10 161

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-2 t+3-2 t^{-1} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+6 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 5, -4 }
Jones polynomial	[math]\displaystyle{ q^{-3} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-a^{10}-z^2 a^8-a^8+z^6 a^6+6 z^4 a^6+9 z^2 a^6+3 a^6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-4 z^3 a^{13}+3 z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^5 a^{11}-3 z^3 a^{11}+z a^{11}+z^4 a^{10}-3 z^2 a^{10}+a^{10}-z^4 a^8+3 z^2 a^8-a^8-z^3 a^7+2 z a^7+z^6 a^6-6 z^4 a^6+9 z^2 a^6-3 a^6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}-q^{30}-q^{28}+q^{22}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{176}+q^{172}-q^{170}+q^{166}-q^{164}+q^{162}+q^{160}-q^{158}+q^{156}-q^{154}-q^{152}+2 q^{150}-4 q^{148}-2 q^{142}+q^{140}-2 q^{138}-q^{136}-2 q^{132}-q^{130}-q^{126}+q^{124}+q^{118}+q^{116}-q^{112}+2 q^{110}-q^{108}+2 q^{106}-2 q^{102}+2 q^{100}-q^{98}-q^{96}-2 q^{92}+q^{88}-2 q^{86}+2 q^{84}+q^{78}-q^{76}+2 q^{74}+2 q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+2 q^{62}+q^{58}+q^{56}+q^{52}+q^{50} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_161.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}+q^{11}+q^7+q^5 }[/math]
2	[math]\displaystyle{ q^{64}-q^{60}+q^{56}-q^{52}+q^{48}-q^{46}-q^{44}-q^{40}-q^{32}+q^{28}+q^{22}+q^{20}+q^{14}+q^{12}+q^{10} }[/math]
3	[math]\displaystyle{ -q^{123}+q^{119}+q^{117}-2 q^{113}-q^{111}+q^{109}+2 q^{107}+q^{105}-q^{103}-2 q^{101}-q^{99}+2 q^{97}+2 q^{95}-q^{93}-2 q^{91}+q^{89}+3 q^{87}-q^{83}+q^{81}+q^{79}-q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}-q^{65}+q^{61}-2 q^{57}-q^{55}+3 q^{53}+2 q^{51}-2 q^{49}-3 q^{47}-q^{45}+3 q^{43}+q^{41}-q^{39}-q^{37}+2 q^{33}+q^{31}+q^{29}+q^{21}+q^{19}+q^{17}+q^{15} }[/math]
5	[math]\displaystyle{ -q^{295}+q^{291}+q^{289}+q^{287}-q^{283}-3 q^{281}-2 q^{279}+2 q^{275}+4 q^{273}+4 q^{271}+q^{269}-3 q^{267}-4 q^{265}-5 q^{263}-4 q^{261}+5 q^{257}+7 q^{255}+7 q^{253}+3 q^{251}-5 q^{249}-10 q^{247}-10 q^{245}-3 q^{243}+6 q^{241}+14 q^{239}+13 q^{237}+2 q^{235}-11 q^{233}-16 q^{231}-11 q^{229}+13 q^{225}+15 q^{223}+8 q^{221}-5 q^{219}-15 q^{217}-12 q^{215}-2 q^{213}+11 q^{211}+14 q^{209}+5 q^{207}-7 q^{205}-13 q^{203}-7 q^{201}+3 q^{199}+9 q^{197}+6 q^{195}-2 q^{193}-7 q^{191}-5 q^{189}+2 q^{187}+4 q^{185}+2 q^{183}-q^{181}-2 q^{179}+2 q^{175}+2 q^{173}+q^{171}+q^{169}+q^{167}+q^{165}+q^{163}+q^{161}-q^{157}-q^{155}-q^{153}-q^{151}+2 q^{149}+4 q^{147}+2 q^{145}-q^{143}-7 q^{141}-9 q^{139}+11 q^{135}+14 q^{133}+3 q^{131}-11 q^{129}-18 q^{127}-11 q^{125}+6 q^{123}+18 q^{121}+15 q^{119}+q^{117}-12 q^{115}-17 q^{113}-12 q^{111}+q^{109}+11 q^{107}+12 q^{105}+6 q^{103}-2 q^{101}-9 q^{99}-11 q^{97}-6 q^{95}+q^{93}+7 q^{91}+8 q^{89}+6 q^{87}-5 q^{83}-5 q^{81}-3 q^{79}-q^{77}+q^{75}+3 q^{73}+2 q^{71}+2 q^{69}-q^{65}-2 q^{63}-2 q^{61}-q^{59}+2 q^{55}+2 q^{53}+2 q^{51}+q^{49}+q^{47}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25} }[/math]
