9 20

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-9 t+11-9 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	{ 41, -4 }
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -5 q^{-3} +7 q^{-4} -7 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^8-a^8+2 z^4 a^6+5 z^2 a^6+2 a^6-z^6 a^4-4 z^4 a^4-5 z^2 a^4-2 a^4+z^4 a^2+3 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}-z^2 a^{10}+5 z^5 a^9-5 z^3 a^9+2 z a^9+5 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+3 z^7 a^7-7 z^3 a^7+2 z a^7+z^8 a^6+5 z^6 a^6-16 z^4 a^6+10 z^2 a^6-2 a^6+5 z^7 a^5-12 z^5 a^5+5 z^3 a^5+z^8 a^4+z^6 a^4-11 z^4 a^4+11 z^2 a^4-2 a^4+2 z^7 a^3-7 z^5 a^3+6 z^3 a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2 q^{10}-q^8+q^6+q^4+1 }[/math]
The G2 invariant	[math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+2 q^{140}-q^{138}-2 q^{136}+9 q^{134}-12 q^{132}+15 q^{130}-13 q^{128}+5 q^{126}+2 q^{124}-13 q^{122}+23 q^{120}-27 q^{118}+23 q^{116}-13 q^{114}-q^{112}+15 q^{110}-23 q^{108}+25 q^{106}-21 q^{104}+5 q^{102}+7 q^{100}-17 q^{98}+16 q^{96}-5 q^{94}-9 q^{92}+23 q^{90}-24 q^{88}+14 q^{86}+3 q^{84}-26 q^{82}+44 q^{80}-44 q^{78}+30 q^{76}-4 q^{74}-21 q^{72}+43 q^{70}-45 q^{68}+33 q^{66}-17 q^{64}-6 q^{62}+22 q^{60}-28 q^{58}+22 q^{56}-6 q^{54}-9 q^{52}+19 q^{50}-19 q^{48}+7 q^{46}+8 q^{44}-23 q^{42}+31 q^{40}-28 q^{38}+12 q^{36}+10 q^{34}-26 q^{32}+36 q^{30}-30 q^{28}+17 q^{26}-q^{24}-13 q^{22}+21 q^{20}-19 q^{18}+14 q^{16}-3 q^{14}-2 q^{12}+5 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_20.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{19}+2 q^{17}-2 q^{15}+q^{13}-q^{11}+2 q^7-q^5+2 q^3-q+ q^{-1} }[/math]
2	[math]\displaystyle{ q^{52}-2 q^{50}+5 q^{46}-6 q^{44}-q^{42}+8 q^{40}-8 q^{38}-q^{36}+9 q^{34}-4 q^{32}-4 q^{30}+4 q^{28}+2 q^{26}-5 q^{24}-3 q^{22}+6 q^{20}-q^{18}-7 q^{16}+8 q^{14}+3 q^{12}-8 q^{10}+5 q^8+5 q^6-6 q^4+q^2+4-2 q^{-2} - q^{-4} + q^{-6} }[/math]
3	[math]\displaystyle{ -q^{99}+2 q^{97}-3 q^{93}+5 q^{89}+2 q^{87}-10 q^{85}+13 q^{81}-2 q^{79}-18 q^{77}+2 q^{75}+25 q^{73}-2 q^{71}-28 q^{69}-2 q^{67}+30 q^{65}+7 q^{63}-26 q^{61}-13 q^{59}+16 q^{57}+19 q^{55}-6 q^{53}-20 q^{51}-5 q^{49}+21 q^{47}+13 q^{45}-16 q^{43}-22 q^{41}+13 q^{39}+23 q^{37}-9 q^{35}-27 q^{33}+2 q^{31}+28 q^{29}+4 q^{27}-27 q^{25}-11 q^{23}+26 q^{21}+17 q^{19}-19 q^{17}-20 q^{15}+13 q^{13}+23 q^{11}-3 q^9-20 q^7-2 q^5+14 q^3+7 q-8 q^{-1} -7 q^{-3} +3 q^{-5} +5 q^{-7} -2 q^{-11} - q^{-13} + q^{-15} }[/math]
