9 8

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^3-2 q^2+3 q-4+5 q^{-1} -5 q^{-2} +5 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^6+2 z^2 a^4+2 a^4-z^4 a^2-z^2 a^2-z^4-2 z^2-1+z^2 a^{-2} + a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+4 a z^7+2 z^7 a^{-1} +2 a^4 z^6+z^6 a^{-2} -z^6+2 a^5 z^5-3 a^3 z^5-13 a z^5-8 z^5 a^{-1} +2 a^6 z^4-4 a^2 z^4-4 z^4 a^{-2} -6 z^4+a^7 z^3+2 a^3 z^3+11 a z^3+8 z^3 a^{-1} -2 a^6 z^2-3 a^4 z^2+2 a^2 z^2+4 z^2 a^{-2} +7 z^2-a^7 z-a^5 z-a^3 z-3 a z-2 z a^{-1} +a^6+2 a^4- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{20}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6-q^4- q^{-2} + q^{-4} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+4 q^{86}-5 q^{84}+4 q^{82}-4 q^{80}+3 q^{76}-5 q^{74}+6 q^{72}-7 q^{70}+5 q^{68}-4 q^{66}+3 q^{62}-5 q^{60}+9 q^{58}-6 q^{56}+5 q^{54}-q^{52}-2 q^{50}+7 q^{48}-5 q^{46}+4 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-3 q^{36}-3 q^{34}+11 q^{32}-13 q^{30}+10 q^{28}-5 q^{26}-5 q^{24}+14 q^{22}-17 q^{20}+14 q^{18}-10 q^{16}+7 q^{12}-12 q^{10}+12 q^8-9 q^6+3 q^4+3 q^2-7+7 q^{-2} -3 q^{-4} -2 q^{-6} +8 q^{-8} -10 q^{-10} +7 q^{-12} -8 q^{-16} +14 q^{-18} -15 q^{-20} +10 q^{-22} -2 q^{-24} -6 q^{-26} +11 q^{-28} -11 q^{-30} +10 q^{-32} -3 q^{-34} - q^{-36} +3 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_8.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+q^{11}-q^9+2 q^7+q- q^{-1} + q^{-3} - q^{-5} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{36}-q^{34}-q^{32}+2 q^{30}-2 q^{28}+3 q^{24}-4 q^{22}+4 q^{18}-3 q^{16}-q^{14}+3 q^{12}+q^{10}-2 q^8+3 q^4-q^2-2+4 q^{-2} -4 q^{-6} +3 q^{-8} +2 q^{-10} -4 q^{-12} + q^{-14} +3 q^{-16} -2 q^{-18} - q^{-20} + q^{-22} }[/math]
3	[math]\displaystyle{ -q^{69}+q^{67}+q^{65}-2 q^{61}+2 q^{57}-q^{51}-q^{49}-q^{47}+4 q^{45}+2 q^{43}-6 q^{41}-5 q^{39}+8 q^{37}+6 q^{35}-5 q^{33}-10 q^{31}+2 q^{29}+9 q^{27}+2 q^{25}-5 q^{23}-6 q^{21}+3 q^{19}+8 q^{17}+q^{15}-9 q^{13}-2 q^{11}+8 q^9+4 q^7-8 q^5-4 q^3+8 q+7 q^{-1} -5 q^{-3} -7 q^{-5} +3 q^{-7} +9 q^{-9} -10 q^{-13} -4 q^{-15} +7 q^{-17} +8 q^{-19} -4 q^{-21} -9 q^{-23} + q^{-25} +8 q^{-27} +3 q^{-29} -6 q^{-31} -5 q^{-33} +3 q^{-35} +4 q^{-37} -2 q^{-41} - q^{-43} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{112}-q^{110}-q^{108}+4 q^{102}-2 q^{100}-2 q^{98}-2 q^{96}-2 q^{94}+9 q^{92}+q^{90}-q^{88}-7 q^{86}-9 