6	[math]\displaystyle{ q^{408}-q^{404}-q^{402}-q^{400}+2 q^{394}+3 q^{392}+2 q^{390}-2 q^{386}-4 q^{384}-5 q^{382}-3 q^{380}+4 q^{376}+6 q^{374}+7 q^{372}+5 q^{370}+q^{368}-4 q^{366}-8 q^{364}-10 q^{362}-9 q^{360}-4 q^{358}+3 q^{356}+12 q^{354}+15 q^{352}+14 q^{350}+7 q^{348}-5 q^{346}-18 q^{344}-24 q^{342}-18 q^{340}-6 q^{338}+12 q^{336}+28 q^{334}+31 q^{332}+17 q^{330}-3 q^{328}-22 q^{326}-34 q^{324}-31 q^{322}-10 q^{320}+15 q^{318}+32 q^{316}+38 q^{314}+23 q^{312}-4 q^{310}-31 q^{308}-40 q^{306}-30 q^{304}-5 q^{302}+26 q^{300}+42 q^{298}+34 q^{296}+6 q^{294}-22 q^{292}-39 q^{290}-31 q^{288}-5 q^{286}+22 q^{284}+36 q^{282}+24 q^{280}+q^{278}-23 q^{276}-29 q^{274}-14 q^{272}+7 q^{270}+21 q^{268}+17 q^{266}+2 q^{264}-12 q^{262}-16 q^{260}-8 q^{258}+4 q^{256}+9 q^{254}+5 q^{252}-q^{250}-6 q^{248}-5 q^{246}+3 q^{242}+3 q^{240}-q^{234}+q^{230}+q^{228}-q^{224}+q^{220}+q^{218}+q^{216}+q^{214}+3 q^{212}+3 q^{210}-2 q^{206}-3 q^{204}-6 q^{202}-4 q^{200}+5 q^{198}+14 q^{196}+14 q^{194}+5 q^{192}-9 q^{190}-25 q^{188}-25 q^{186}-5 q^{184}+22 q^{182}+37 q^{180}+31 q^{178}+6 q^{176}-30 q^{174}-48 q^{172}-36 q^{170}-2 q^{168}+31 q^{166}+48 q^{164}+40 q^{162}+6 q^{160}-28 q^{158}-46 q^{156}-38 q^{154}-16 q^{152}+15 q^{150}+36 q^{148}+36 q^{146}+21 q^{144}-3 q^{142}-21 q^{140}-32 q^{138}-25 q^{136}-9 q^{134}+8 q^{132}+18 q^{130}+19 q^{128}+14 q^{126}+2 q^{124}-7 q^{122}-13 q^{120}-12 q^{118}-9 q^{116}-4 q^{114}+2 q^{112}+7 q^{110}+9 q^{108}+7 q^{106}+5 q^{104}-5 q^{100}-6 q^{98}-5 q^{96}-3 q^{94}-q^{92}+2 q^{90}+4 q^{88}+3 q^{86}+2 q^{84}+2 q^{82}-2 q^{78}-2 q^{76}-2 q^{74}-2 q^{72}-q^{70}+2 q^{66}+2 q^{64}+2 q^{62}+2 q^{60}+q^{58}+q^{56}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{34}-q^{30}-q^{28}+q^{22}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10} }[/math]
1,1	[math]\displaystyle{ q^{92}+2 q^{88}-2 q^{86}-2 q^{80}+4 q^{78}-6 q^{76}+4 q^{74}-2 q^{72}+2 q^{70}-2 q^{66}+4 q^{64}-2 q^{62}+4 q^{60}-4 q^{58}+4 q^{56}-6 q^{54}-4 q^{50}-6 q^{48}-5 q^{44}+2 q^{42}-2 q^{40}+2 q^{38}+4 q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+3 q^{28}+4 q^{26}+2 q^{24}+2 q^{22}+q^{20} }[/math]
2,0	[math]\displaystyle{ q^{86}+q^{84}-q^{72}+q^{70}+q^{66}-q^{62}-q^{60}-2 q^{58}-3 q^{56}-4 q^{54}-2 q^{52}-2 q^{50}+q^{44}+2 q^{42}+2 q^{40}+2 q^{38}+q^{36}+q^{34}+2 q^{32}+q^{30}+q^{28}+q^{26}+2 q^{24}+q^{22}+q^{20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{74}+q^{70}-q^{62}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-2 q^{44}-q^{42}-q^{40}+q^{36}+q^{34}+3 q^{32}+2 q^{30}+3 q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{20} }[/math]