4	[math]\displaystyle{ q^{160}-2 q^{158}+3 q^{154}-2 q^{152}+q^{150}-6 q^{148}+3 q^{146}+9 q^{144}-8 q^{142}+2 q^{140}-10 q^{138}+11 q^{136}+18 q^{134}-25 q^{132}-10 q^{130}-9 q^{128}+40 q^{126}+37 q^{124}-53 q^{122}-46 q^{120}-15 q^{118}+79 q^{116}+79 q^{114}-60 q^{112}-93 q^{110}-51 q^{108}+86 q^{106}+122 q^{104}-15 q^{102}-92 q^{100}-97 q^{98}+30 q^{96}+111 q^{94}+48 q^{92}-28 q^{90}-96 q^{88}-45 q^{86}+39 q^{84}+78 q^{82}+50 q^{80}-51 q^{78}-87 q^{76}-28 q^{74}+73 q^{72}+85 q^{70}-5 q^{68}-93 q^{66}-68 q^{64}+60 q^{62}+92 q^{60}+24 q^{58}-90 q^{56}-92 q^{54}+44 q^{52}+92 q^{50}+58 q^{48}-70 q^{46}-114 q^{44}+3 q^{42}+76 q^{40}+98 q^{38}-20 q^{36}-112 q^{34}-52 q^{32}+26 q^{30}+114 q^{28}+46 q^{26}-63 q^{24}-76 q^{22}-40 q^{20}+70 q^{18}+73 q^{16}+6 q^{14}-44 q^{12}-68 q^{10}+7 q^8+44 q^6+35 q^4+4 q^2-40-19 q^{-2} +3 q^{-4} +19 q^{-6} +18 q^{-8} -8 q^{-10} -9 q^{-12} -7 q^{-14} + q^{-16} +7 q^{-18} + q^{-20} -2 q^{-24} - q^{-26} + q^{-28} }[/math]
5	[math]\displaystyle{ -q^{235}+2 q^{233}-3 q^{229}+2 q^{227}+q^{225}+q^{221}-2 q^{219}-4 q^{217}+2 q^{215}+6 q^{213}-q^{211}-8 q^{209}-5 q^{207}+10 q^{205}+15 q^{203}+8 q^{201}-18 q^{199}-46 q^{197}-15 q^{195}+50 q^{193}+81 q^{191}+32 q^{189}-77 q^{187}-147 q^{185}-73 q^{183}+114 q^{181}+236 q^{179}+133 q^{177}-137 q^{175}-326 q^{173}-232 q^{171}+119 q^{169}+420 q^{167}+350 q^{165}-68 q^{163}-460 q^{161}-464 q^{159}-46 q^{157}+443 q^{155}+556 q^{153}+176 q^{151}-353 q^{149}-573 q^{147}-306 q^{145}+198 q^{143}+514 q^{141}+400 q^{139}-23 q^{137}-388 q^{135}-422 q^{133}-140 q^{131}+213 q^{129}+390 q^{127}+262 q^{125}-44 q^{123}-304 q^{121}-338 q^{119}-100 q^{117}+215 q^{115}+354 q^{113}+205 q^{111}-132 q^{109}-361 q^{107}-261 q^{105}+87 q^{103}+349 q^{101}+288 q^{99}-49 q^{97}-356 q^{95}-311 q^{93}+44 q^{91}+362 q^{89}+335 q^{87}-17 q^{85}-376 q^{83}-381 q^{81}-16 q^{79}+378 q^{77}+426 q^{75}+83 q^{73}-346 q^{71}-472 q^{69}-180 q^{67}+280 q^{65}+491 q^{63}+277 q^{61}-165 q^{59}-467 q^{57}-379 q^{55}+21 q^{53}+390 q^{51}+433 q^{49}+129 q^{47}-259 q^{45}-429 q^{43}-260 q^{41}+102 q^{39}+361 q^{37}+336 q^{35}+55 q^{33}-241 q^{31}-332 q^{29}-176 q^{27}+97 q^{25}+275 q^{23}+234 q^{21}+28 