10 141

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-3 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 21, 0 }
Jones polynomial	[math]\displaystyle{ q^2-2 q+3-3 q^{-1} +4 q^{-2} -3 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^6+a^4 z^4-5 a^2 z^4+z^4+3 a^4 z^2-7 a^2 z^2+3 z^2+a^4-2 a^2+2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^4 z^8+a^2 z^8+2 a^5 z^7+3 a^3 z^7+a z^7+a^6 z^6-3 a^4 z^6-4 a^2 z^6-9 a^5 z^5-12 a^3 z^5-3 a z^5-4 a^6 z^4+a^4 z^4+8 a^2 z^4+3 z^4+10 a^5 z^3+13 a^3 z^3+5 a z^3+2 z^3 a^{-1} +3 a^6 z^2-a^4 z^2-9 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^4+2 a^2+2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}-q^{12}-q^{10}+q^8+q^4+ q^{-2} + q^{-6} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-q^{92}+2 q^{90}-3 q^{88}+q^{86}-q^{84}-2 q^{82}+6 q^{80}-7 q^{78}+6 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+4 q^{66}-3 q^{62}+7 q^{60}-q^{58}-2 q^{56}+4 q^{54}-8 q^{52}+7 q^{50}-7 q^{46}+3 q^{44}-4 q^{42}+9 q^{40}-4 q^{38}-2 q^{36}-q^{34}-3 q^{32}+7 q^{30}-6 q^{28}-q^{26}+q^{22}+5 q^{20}-3 q^{18}-3 q^{16}+5 q^{14}-5 q^{12}+4 q^{10}-6 q^6+8 q^4-3 q^2+2+2 q^{-2} -3 q^{-4} +2 q^{-6} - q^{-8} + q^{-10} +2 q^{-12} +2 q^{-18} +2 q^{-24} -2 q^{-26} + q^{-28} - q^{-30} - q^{-32} + q^{-34} - q^{-36} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_141.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-q^{11}-q^7+q^5+q^3+ q^{-1} - q^{-3} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{38}-q^{36}-2 q^{34}+2 q^{32}+q^{30}-2 q^{28}+q^{26}+3 q^{24}-q^{22}-2 q^{20}-q^{16}-2 q^{14}+q^{12}+2 q^{10}-q^8+q^6+3 q^4+q^2-2+ q^{-2} + q^{-4} -2 q^{-6} + q^{-14} }[/math]
3	[math]\displaystyle{ q^{75}-q^{73}-2 q^{71}+3 q^{67}+3 q^{65}-3 q^{63}-4 q^{61}+q^{59}+5 q^{57}+2 q^{55}-5 q^{53}-6 q^{51}+q^{49}+5 q^{47}+3 q^{45}-3 q^{43}-4 q^{41}+3 q^{39}+7 q^{37}+2 q^{35}-7 q^{33}-4 q^{31}+4 q^{29}+3 q^{27}-6 q^{25}-3 q^{23}+4 q^{21}+2 q^{19}-4 q^{17}-2 q^{15}+4 q^{13}+4 q^{11}-3 q^7-2 q^5+4 q^3+7 q-2 q^{-1} -8 q^{-3} -2 q^{-5} +8 q^{-7} +5 q^{-9} -5 q^{-11} -5 q^{-13} + q^{-15} +5 q^{-17} + q^{-19} -3 q^{-21} -2 q^{-23} +2 q^{-27} }[/math]
5	[math]\displaystyle{ q^{185}-q^{183}-2 q^{181}+q^{177}+3 q^{175}+3 q^{173}+q^{171}-5 q^{169}-7 q^{167}-4 q^{165}+q^{163}+8 q^{161}+12 q^{159}+7 q^{157}-8 q^{155}-15 q^{153}-14 q^{151}-5 q^{149}+12 q^{147}+23 q^{145}+19 q^{143}+q^{141}-16 q^{139}-27 q^{137}-23 q^{135}-q^{133}+20 q^{131}+30 q^{129}+22 q^{127}+3 q^{125}-22 q^{123}-37 q^{121}-30 q^{119}-3 q^{117}+31 q^{115}+50 q^{113}+40 q^{111}-2 q^{109}-47 q^{107}-65 q^{105}-36 q^{103}+26 q^{101}+71 q^{99}+68 q^{97}+12 q^{95}-64 q^{93}-91 q^{91}-48 