9 17

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+9 t-9+9 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 39, -2 }
Jones polynomial	[math]\displaystyle{ q^3-2 q^2+4 q-5+6 q^{-1} -7 q^{-2} +6 q^{-3} -4 q^{-4} +3 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-2 z^4-2 a^4 z^2+5 a^2 z^2+z^2 a^{-2} -6 z^2+2 a^2+2 a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +4 a^4 z^6+4 a^2 z^6+z^6 a^{-2} +z^6+4 a^5 z^5-2 a^3 z^5-13 a z^5-7 z^5 a^{-1} +3 a^6 z^4-3 a^4 z^4-14 a^2 z^4-4 z^4 a^{-2} -12 z^4+a^7 z^3-3 a^5 z^3-4 a^3 z^3+6 a z^3+6 z^3 a^{-1} -2 a^6 z^2-a^4 z^2+9 a^2 z^2+5 z^2 a^{-2} +13 z^2+a^5 z+3 a^3 z+a z-z a^{-1} -2 a^2-2 a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{18}+q^{16}+q^{12}+2 q^{10}-q^8+q^6-2 q^4- q^{-2} + q^{-4} + q^{-8} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+q^{92}-3 q^{88}+8 q^{86}-9 q^{84}+12 q^{82}-9 q^{80}+3 q^{78}+4 q^{76}-9 q^{74}+14 q^{72}-20 q^{70}+17 q^{68}-12 q^{66}+q^{64}+10 q^{62}-18 q^{60}+22 q^{58}-16 q^{56}+8 q^{54}+q^{52}-14 q^{50}+16 q^{48}-7 q^{46}-2 q^{44}+15 q^{42}-18 q^{40}+15 q^{38}+3 q^{36}-17 q^{34}+30 q^{32}-35 q^{30}+26 q^{28}-6 q^{26}-14 q^{24}+33 q^{22}-38 q^{20}+32 q^{18}-16 q^{16}-4 q^{14}+15 q^{12}-26 q^{10}+22 q^8-12 q^6-4 q^4+14 q^2-17+10 q^{-2} +3 q^{-4} -17 q^{-6} +23 q^{-8} -24 q^{-10} +11 q^{-12} +5 q^{-14} -21 q^{-16} +32 q^{-18} -27 q^{-20} +17 q^{-22} - q^{-24} -11 q^{-26} +19 q^{-28} -18 q^{-30} +14 q^{-32} -4 q^{-34} -2 q^{-36} +5 q^{-38} -5 q^{-40} +4 q^{-42} - q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_17.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+2 q^{11}-q^9+2 q^7-q^5-q^3+q- q^{-1} +2 q^{-3} - q^{-5} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{36}-2 q^{34}-q^{32}+4 q^{30}-4 q^{28}+q^{26}+7 q^{24}-7 q^{22}-q^{20}+7 q^{18}-5 q^{16}-4 q^{14}+4 q^{12}+2 q^{10}-3 q^8-q^6+6 q^4-q^2-5+7 q^{-2} + q^{-4} -8 q^{-6} +4 q^{-8} +4 q^{-10} -6 q^{-12} + q^{-14} +4 q^{-16} -2 q^{-18} - q^{-20} + q^{-22} }[/math]
3	[math]\displaystyle{ -q^{69}+2 q^{67}+q^{65}-2 q^{63}-2 q^{61}+q^{59}+2 q^{57}-5 q^{55}+3 q^{53}+7 q^{51}-3 q^{49}-12 q^{47}+7 q^{45}+17 q^{43}-8 q^{41}-22 q^{39}+4 q^{37}+24 q^{35}-22 q^{31}-8 q^{29}+18 q^{27}+12 q^{25}-6 q^{23}-17 q^{21}+18 q^{17}+7 q^{15}-16 q^{13}-13 q^{11}+15 q^9+16 q^7-11 q^5-21 q^3+9 q+22 q^{-1} -2 q^{-3} -23 q^{-5} -4 q^{-7} +23 q^{-9} +11 q^{-11} -19 q^{-13} -15 q^{-15} +12 