q^{84}+12 q^{82}+10 q^{80}+5 q^{78}-14 q^{76}-24 q^{74}+9 q^{72}+23 q^{70}+21 q^{68}-15 q^{66}-44 q^{64}-6 q^{62}+29 q^{60}+42 q^{58}-46 q^{54}-25 q^{52}+9 q^{50}+40 q^{48}+23 q^{46}-20 q^{44}-27 q^{42}-17 q^{40}+12 q^{38}+24 q^{36}+10 q^{34}-6 q^{32}-25 q^{30}-14 q^{28}+14 q^{26}+25 q^{24}+7 q^{22}-21 q^{20}-22 q^{18}+9 q^{16}+31 q^{14}+9 q^{12}-22 q^{10}-27 q^8+7 q^6+36 q^4+12 q^2-19-33 q^{-2} -4 q^{-4} +34 q^{-6} +21 q^{-8} -5 q^{-10} -32 q^{-12} -20 q^{-14} +17 q^{-16} +22 q^{-18} +18 q^{-20} -13 q^{-22} -26 q^{-24} -7 q^{-26} +5 q^{-28} +26 q^{-30} +11 q^{-32} -10 q^{-34} -16 q^{-36} -19 q^{-38} +11 q^{-40} +18 q^{-42} +10 q^{-44} -2 q^{-46} -21 q^{-48} -7 q^{-50} +4 q^{-52} +12 q^{-54} +11 q^{-56} -7 q^{-58} -7 q^{-60} -5 q^{-62} + q^{-64} +6 q^{-66} + q^{-68} -2 q^{-72} - q^{-74} + q^{-76} }[/math]
5	[math]\displaystyle{ -q^{165}+q^{163}+q^{161}-2 q^{155}-2 q^{153}+2 q^{151}+4 q^{149}+q^{147}-6 q^{143}-7 q^{141}+q^{139}+9 q^{137}+10 q^{135}+q^{133}-11 q^{131}-17 q^{129}-6 q^{127}+18 q^{125}+28 q^{123}+8 q^{121}-24 q^{119}-37 q^{117}-18 q^{115}+31 q^{113}+62 q^{111}+26 q^{109}-44 q^{107}-84 q^{105}-45 q^{103}+49 q^{101}+113 q^{99}+72 q^{97}-51 q^{95}-138 q^{93}-107 q^{91}+36 q^{89}+156 q^{87}+136 q^{85}-q^{83}-145 q^{81}-163 q^{79}-38 q^{77}+119 q^{75}+163 q^{73}+73 q^{71}-63 q^{69}-142 q^{67}-102 q^{65}+12 q^{63}+99 q^{61}+100 q^{59}+32 q^{57}-45 q^{55}-87 q^{53}-63 q^{51}+3 q^{49}+61 q^{47}+72 q^{45}+31 q^{43}-37 q^{41}-79 q^{39}-46 q^{37}+29 q^{35}+75 q^{33}+53 q^{31}-24 q^{29}-82 q^{27}-54 q^{25}+31 q^{23}+89 q^{21}+60 q^{19}-33 q^{17}-100 q^{15}-70 q^{13}+34 q^{11}+111 q^9+87 q^7-24 q^5-118 q^3-103 q+4 q^{-1} +113 q^{-3} +123 q^{-5} +23 q^{-7} -97 q^{-9} -131 q^{-11} -53 q^{-13} +66 q^{-15} +128 q^{-17} +83 q^{-19} -27 q^{-21} -110 q^{-23} -98 q^{-25} -12 q^{-27} +71 q^{-29} +97 q^{-31} +48 q^{-33} -29 q^{-35} -77 q^{-37} -64 q^{-39} -11 q^{-41} +39 q^{-43} +61 q^{-45} +41 q^{-47} - q^{-49} -39 q^{-51} -48 q^{-53} -28 q^{-55} +4 q^{-57} +36 q^{-59} +43 q^{-61} +22 q^{-63} -13 q^{-65} -35 q^{-67} -34 q^{-69} -13 q^{-71} +18 q^{-73} +33 q^{-75} +25 q^{-77} -19 q^{-81} -24 q^{-83} -15 q^{-85} +6 q^{-87} +17 q^{-89} +14 q^{-91} +3 q^{-93} -5 q^{-95} -9 q^{-97} -7 q^{-99} + q^{-101} +4 q^{-103} +3 q^{-105} + q^{-107} -2 