+13 q^{30}-12 q^{28}+2 q^{26}+12 q^{24}-19 q^{22}+22 q^{20}-13 q^{18}+12 q^{14}-21 q^{12}+24 q^{10}-18 q^8+8 q^6+2 q^4-9 q^2+12-10 q^{-2} +9 q^{-4} -3 q^{-6} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_14.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-2 q^{13}+q^{11}-q^9+q^7+q^5-q^3+2 q- q^{-1} + q^{-3} }[/math]
2	[math]\displaystyle{ q^{42}-2 q^{40}-2 q^{38}+6 q^{36}-q^{34}-6 q^{32}+7 q^{30}+q^{28}-8 q^{26}+4 q^{24}+2 q^{22}-5 q^{20}+3 q^{16}+2 q^{14}-5 q^{12}+2 q^{10}+7 q^8-7 q^6-q^4+8 q^2-4-2 q^{-2} +4 q^{-4} - q^{-6} - q^{-8} + q^{-10} }[/math]
3	[math]\displaystyle{ q^{81}-2 q^{79}-2 q^{77}+3 q^{75}+6 q^{73}-q^{71}-13 q^{69}-q^{67}+16 q^{65}+7 q^{63}-19 q^{61}-15 q^{59}+19 q^{57}+22 q^{55}-16 q^{53}-25 q^{51}+12 q^{49}+26 q^{47}-5 q^{45}-25 q^{43}+2 q^{41}+19 q^{39}+3 q^{37}-15 q^{35}-8 q^{33}+6 q^{31}+13 q^{29}-18 q^{25}-7 q^{23}+20 q^{21}+16 q^{19}-20 q^{17}-21 q^{15}+19 q^{13}+25 q^{11}-12 q^9-25 q^7+6 q^5+23 q^3-q-16 q^{-1} -2 q^{-3} +11 q^{-5} +3 q^{-7} -6 q^{-9} -2 q^{-11} +3 q^{-13} + q^{-15} - q^{-17} - q^{-19} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{132}-2 q^{130}-2 q^{128}+3 q^{126}+3 q^{124}+6 q^{122}-8 q^{120}-13 q^{118}-q^{116}+8 q^{114}+31 q^{112}-2 q^{110}-33 q^{108}-27 q^{106}-2 q^{104}+65 q^{102}+34 q^{100}-30 q^{98}-68 q^{96}-50 q^{94}+74 q^{92}+84 q^{90}+9 q^{88}-84 q^{86}-101 q^{84}+43 q^{82}+102 q^{80}+57 q^{78}-60 q^{76}-116 q^{74}+3 q^{72}+81 q^{70}+73 q^{68}-23 q^{66}-91 q^{64}-24 q^{62}+41 q^{60}+62 q^{58}+8 q^{56}-49 q^{54}-42 q^{52}-q^{50}+45 q^{48}+42 q^{46}-q^{44}-65 q^{42}-46 q^{40}+28 q^{38}+75 q^{36}+49 q^{34}-74 q^{32}-88 q^{30}-6 q^{28}+90 q^{26}+99 q^{24}-53 q^{22}-104 q^{20}-48 q^{18}+64 q^{16}+115 q^{14}-7 q^{12}-73 q^{10}-68 q^8+14 q^6+85 q^4+22 q^2-24-49 q^{-2} -15 q^{-4} +37 q^{-6} +17 q^{-8} +2 q^{-10} -19 q^{-12} -12 q^{-14} +11 q^{-16} +4 q^{-18} +4 q^{-20} -4 q^{-22} -4 q^{-24} +3 q^{-26} + q^{-30} - q^{-32} - q^{-34} + q^{-36} }[/math]
5	[math]\displaystyle{ q^{195}-2 q^{193}-2 q^{191}+3 q^{189}+3 q^{187}+3 q^{185}-q^{183}-8 q^{181}-13 q^{179}-q^{177}+17 q^{175}+23 q^{173}+13 q^{171}-16 q^{169}-43 q^{167}-44 q^{165}+10 q^{163}+70 q^{161}+78 q^{159}+23 q^{157}-76 q^{155}-138 q^{153}-86 q^{151}+67 q^{149}+189 q^{147}+165 q^{145}-8 q^{143}-220 q^{141}-266 q^{139}-76 q^{137}+215 q^{135}+352 q^{133}+184 q^{131}-165 q^{129}-404 q^{127}-295 q^{125}+85 q^{123}+414 q^{121}+381 q^{119}+7 q^{117}-379 q^{115}-432 q^{113}-94 q^{111}+316 q^{109}+431 q^{107}+164 q^{105}-241 q^{103}-402 q^{101}-194 q^{99}+161 q^{97}+341 q^{95}+211 q^{93}-89 q^{91}-279 q^{89}-199 q^{87}+30 q^{85}+203 q^{83}+196 q^{81}+28 q^{79}-146 q^{77}-184 q^{75}-83 q^{73}+84 q^{71}+187 q^{69}+147 q^{67}-27 q^{65}-200 q^{63}-214 q^{61}-37 q^{59}+202 q^{57}+287 q^{55}+114 q^{53}-199 q^{51}-361 q^{49}-193 q^{47}+170 q^{45}+411 q^{43}+284 q^{41}-114 q^{39}-432 q^{37}-367 q^{35}+39 q^{33}+407 q^{31}+416 q^{29}+57 q^{27}-339 q^{25}-427 q^{23}-145 q^{21}+245 q^{19}+392 q^{17}+203 q^{15}-130 q^{13}-316 q^{11}-226 q^9+33 q^7+228 q^5+205 q^3+31 q-132 q^{-1} -161 q^{-3} -62 