10 29

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-7 t^2+15 t-17+15 t^{-1} -7 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6-z^4-4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 63, -2 }
Jones polynomial	[math]\displaystyle{ q^3-2 q^2+5 q-7+9 q^{-1} -11 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6+a^6-2 z^4 a^4-4 z^2 a^4-a^4+z^6 a^2+3 z^4 a^2+3 z^2 a^2+a^2-2 z^4-5 z^2-2+z^2 a^{-2} +2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+5 a^2 z^8+2 z^8+5 a^5 z^7+5 a^3 z^7+2 a z^7+2 z^7 a^{-1} +5 a^6 z^6-9 a^2 z^6+z^6 a^{-2} -3 z^6+3 a^7 z^5-7 a^5 z^5-12 a^3 z^5-8 a z^5-6 z^5 a^{-1} +a^8 z^4-7 a^6 z^4-5 a^4 z^4+3 a^2 z^4-4 z^4 a^{-2} -4 z^4-3 a^7 z^3+6 a^5 z^3+7 a^3 z^3+2 a z^3+4 z^3 a^{-1} -a^8 z^2+4 a^6 z^2+4 a^4 z^2+5 z^2 a^{-2} +6 z^2-2 a^5 z+2 a z-a^6-a^4-a^2-2 a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}-q^{18}+2 q^{16}-q^{14}+q^{12}+q^{10}-2 q^8+q^6-3 q^4+q^2- q^{-2} +2 q^{-4} + q^{-8} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+5 q^{106}-3 q^{104}-2 q^{102}+12 q^{100}-20 q^{98}+28 q^{96}-29 q^{94}+16 q^{92}+q^{90}-25 q^{88}+50 q^{86}-63 q^{84}+64 q^{82}-42 q^{80}+5 q^{78}+38 q^{76}-70 q^{74}+85 q^{72}-76 q^{70}+41 q^{68}+4 q^{66}-46 q^{64}+69 q^{62}-55 q^{60}+21 q^{58}+25 q^{56}-55 q^{54}+52 q^{52}-24 q^{50}-32 q^{48}+83 q^{46}-109 q^{44}+97 q^{42}-42 q^{40}-34 q^{38}+103 q^{36}-137 q^{34}+128 q^{32}-82 q^{30}+9 q^{28}+56 q^{26}-94 q^{24}+105 q^{22}-71 q^{20}+17 q^{18}+34 q^{16}-62 q^{14}+53 q^{12}-22 q^{10}-28 q^8+64 q^6-75 q^4+52 q^2-5-52 q^{-2} +93 q^{-4} -99 q^{-6} +70 q^{-8} -25 q^{-10} -28 q^{-12} +64 q^{-14} -74 q^{-16} +66 q^{-18} -35 q^{-20} +6 q^{-22} +19 q^{-24} -30 q^{-26} +30 q^{-28} -21 q^{-30} +12 q^{-32} - q^{-34} -4 q^{-36} +6 q^{-38} -5 q^{-40} +4 q^{-42} - q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_29.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-2 q^{13}+3 q^{11}-2 q^9+2 q^7-q^5-2 q^3+2 q-2 q^{-1} +3 q^{-3} - q^{-5} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{42}-2 q^{40}+6 q^{36}-7 q^{34}-3 q^{32}+14 q^{30}-11 q^{28}-7 q^{26}+20 q^{24}-9 q^{22}-12 q^{20}+14 q^{18}-10 q^{14}+q^{12}+11 q^{10}-q^8-12 q^6+13 q^4+6 q^2-19+8 q^{-2} +11 q^{-4} -16 q^{-6} + q^{-8} +11 q^{-10} -7 q^{-12} -2 q^{-14} +5 q^{-16} - q^{-18} - q^{-20} + q^{-22} }[/math]