q^{19}-167 q^{17}-220 q^{15}-117 q^{13}+61 q^{11}+167 q^9+140 q^7+22 q^5-87 q^3-119 q-65 q^{-1} +23 q^{-3} +75 q^{-5} +67 q^{-7} +15 q^{-9} -32 q^{-11} -45 q^{-13} -29 q^{-15} +4 q^{-17} +25 q^{-19} +22 q^{-21} +5 q^{-23} -6 q^{-25} -11 q^{-27} -9 q^{-29} + q^{-31} +5 q^{-33} +3 q^{-35} + q^{-37} -2 q^{-41} - q^{-43} + q^{-45} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2 q^{10}-q^8+q^6+q^4+1 }[/math]
1,1	[math]\displaystyle{ q^{76}-4 q^{74}+8 q^{72}-12 q^{70}+22 q^{68}-36 q^{66}+46 q^{64}-56 q^{62}+72 q^{60}-84 q^{58}+84 q^{56}-82 q^{54}+77 q^{52}-62 q^{50}+32 q^{48}+4 q^{46}-43 q^{44}+90 q^{42}-132 q^{40}+168 q^{38}-192 q^{36}+198 q^{34}-190 q^{32}+158 q^{30}-126 q^{28}+82 q^{26}-30 q^{24}-18 q^{22}+57 q^{20}-82 q^{18}+106 q^{16}-110 q^{14}+104 q^{12}-84 q^{10}+70 q^8-48 q^6+29 q^4-16 q^2+8-2 q^{-2} + q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{70}-q^{66}+q^{62}-3 q^{58}+q^{56}+2 q^{54}-3 q^{52}-q^{50}+4 q^{48}+3 q^{46}-4 q^{44}-q^{42}+3 q^{40}-2 q^{38}-4 q^{36}+2 q^{34}+q^{32}-4 q^{30}+q^{28}+2 q^{26}-2 q^{24}-2 q^{22}+4 q^{20}+3 q^{18}-2 q^{16}+q^{14}+5 q^{12}-2 q^8+q^6+2 q^4-1+ q^{-4} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{62}-2 q^{60}-q^{58}+5 q^{56}-3 q^{54}-3 q^{52}+8 q^{50}-3 q^{48}-6 q^{46}+6 q^{44}-2 q^{42}-6 q^{40}+4 q^{38}+q^{36}-2 q^{34}-q^{32}+3 q^{30}+3 q^{28}-6 q^{26}+3 q^{24}+5 q^{22}-7 q^{20}+5 q^{16}-5 q^{14}+2 q^{12}+5 q^{10}-2 q^8+2 q^6+2 q^4-q^2+1 }[/math]
1,0,0	[math]\displaystyle{ -q^{37}-q^{33}+q^{31}-q^{29}+2 q^{27}-q^{25}+q^{23}-q^{15}+q^{13}-q^{11}+2 q^9+2 q^5+q }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{80}-2 q^{76}-q^{74}+2 q^{72}+2 q^{70}-2 q^{68}+5 q^{64}-6 q^{60}+2 q^{56}-6 q^{54}-3 q^{52}+3 q^{50}+q^{48}-4 q^{46}+3 q^{44}+5 q^{42}-4 q^{40}-q^{38}+7 q^{36}-7 q^{32}+4 q^{30}+3 q^{28}-4 q^{26}-3 q^{24}+3 q^{22}+q^{20}-2 q^{18}+2 q^{16}+4 q^{14}+q^{12}+q^{10}+3 q^8+q^6+q^2 }[/math]
1,0,0,0	[math]\displaystyle{ -q^{46}-q^{42}-q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30}+q^{28}-q^{22}-2 q^{18}+q^{16}-q^{14}+2 q^{12}+q^{10}+q^8+2 q^6+q^2 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{62}+2 q^{60}-3 q^{58}+5 q^{56}-7 q^{54}+7 q^{52}-8 q^{50}+7 q^{48}-6 q^{46}+4 q^{44}-4 q^{40}+8 q^{38}-11 q^{36}+14 q^{34}-15 q^{32}+15 q^{30}-13 q^{28}+10 q^{26}-7 q^{24}+3 q^{22}+q^{20}-4 q^{18}+7 q^{16}-7 q^{14}+8 q^{12}-7 q^{10}+6 q^8-4 q^6+4 q^4-q^2+1 }[/math]