q^{-17} +18 q^{-19} -4 q^{-21} -18 q^{-23} - q^{-25} +13 q^{-27} +6 q^{-29} -8 q^{-31} -7 q^{-33} +3 q^{-35} +5 q^{-37} -2 q^{-41} - q^{-43} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{112}-2 q^{110}-q^{108}+2 q^{106}+5 q^{102}-4 q^{100}-q^{98}-9 q^{94}+9 q^{92}+9 q^{88}+q^{86}-28 q^{84}+q^{82}+12 q^{80}+36 q^{78}+9 q^{76}-60 q^{74}-30 q^{72}+17 q^{70}+76 q^{68}+39 q^{66}-80 q^{64}-74 q^{62}-6 q^{60}+96 q^{58}+80 q^{56}-50 q^{54}-86 q^{52}-55 q^{50}+56 q^{48}+93 q^{46}+14 q^{44}-44 q^{42}-75 q^{40}-14 q^{38}+51 q^{36}+57 q^{34}+19 q^{32}-56 q^{30}-64 q^{28}+q^{26}+70 q^{24}+53 q^{22}-30 q^{20}-80 q^{18}-29 q^{16}+71 q^{14}+66 q^{12}-12 q^{10}-85 q^8-51 q^6+64 q^4+77 q^2+18-78 q^{-2} -78 q^{-4} +34 q^{-6} +76 q^{-8} +58 q^{-10} -45 q^{-12} -91 q^{-14} -18 q^{-16} +40 q^{-18} +85 q^{-20} +13 q^{-22} -64 q^{-24} -52 q^{-26} -16 q^{-28} +64 q^{-30} +50 q^{-32} -7 q^{-34} -38 q^{-36} -52 q^{-38} +12 q^{-40} +38 q^{-42} +27 q^{-44} + q^{-46} -37 q^{-48} -16 q^{-50} +4 q^{-52} +18 q^{-54} +17 q^{-56} -8 q^{-58} -9 q^{-60} -7 q^{-62} + q^{-64} +7 q^{-66} + q^{-68} -2 q^{-72} - q^{-74} + q^{-76} }[/math]
5	[math]\displaystyle{ -q^{165}+2 q^{163}+q^{161}-2 q^{159}-3 q^{155}-2 q^{153}+3 q^{151}+6 q^{149}+3 q^{147}+2 q^{145}-6 q^{143}-13 q^{141}-9 q^{139}+8 q^{137}+27 q^{135}+16 q^{133}-9 q^{131}-40 q^{129}-46 q^{127}+10 q^{125}+78 q^{123}+71 q^{121}-7 q^{119}-103 q^{117}-128 q^{115}-13 q^{113}+159 q^{111}+194 q^{109}+37 q^{107}-194 q^{105}-279 q^{103}-97 q^{101}+218 q^{99}+372 q^{97}+176 q^{95}-211 q^{93}-442 q^{91}-274 q^{89}+152 q^{87}+468 q^{85}+377 q^{83}-56 q^{81}-433 q^{79}-435 q^{77}-72 q^{75}+338 q^{73}+439 q^{71}+185 q^{69}-189 q^{67}-386 q^{65}-271 q^{63}+43 q^{61}+283 q^{59}+291 q^{57}+103 q^{55}-157 q^{53}-285 q^{51}-193 q^{49}+43 q^{47}+239 q^{45}+251 q^{43}+50 q^{41}-205 q^{39}-281 q^{37}-101 q^{35}+174 q^{33}+292 q^{31}+133 q^{29}-160 q^{27}-311 q^{25}-151 q^{23}+168 q^{21}+325 q^{19}+176 q^{17}-153 q^{15}-356 q^{13}-214 q^{11}+143 q^9+372 q^7+260 q^5-95 q^3-375 q-322 q^{-1} +27 q^{-3} +348 q^{-5} +369 q^{-7} +69 q^{-9} -286 q^{-11} -391 q^{-13} -169 q^{-15} +186 q^{-17} +376 q^{-19} +257 q^{-21} -64 q^{-23} -313 q^{-25} -300 q^{-27} -65 q^{-29} +206 q^{-31} +300 q^{-33} +166 q^{-35} -81 q^{-37} -244 q^{-39} -212 q^{-41} -42 q^{-43} +146 q^{-45} +211 q^{-47} +122 q^{-49} -43 q^{-51} -154 q^{-53} -147 q^{-55} -50 