3	[math]\displaystyle{ q^{81}-2 q^{79}+3 q^{75}+q^{73}-6 q^{71}-4 q^{69}+12 q^{67}+6 q^{65}-18 q^{63}-9 q^{61}+26 q^{59}+17 q^{57}-40 q^{55}-27 q^{53}+50 q^{51}+43 q^{49}-56 q^{47}-60 q^{45}+53 q^{43}+73 q^{41}-38 q^{39}-80 q^{37}+19 q^{35}+74 q^{33}+7 q^{31}-61 q^{29}-27 q^{27}+39 q^{25}+50 q^{23}-19 q^{21}-61 q^{19}-4 q^{17}+65 q^{15}+24 q^{13}-72 q^{11}-40 q^9+65 q^7+60 q^5-56 q^3-68 q+39 q^{-1} +79 q^{-3} -18 q^{-5} -77 q^{-7} -3 q^{-9} +68 q^{-11} +20 q^{-13} -50 q^{-15} -30 q^{-17} +30 q^{-19} +31 q^{-21} -16 q^{-23} -23 q^{-25} +3 q^{-27} +16 q^{-29} + q^{-31} -8 q^{-33} -2 q^{-35} +4 q^{-37} + q^{-39} - q^{-41} - q^{-43} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{132}-2 q^{130}+3 q^{126}-2 q^{124}+2 q^{122}-7 q^{120}+q^{118}+11 q^{116}-6 q^{114}+6 q^{112}-19 q^{110}+2 q^{108}+30 q^{106}-14 q^{104}-3 q^{102}-46 q^{100}+21 q^{98}+87 q^{96}-12 q^{94}-50 q^{92}-134 q^{90}+30 q^{88}+214 q^{86}+75 q^{84}-101 q^{82}-313 q^{80}-57 q^{78}+333 q^{76}+269 q^{74}-34 q^{72}-461 q^{70}-255 q^{68}+285 q^{66}+419 q^{64}+160 q^{62}-402 q^{60}-399 q^{58}+57 q^{56}+361 q^{54}+324 q^{52}-153 q^{50}-355 q^{48}-173 q^{46}+143 q^{44}+332 q^{42}+115 q^{40}-184 q^{38}-304 q^{36}-73 q^{34}+252 q^{32}+297 q^{30}-25 q^{28}-355 q^{26}-213 q^{24}+161 q^{22}+412 q^{20}+109 q^{18}-366 q^{16}-331 q^{14}+39 q^{12}+464 q^{10}+260 q^8-277 q^6-408 q^4-155 q^2+388+390 q^{-2} -58 q^{-4} -350 q^{-6} -340 q^{-8} +159 q^{-10} +370 q^{-12} +169 q^{-14} -137 q^{-16} -359 q^{-18} -73 q^{-20} +184 q^{-22} +222 q^{-24} +70 q^{-26} -201 q^{-28} -142 q^{-30} -2 q^{-32} +119 q^{-34} +122 q^{-36} -45 q^{-38} -75 q^{-40} -54 q^{-42} +18 q^{-44} +65 q^{-46} +8 q^{-48} -11 q^{-50} -28 q^{-52} -9 q^{-54} +18 q^{-56} +4 q^{-58} +3 q^{-60} -6 q^{-62} -4 q^{-64} +4 q^{-66} + q^{-70} - q^{-72} - q^{-74} + q^{-76} }[/math]
5	[math]\displaystyle{ q^{195}-2 q^{193}+3 q^{189}-2 q^{187}-q^{185}+q^{183}-2 q^{181}+6 q^{177}-6 q^{173}+q^{171}+q^{169}+q^{167}-4 q^{163}-11 q^{161}-q^{159}+30 q^{157}+34 q^{155}-q^{153}-61 q^{151}-92 q^{149}-36 q^{147}+110 q^{145}+218 q^{143}+123 q^{141}-159 q^{139}-397 q^{137}-311 q^{135}+140 q^{133}+635 q^{131}+650 q^{129}-24 q^{127}-879 q^{125}-1091 q^{123}-284 q^{121}+1018 q^{119}+1635 q^{117}+779 q^{115}-995 q^{113}-2119 q^{111}-1417 q^{109}+706 q^{107}+2428 q^{105}+2102 q^{103}-177 q^{101}-2458 q^{99}-2646 