q^{89}+33 q^{87}+88 q^{85}+76 q^{83}-77 q^{79}-87 q^{77}-23 q^{75}+57 q^{73}+87 q^{71}+46 q^{69}-34 q^{67}-79 q^{65}-50 q^{63}+19 q^{61}+64 q^{59}+51 q^{57}-7 q^{55}-52 q^{53}-43 q^{51}+4 q^{49}+35 q^{47}+29 q^{45}-7 q^{43}-30 q^{41}-18 q^{39}+11 q^{37}+24 q^{35}+11 q^{33}-15 q^{31}-24 q^{29}-16 q^{27}+9 q^{25}+31 q^{23}+30 q^{21}+9 q^{19}-25 q^{17}-45 q^{15}-36 q^{13}+9 q^{11}+59 q^9+71 q^7+22 q^5-55 q^3-96 q-63 q^{-1} +29 q^{-3} +110 q^{-5} +101 q^{-7} +6 q^{-9} -97 q^{-11} -115 q^{-13} -43 q^{-15} +62 q^{-17} +111 q^{-19} +63 q^{-21} -28 q^{-23} -82 q^{-25} -61 q^{-27} -2 q^{-29} +45 q^{-31} +48 q^{-33} +13 q^{-35} -19 q^{-37} -25 q^{-39} -11 q^{-41} +4 q^{-43} +9 q^{-45} +7 q^{-47} -2 q^{-49} -3 q^{-51} + q^{-53} +3 q^{-55} + q^{-57} - q^{-59} -3 q^{-61} -3 q^{-63} + q^{-65} + q^{-67} + q^{-69} }[/math]
6	[math]\displaystyle{ q^{258}-q^{256}-2 q^{254}+q^{250}+3 q^{248}+q^{246}+3 q^{244}-q^{242}-7 q^{240}-6 q^{238}-4 q^{236}+2 q^{234}+5 q^{232}+15 q^{230}+11 q^{228}+q^{226}-10 q^{224}-19 q^{222}-18 q^{220}-15 q^{218}+12 q^{216}+27 q^{214}+31 q^{212}+22 q^{210}-24 q^{206}-48 q^{204}-39 q^{202}-17 q^{200}+17 q^{198}+43 q^{196}+56 q^{194}+45 q^{192}+4 q^{190}-32 q^{188}-61 q^{186}-61 q^{184}-41 q^{182}+q^{180}+51 q^{178}+80 q^{176}+79 q^{174}+43 q^{172}-14 q^{170}-82 q^{168}-121 q^{166}-102 q^{164}-32 q^{162}+62 q^{160}+142 q^{158}+165 q^{156}+94 q^{154}-39 q^{152}-158 q^{150}-211 q^{148}-158 q^{146}-6 q^{144}+169 q^{142}+259 q^{140}+209 q^{138}+49 q^{136}-157 q^{134}-289 q^{132}-257 q^{130}-74 q^{128}+154 q^{126}+300 q^{124}+289 q^{122}+95 q^{120}-158 q^{118}-319 q^{116}-293 q^{114}-97 q^{112}+154 q^{110}+327 q^{108}+286 q^{106}+68 q^{104}-183 q^{102}-312 q^{100}-249 q^{98}-34 q^{96}+211 q^{94}+301 q^{92}+189 q^{90}-35 q^{88}-217 q^{86}-249 q^{84}-119 q^{82}+97 q^{80}+220 q^{78}+179 q^{76}+28 q^{74}-122 q^{72}-174 q^{70}-104 q^{68}+37 q^{66}+126 q^{64}+106 q^{62}+19 q^{60}-64 q^{58}-87 q^{56}-42 q^{54}+28 q^{52}+59 q^{50}+33 q^{48}-19 q^{46}-53 q^{44}-41 q^{42}+6 q^{40}+53 q^{38}+67 q^{36}+37 q^{34}-23 q^{32}-76 q^{30}-91 q^{28}-52 q^{26}+22 q^{24}+106 q^{22}+147 q^{20}+102 q^{18}-17 q^{16}-150 q^{14}-215 q^{12}-166 q^{10}+10 q^8+212 q^6+306 q^4+218 q^2-19-267 q^{-2} -389 q^{-4} -259 q^{-6} +44 q^{-8} +329 q^{-10} +419 q^{-12} +255 q^{-14} -60 q^{-16} -354 q^{-18} -407 q^{-20} -213 q^{-22} +91 q^{-24} +309 q^{-26} +335 q^{-28} +170 q^{-30} -90 q^{-32} -241 q^{-34} -237 q^{-36} -104 q^{-38} +50 q^{-40} +152 q^{-42} +156 q^{-44} +63 q^{-46} -23 q^{-48} -77 q^{-50} -78 q^{-52} -50 q^{-54} +35 q^{-58} +33 q^{-60} +28 q^{-62} +9 q^{-64} -4 q^{-66} -18 q^{-68} -15 q^{-70} -7 q^{-72} -3 q^{-74} +7 q^{-76} +8 q^{-78} +9 q^{-80} + q^{-82} -2 q^{-84} -4 q^{-86} -5 q^{-88} - q^{-90} +2 q^{-94} + q^{-96} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{18}-q^{12}-q^{10}+q^8+q^4+ q^{-2} + q^{-6} }[/math]