q^{-57} +76 q^{-59} +132 q^{-61} +90 q^{-63} -3 q^{-65} -79 q^{-67} -92 q^{-69} -44 q^{-71} +27 q^{-73} +66 q^{-75} +55 q^{-77} +9 q^{-79} -31 q^{-81} -41 q^{-83} -26 q^{-85} +5 q^{-87} +24 q^{-89} +21 q^{-91} +5 q^{-93} -6 q^{-95} -11 q^{-97} -9 q^{-99} + q^{-101} +5 q^{-103} +3 q^{-105} + q^{-107} -2 q^{-111} - q^{-113} + q^{-115} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{18}+q^{16}+q^{12}+2 q^{10}-q^8+q^6-2 q^4- q^{-2} + q^{-4} + q^{-8} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{52}-4 q^{50}+8 q^{48}-12 q^{46}+18 q^{44}-28 q^{42}+36 q^{40}-40 q^{38}+45 q^{36}-56 q^{34}+60 q^{32}-56 q^{30}+60 q^{28}-50 q^{26}+40 q^{24}-16 q^{22}-17 q^{20}+44 q^{18}-88 q^{16}+114 q^{14}-141 q^{12}+160 q^{10}-156 q^8+148 q^6-117 q^4+92 q^2-50+6 q^{-2} +31 q^{-4} -62 q^{-6} +80 q^{-8} -94 q^{-10} +92 q^{-12} -78 q^{-14} +64 q^{-16} -46 q^{-18} +29 q^{-20} -16 q^{-22} +8 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{46}-q^{44}-q^{42}-q^{38}+2 q^{34}+3 q^{32}-2 q^{30}+4 q^{26}+2 q^{24}-6 q^{22}-q^{20}+3 q^{18}-3 q^{16}-3 q^{14}+2 q^{10}-2 q^8+2 q^6+4 q^4+q^2+1+4 q^{-2} -5 q^{-6} +2 q^{-10} - q^{-12} -3 q^{-14} + q^{-16} +3 q^{-18} -2 q^{-22} + q^{-26} + q^{-28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{42}-2 q^{40}-q^{38}+4 q^{36}-3 q^{34}-2 q^{32}+7 q^{30}-q^{28}-5 q^{26}+6 q^{24}-2 q^{22}-6 q^{20}+3 q^{18}+q^{16}+q^{12}+5 q^{10}+3 q^8-5 q^6+q^4+3 q^2-7- q^{-2} +4 q^{-4} -5 q^{-6} + q^{-8} +4 q^{-10} -2 q^{-12} +2 q^{-14} +2 q^{-16} - q^{-18} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ -q^{23}+q^{21}-q^{19}+2 q^{17}+2 q^{13}+q^9-q^5-2 q-2 q^{-3} + q^{-5} +2 q^{-9} + q^{-11} + q^{-13} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{52}-q^{50}-2 q^{48}+2 q^{46}+q^{44}-4 q^{42}-q^{40}+5 q^{38}+2 q^{36}-2 q^{34}+3 q^{32}+6 q^{30}-3 q^{28}-5 q^{26}+q^{24}-3 q^{22}-7 q^{20}+2 q^{18}+4 q^{16}-q^{14}+3 q^{12}+8 q^{10}+3 q^8-3 q^6+2 q^4+4 q^2-4-5 q^{-2} - q^{-6} -4 q^{-8} - q^{-10} + q^{-12} + q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} + q^{-22} + q^{-24} + q^{-26} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{28}+q^{26}-q^{24}+q^{22}+q^{20}+2 q^{16}+2 q^{12}+q^8-q^6-2 q^2-2- q^{-2} -2 q^{-4} + q^{-6} +2 q^{-10} +2 q^{-12} + q^{-14} + q^{-16} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}+2 q^{40}-3 q^{38}+4 q^{36}-5 q^{34}+6 q^{32}-7 q^{30}+7 q^{28}-5 q^{26}+4 q^{24}-2 q^{20}+7 q^{18}-9 q^{16}+12 q^{14}-13 q^{12}+13 q^{10}-13 q^8+9 q^6-7 q^4+3 q^2-1-3 q^{-2} +6 q^{-4} -7 q^{-6} +7 q^{-8} -6 q^{-10} +6 q^{-12} -4 q^{-14} +4 q^{-16} - q^{-18} + q^{-20} }[/math]