1,0	[math]\displaystyle{ q^{100}-2 q^{96}-2 q^{94}+q^{92}+5 q^{90}+2 q^{88}-5 q^{86}-5 q^{84}+2 q^{82}+8 q^{80}+2 q^{78}-7 q^{76}-6 q^{74}+3 q^{72}+7 q^{70}-q^{68}-7 q^{66}-2 q^{64}+6 q^{62}+4 q^{60}-5 q^{58}-5 q^{56}+3 q^{54}+5 q^{52}-2 q^{50}-5 q^{48}+q^{46}+5 q^{44}-6 q^{40}-q^{38}+7 q^{36}+5 q^{34}-5 q^{32}-8 q^{30}+q^{28}+9 q^{26}+3 q^{24}-6 q^{22}-6 q^{20}+4 q^{18}+7 q^{16}+q^{14}-4 q^{12}-2 q^{10}+3 q^8+3 q^6-q^4-q^2+ q^{-2} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{86}-2 q^{84}+q^{82}-2 q^{80}+5 q^{78}-5 q^{76}+4 q^{74}-5 q^{72}+8 q^{70}-5 q^{68}+4 q^{66}-6 q^{64}+3 q^{62}-2 q^{60}-2 q^{58}-6 q^{54}+7 q^{52}-8 q^{50}+11 q^{48}-11 q^{46}+13 q^{44}-9 q^{42}+12 q^{40}-9 q^{38}+8 q^{36}-5 q^{34}+3 q^{32}-3 q^{30}-2 q^{28}+2 q^{26}-5 q^{24}+4 q^{22}-6 q^{20}+8 q^{18}-4 q^{16}+7 q^{14}-3 q^{12}+6 q^{10}-2 q^8+3 q^6-q^4+q^2 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+2 q^{140}-q^{138}-2 q^{136}+9 q^{134}-12 q^{132}+15 q^{130}-13 q^{128}+5 q^{126}+2 q^{124}-13 q^{122}+23 q^{120}-27 q^{118}+23 q^{116}-13 q^{114}-q^{112}+15 q^{110}-23 q^{108}+25 q^{106}-21 q^{104}+5 q^{102}+7 q^{100}-17 q^{98}+16 q^{96}-5 q^{94}-9 q^{92}+23 q^{90}-24 q^{88}+14 q^{86}+3 q^{84}-26 q^{82}+44 q^{80}-44 q^{78}+30 q^{76}-4 q^{74}-21 q^{72}+43 q^{70}-45 q^{68}+33 q^{66}-17 q^{64}-6 q^{62}+22 q^{60}-28 q^{58}+22 q^{56}-6 q^{54}-9 q^{52}+19 q^{50}-19 q^{48}+7 q^{46}+8 q^{44}-23 q^{42}+31 q^{40}-28 q^{38}+12 q^{36}+10 q^{34}-26 q^{32}+36 q^{30}-30 q^{28}+17 q^{26}-q^{24}-13 q^{22}+21 q^{20}-19 q^{18}+14 q^{16}-3 q^{14}-2 q^{12}+5 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/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 20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-9 t+11-9 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]=

{ 41, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -5 q^{-3} +7 q^{-4} -7 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}-z^2 a^{10}+5 z^5 a^9-5 z^3 a^9+2 z a^9+5 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+3 z^7 a^7-7 z^3 a^7+2 z a^7+z^8 a^6+5 z^6 a^6-16 z^4 a^6+10 z^2 a^6-2 a^6+5 z^7 a^5-12 z^5 a^5+5 z^3 a^5+z^8 a^4+z^6 a^4-11 z^4 a^4+11 z^2 a^4-2 a^4+2 z^7 a^3-7 z^5 a^3+6 z^3 a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2 }[/math]