q^{-5} +63 q^{-7} +109 q^{-9} +60 q^{-11} -19 q^{-13} -61 q^{-15} -44 q^{-17} - q^{-19} +29 q^{-21} +28 q^{-23} +5 q^{-25} -13 q^{-27} -13 q^{-29} -3 q^{-31} +2 q^{-33} +6 q^{-35} +4 q^{-37} -3 q^{-39} -2 q^{-41} + q^{-43} + q^{-49} - q^{-51} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^6-q^4+2 q^2+ q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+32 q^{52}-48 q^{50}+66 q^{48}-78 q^{46}+83 q^{44}-78 q^{42}+64 q^{40}-36 q^{38}-q^{36}+42 q^{34}-84 q^{32}+116 q^{30}-145 q^{28}+156 q^{26}-154 q^{24}+138 q^{22}-105 q^{20}+68 q^{18}-26 q^{16}-12 q^{14}+47 q^{12}-68 q^{10}+78 q^8-76 q^6+70 q^4-56 q^2+42-28 q^{-2} +19 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{56}-q^{54}-2 q^{52}+3 q^{48}+2 q^{46}-4 q^{44}+4 q^{40}+q^{38}-4 q^{36}-q^{34}+3 q^{32}-2 q^{30}-4 q^{28}+q^{24}-2 q^{22}+3 q^{20}+3 q^{18}-q^{16}+q^{14}+4 q^{12}-5 q^8+5 q^4-q^2-3+2 q^{-2} +3 q^{-4} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-2 q^{46}+3 q^{42}-5 q^{40}+2 q^{38}+5 q^{36}-6 q^{34}+q^{32}+5 q^{30}-4 q^{28}-q^{26}+3 q^{24}-q^{22}-2 q^{20}-2 q^{18}+3 q^{16}-q^{14}-4 q^{12}+7 q^{10}+q^8-5 q^6+6 q^4+q^2-3+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{29}-q^{27}-q^{23}+q^{21}-q^{19}+q^{17}+q^7-q^5+2 q^3+ q^{-1} + q^{-5} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-2 q^{46}+4 q^{44}-5 q^{42}+5 q^{40}-6 q^{38}+5 q^{36}-4 q^{34}+q^{32}+q^{30}-4 q^{28}+7 q^{26}-9 q^{24}+11 q^{22}-10 q^{20}+10 q^{18}-7 q^{16}+5 q^{14}-2 q^{12}-q^{10}+3 q^8-5 q^6+6 q^4-5 q^2+5-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+4 q^{68}-q^{66}-5 q^{64}-2 q^{62}+6 q^{60}+5 q^{58}-3 q^{56}-6 q^{54}+5 q^{50}+2 q^{48}-4 q^{46}-3 q^{44}+3 q^{42}+3 q^{40}-2 q^{38}-4 q^{36}+q^{34}+4 q^{32}-5 q^{28}-q^{26}+4 q^{24}+2 q^{22}-4 q^{20}-3 q^{18}+4 q^{16}+6 q^{14}-q^{12}-6 q^{10}-q^8+6 q^6+4 q^4-2 q^2-4+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}-2 q^{64}+2 q^{62}-3 q^{60}+4 q^{58}-5 q^{56}+4 q^{54}-4 q^{52}+5 q^{50}-3 q^{48}+q^{46}+q^{42}+3 q^{40}-5 q^{38}+5 q^{36}-6 q^{34}+8 q^{32}-9 q^{30}+6 q^{28}-9 q^{26}+7 q^{24}-5 q^{22}+3 q^{20}-3 q^{18}+2 q^{16}+3 q^{14}-q^{12}+3 q^{10}-4 q^8+6 q^6-3 q^4+4 q^2-3+4 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} + q^{-10} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+3 q^{106}-6 q^{102}+14 q^{100}-16 q^{98}+17 q^{96}-11 q^{94}-4 q^{92}+17 q^{90}-25 q^{88}+25 q^{86}-17 q^{84}+4 q^{82}+11 q^{80}-17 q^{78}+17 q^{76}-9 q^{74}-3 q^{72}+13 q^{70}-16 q^{68}+5 q^{66}+7 q^{64}-19 q^{62}+28 q^{60}-26 q^{58}+15 q^{56}+q^{54}-21 q^{52}+33 q^{50}-37 q^{48}+28 q^{46}-11 q^{44}-7 q^{42}+21 q^{40}-23 q^{38}+19 q^{36}-7 q^{34}-6 q^{32}+13 q^{30}-12 q^{28}+2 q^{26}+12 q^{24}-19 q^{22}+22 q^{20}-13 q^{18}+12 q^{14}-21 q^{12}+24 q^{10}-18 q^8+8 q^6+2 q^4-9 q^2+12-10 q^{-2} +9 q^{-4} -3 q^{-6} +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 14"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-2 z^4 }[/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]=

{ 31, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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