1,0	[math]\displaystyle{ q^{68}-2 q^{64}-2 q^{62}+q^{60}+4 q^{58}+q^{56}-4 q^{54}-4 q^{52}+2 q^{50}+7 q^{48}+3 q^{46}-5 q^{44}-5 q^{42}+2 q^{40}+7 q^{38}-q^{36}-7 q^{34}-3 q^{32}+5 q^{30}+3 q^{28}-4 q^{26}-4 q^{24}+3 q^{22}+6 q^{20}-3 q^{16}+q^{14}+5 q^{12}-5 q^8-2 q^6+5 q^4+4 q^2-4-8 q^{-2} +8 q^{-6} +3 q^{-8} -6 q^{-10} -6 q^{-12} +3 q^{-14} +6 q^{-16} + q^{-18} -4 q^{-20} -2 q^{-22} +3 q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-34} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{58}-2 q^{56}+q^{54}-2 q^{52}+4 q^{50}-4 q^{48}+3 q^{46}-4 q^{44}+7 q^{42}-4 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-2 q^{32}-q^{30}-5 q^{26}+6 q^{24}-6 q^{22}+10 q^{20}-8 q^{18}+13 q^{16}-7 q^{14}+12 q^{12}-8 q^{10}+7 q^8-6 q^6+2 q^4-5 q^2-2-5 q^{-4} +3 q^{-6} -6 q^{-8} +7 q^{-10} -4 q^{-12} +6 q^{-14} -3 q^{-16} +6 q^{-18} -2 q^{-20} +3 q^{-22} - q^{-24} + q^{-26} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+q^{92}-3 q^{88}+8 q^{86}-9 q^{84}+12 q^{82}-9 q^{80}+3 q^{78}+4 q^{76}-9 q^{74}+14 q^{72}-20 q^{70}+17 q^{68}-12 q^{66}+q^{64}+10 q^{62}-18 q^{60}+22 q^{58}-16 q^{56}+8 q^{54}+q^{52}-14 q^{50}+16 q^{48}-7 q^{46}-2 q^{44}+15 q^{42}-18 q^{40}+15 q^{38}+3 q^{36}-17 q^{34}+30 q^{32}-35 q^{30}+26 q^{28}-6 q^{26}-14 q^{24}+33 q^{22}-38 q^{20}+32 q^{18}-16 q^{16}-4 q^{14}+15 q^{12}-26 q^{10}+22 q^8-12 q^6-4 q^4+14 q^2-17+10 q^{-2} +3 q^{-4} -17 q^{-6} +23 q^{-8} -24 q^{-10} +11 q^{-12} +5 q^{-14} -21 q^{-16} +32 q^{-18} -27 q^{-20} +17 q^{-22} - q^{-24} -11 q^{-26} +19 q^{-28} -18 q^{-30} +14 q^{-32} -4 q^{-34} -2 q^{-36} +5 q^{-38} -5 q^{-40} +4 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 17"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-5 t^2+9 t-9+9 t^{-1} -5 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+z^4-2 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]=

{ 39, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +4 a^4 z^6+4 a^2 z^6+z^6 a^{-2} +z^6+4 a^5 z^5-2 a^3 z^5-13 a z^5-7 z^5 a^{-1} +3 a^6 z^4-3 a^4 z^4-14 a^2 z^4-4 z^4 a^{-2} -12 z^4+a^7 z^3-3 a^5 z^3-4 a^3 z^3+6 a z^3+6 z^3 a^{-1} -2 a^6 z^2-a^4 z^2+9 a^2 z^2+5 z^2 a^{-2} +13 z^2+a^5 z+3 a^3 z+a z-z a^{-1} -2 a^2-2 a^{-2} -3 }[/math]