1,0,0	[math]\displaystyle{ -q^{45}-q^{41}-q^{39}-q^{37}-q^{35}+q^{29}+q^{27}+q^{25}+q^{23}+2 q^{21}+q^{19}+q^{17}+q^{15} }[/math]
1,0,1	[math]\displaystyle{ q^{120}+2 q^{116}+q^{112}-2 q^{110}-q^{108}-2 q^{106}-q^{104}+2 q^{102}-3 q^{100}+4 q^{98}-2 q^{96}+4 q^{94}-3 q^{92}+q^{90}-2 q^{86}+5 q^{84}-q^{82}+8 q^{80}-q^{78}+8 q^{76}-q^{74}-3 q^{72}-3 q^{70}-10 q^{68}-8 q^{66}-13 q^{64}-6 q^{62}-9 q^{60}-3 q^{58}-3 q^{56}-q^{54}+3 q^{52}+3 q^{50}+7 q^{48}+5 q^{46}+7 q^{44}+8 q^{42}+6 q^{40}+7 q^{38}+5 q^{36}+3 q^{34}+2 q^{32}+q^{30} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ 2 q^{92}+2 q^{90}+2 q^{88}+2 q^{86}+2 q^{84}-2 q^{80}-3 q^{78}-3 q^{76}-4 q^{74}-4 q^{72}-2 q^{70}-2 q^{68}-2 q^{66}-q^{62}-2 q^{60}-2 q^{58}-2 q^{56}-2 q^{54}-2 q^{52}+q^{50}+3 q^{48}+4 q^{46}+4 q^{44}+6 q^{42}+4 q^{40}+4 q^{38}+3 q^{36}+2 q^{34}+q^{32}+q^{30} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{56}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}+q^{36}+q^{34}+2 q^{32}+q^{30}+2 q^{28}+2 q^{26}+q^{24}+q^{22}+q^{20} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{74}-q^{70}+q^{62}+q^{58}-q^{56}+q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{42}+q^{40}+q^{36}+q^{34}+q^{32}+q^{28}+2 q^{26}+q^{22}+q^{20} }[/math]
1,0	[math]\displaystyle{ q^{120}+q^{112}-q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{84}-q^{82}-q^{78}-q^{74}-q^{72}-q^{70}-q^{64}+q^{60}+q^{58}-q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+2 q^{42}+q^{38}+q^{36}+q^{34}+q^{30} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{102}+q^{98}+q^{94}-q^{90}-q^{86}-q^{82}-2 q^{78}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}-q^{64}-q^{62}-2 q^{60}-q^{58}-2 q^{56}-q^{52}+q^{50}+2 q^{48}+2 q^{46}+3 q^{44}+4 q^{42}+3 q^{40}+2 q^{38}+3 q^{36}+q^{34}+q^{32}+q^{30} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{176}+q^{172}-q^{170}+q^{166}-q^{164}+q^{162}+q^{160}-q^{158}+q^{156}-q^{154}-q^{152}+2 q^{150}-4 q^{148}-2 q^{142}+q^{140}-2 q^{138}-q^{136}-2 q^{132}-q^{130}-q^{126}+q^{124}+q^{118}+q^{116}-q^{112}+2 q^{110}-q^{108}+2 q^{106}-2 q^{102}+2 q^{100}-q^{98}-q^{96}-2 q^{92}+q^{88}-2 q^{86}+2 q^{84}+q^{78}-q^{76}+2 q^{74}+2 q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+2 q^{62}+q^{58}+q^{56}+q^{52}+q^{50} }[/math]

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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["10 161"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+6 z^4+7 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]=

{ 5, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-3} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^5 a^{13}-4 z^3 a^{13}+3 z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^5 a^{11}-3 z^3 a^{11}+z a^{11}+z^4 a^{10}-3 z^2 a^{10}+a^{10}-z^4 a^8+3 z^2 a^8-a^8-z^3 a^7+2 z a^7+z^6 a^6-6 z^4 a^6+9 z^2 a^6-3 a^6 }[/math]