1,1	[math]\displaystyle{ q^{52}-2 q^{50}+4 q^{48}-8 q^{46}+11 q^{44}-16 q^{42}+18 q^{40}-14 q^{38}+12 q^{36}-4 q^{34}-4 q^{32}+14 q^{30}-23 q^{28}+22 q^{26}-26 q^{24}+24 q^{22}-21 q^{20}+18 q^{18}-8 q^{16}+4 q^{14}+3 q^{12}-10 q^{10}+12 q^8-14 q^6+14 q^4-4 q^2+4+2 q^{-2} - q^{-4} -2 q^{-8} -4 q^{-10} +5 q^{-12} -2 q^{-14} +4 q^{-16} -2 q^{-18} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{48}-q^{44}-q^{42}-q^{36}+q^{34}+3 q^{32}+4 q^{30}-q^{26}-3 q^{24}-4 q^{22}-3 q^{20}-3 q^{18}+q^{16}+q^{14}+4 q^{12}+4 q^{10}+3 q^8+2 q^6+2 q^4-3-2 q^{-2} - q^{-4} - q^{-8} +2 q^{-12} + q^{-14} + q^{-16} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}-q^{38}-2 q^{32}+q^{30}+q^{26}+2 q^{24}+q^{22}+q^{18}-2 q^{14}-2 q^{10}-q^8-2 q^6+2 q^2+1+3 q^{-2} +3 q^{-4} + q^{-10} - q^{-12} - q^{-14} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ q^{23}+q^{19}-q^{17}-q^{13}+q+2 q^{-3} + q^{-7} }[/math]
1,0,1	[math]\displaystyle{ q^{66}-2 q^{64}+3 q^{62}-3 q^{60}+q^{58}+q^{56}-6 q^{54}+8 q^{52}-7 q^{50}+9 q^{48}-4 q^{46}+4 q^{44}-5 q^{40}+8 q^{38}-15 q^{36}+12 q^{34}-18 q^{32}+13 q^{30}-8 q^{28}+3 q^{26}+11 q^{24}-9 q^{22}+17 q^{20}-13 q^{18}+16 q^{16}-17 q^{14}+8 q^{12}-7 q^{10}-7 q^8+8 q^6-8 q^4+15 q^2-5+12 q^{-2} -2 q^{-4} + q^{-6} -2 q^{-8} -3 q^{-10} -3 q^{-14} +7 q^{-16} -2 q^{-18} +3 q^{-20} + q^{-22} -2 q^{-24} + q^{-26} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{94}-q^{92}+2 q^{90}-3 q^{88}+q^{86}-q^{84}-2 q^{82}+6 q^{80}-7 q^{78}+6 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+4 q^{66}-3 q^{62}+7 q^{60}-q^{58}-2 q^{56}+4 q^{54}-8 q^{52}+7 q^{50}-7 q^{46}+3 q^{44}-4 q^{42}+9 q^{40}-4 q^{38}-2 q^{36}-q^{34}-3 q^{32}+7 q^{30}-6 q^{28}-q^{26}+q^{22}+5 q^{20}-3 q^{18}-3 q^{16}+5 q^{14}-5 q^{12}+4 q^{10}-6 q^6+8 q^4-3 q^2+2+2 q^{-2} -3 q^{-4} +2 q^{-6} - q^{-8} + q^{-10} +2 q^{-12} +2 q^{-18} +2 q^{-24} -2 q^{-26} + q^{-28} - q^{-30} - q^{-32} + q^{-34} - q^{-36} + q^{-38} }[/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["10 141"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-3 z^4-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]=

{ 21, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^4 z^8+a^2 z^8+2 a^5 z^7+3 a^3 z^7+a z^7+a^6 z^6-3 a^4 z^6-4 a^2 z^6-9 a^5 z^5-12 a^3 z^5-3 a z^5-4 a^6 z^4+a^4 z^4+8 a^2 z^4+3 z^4+10 a^5 z^3+13 a^3 z^3+5 a z^3+2 z^3 a^{-1} +3 a^6 z^2-a^4 z^2-9 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^4+2 a^2+2 }[/math]