q^{-111} - q^{-113} + q^{-115} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{20}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6-q^4- q^{-2} + q^{-4} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{52}-2 q^{50}+4 q^{48}-6 q^{46}+9 q^{44}-12 q^{42}+14 q^{40}-16 q^{38}+15 q^{36}-18 q^{34}+14 q^{32}-16 q^{30}+16 q^{28}-14 q^{26}+18 q^{24}-10 q^{22}+10 q^{20}+2 q^{18}-14 q^{16}+26 q^{14}-44 q^{12}+52 q^{10}-62 q^8+64 q^6-59 q^4+56 q^2-38+26 q^{-2} -7 q^{-4} -10 q^{-6} +24 q^{-8} -36 q^{-10} +41 q^{-12} -40 q^{-14} +36 q^{-16} -28 q^{-18} +19 q^{-20} -12 q^{-22} +6 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{50}+q^{48}-2 q^{44}-q^{42}-2 q^{38}-2 q^{36}+2 q^{34}+3 q^{32}-q^{30}-q^{28}+3 q^{26}+3 q^{24}-3 q^{22}+q^{18}-q^{16}-q^{14}-q^8+2 q^6+3 q^4+3 q^{-2} + q^{-4} -3 q^{-6} - q^{-8} + q^{-10} + q^{-12} - q^{-14} - q^{-16} +2 q^{-18} + q^{-20} - q^{-22} - q^{-24} + q^{-28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{42}-q^{40}+2 q^{36}-3 q^{34}-3 q^{32}+2 q^{30}-2 q^{28}-3 q^{26}+5 q^{24}+2 q^{22}+4 q^{18}+2 q^{16}-q^{14}-q^{12}-4 q^6+q^4+3 q^2-2+ q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-10} -2 q^{-12} + q^{-14} + q^{-16} - q^{-18} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ -q^{27}-q^{25}-q^{23}+q^{21}+2 q^{17}+q^{15}+2 q^{13}+q^9-q^5-q- q^{-3} + q^{-5} + q^{-9} + q^{-13} }[/math]

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+4 q^{86}-5 q^{84}+4 q^{82}-4 q^{80}+3 q^{76}-5 q^{74}+6 q^{72}-7 q^{70}+5 q^{68}-4 q^{66}+3 q^{62}-5 q^{60}+9 q^{58}-6 q^{56}+5 q^{54}-q^{52}-2 q^{50}+7 q^{48}-5 q^{46}+4 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-3 q^{36}-3 q^{34}+11 q^{32}-13 q^{30}+10 q^{28}-5 q^{26}-5 q^{24}+14 q^{22}-17 q^{20}+14 q^{18}-10 q^{16}+7 q^{12}-12 q^{10}+12 q^8-9 q^6+3 q^4+3 q^2-7+7 q^{-2} -3 q^{-4} -2 q^{-6} +8 q^{-8} -10 q^{-10} +7 q^{-12} -8 q^{-16} +14 q^{-18} -15 q^{-20} +10 q^{-22} -2 q^{-24} -6 q^{-26} +11 q^{-28} -11 q^{-30} +10 q^{-32} -3 q^{-34} - q^{-36} +3 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} }[/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["9 8"];

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+4 a z^7+2 z^7 a^{-1} +2 a^4 z^6+z^6 a^{-2} -z^6+2 a^5 z^5-3 a^3 z^5-13 a z^5-8 z^5 a^{-1} +2 a^6 z^4-4 a^2 z^4-4 z^4 a^{-2} -6 z^4+a^7 z^3+2 a^3 z^3+11 a z^3+8 z^3 a^{-1} -2 a^6 z^2-3 a^4 z^2+2 a^2 z^2+4 z^2 a^{-2} +7 z^2-a^7 z-a^5 z-a^3 z-3 a z-2 z a^{-1} +a^6+2 a^4- a^{-2} -1 }[/math]