q^{97}-501 q^{95}+2128 q^{93}+2928 q^{91}+1203 q^{89}-1532 q^{87}-2868 q^{85}-1741 q^{83}+758 q^{81}+2478 q^{79}+2045 q^{77}+6 q^{75}-1848 q^{73}-2075 q^{71}-666 q^{69}+1146 q^{67}+1895 q^{65}+1110 q^{63}-450 q^{61}-1588 q^{59}-1427 q^{57}-91 q^{55}+1289 q^{53}+1588 q^{51}+518 q^{49}-1036 q^{47}-1728 q^{45}-845 q^{43}+888 q^{41}+1871 q^{39}+1111 q^{37}-771 q^{35}-2032 q^{33}-1425 q^{31}+627 q^{29}+2220 q^{27}+1755 q^{25}-405 q^{23}-2289 q^{21}-2148 q^{19}+16 q^{17}+2261 q^{15}+2492 q^{13}+485 q^{11}-1964 q^9-2717 q^7-1094 q^5+1466 q^3+2719 q+1651 q^{-1} -750 q^{-3} -2444 q^{-5} -2049 q^{-7} -29 q^{-9} +1883 q^{-11} +2177 q^{-13} +742 q^{-15} -1153 q^{-17} -1994 q^{-19} -1223 q^{-21} +380 q^{-23} +1533 q^{-25} +1421 q^{-27} +263 q^{-29} -948 q^{-31} -1307 q^{-33} -658 q^{-35} +363 q^{-37} +975 q^{-39} +806 q^{-41} +74 q^{-43} -586 q^{-45} -708 q^{-47} -307 q^{-49} +220 q^{-51} +502 q^{-53} +374 q^{-55} - q^{-57} -277 q^{-59} -297 q^{-61} -111 q^{-63} +106 q^{-65} +193 q^{-67} +120 q^{-69} -12 q^{-71} -95 q^{-73} -90 q^{-75} -20 q^{-77} +35 q^{-79} +46 q^{-81} +27 q^{-83} -8 q^{-85} -24 q^{-87} -13 q^{-89} +2 q^{-91} +4 q^{-93} +7 q^{-95} +3 q^{-97} -5 q^{-99} -2 q^{-101} +2 q^{-103} + q^{-109} - q^{-111} - q^{-113} + q^{-115} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}-q^{18}+2 q^{16}-q^{14}+q^{12}+q^{10}-2 q^8+q^6-3 q^4+q^2- q^{-2} +2 q^{-4} + q^{-8} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+38 q^{52}-64 q^{50}+100 q^{48}-142 q^{46}+189 q^{44}-242 q^{42}+290 q^{40}-320 q^{38}+334 q^{36}-322 q^{34}+274 q^{32}-186 q^{30}+56 q^{28}+94 q^{26}-264 q^{24}+430 q^{22}-575 q^{20}+680 q^{18}-726 q^{16}+718 q^{14}-644 q^{12}+534 q^{10}-378 q^8+204 q^6-28 q^4-128 q^2+250-338 q^{-2} +378 q^{-4} -376 q^{-6} +336 q^{-8} -282 q^{-10} +217 q^{-12} -154 q^{-14} +102 q^{-16} -60 q^{-18} +35 q^{-20} -16 q^{-22} +8 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{56}-q^{52}+2 q^{48}+q^{46}-4 q^{44}+5 q^{40}-3 q^{38}-5 q^{36}+4 q^{34}+8 q^{32}-6 q^{30}-8 q^{28}+6 q^{26}+3 q^{24}-9 q^{22}-q^{20}+8 q^{18}-2 q^{16}-q^{14}+6 q^{12}+3 q^{10}-5 q^8+3 q^6+8 q^4-5 q^2-6+6 q^{-2} +2 q^{-4} -9 q^{-6} -3 q^{-8} +5 q^{-10} +2 q^{-12} -5 q^{-14} - q^{-16} +4 q^{-18} +2 q^{-20} - q^{-22} + q^{-26} + q^{-28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-2 q^{46}+5 q^{42}-6 q^{40}-q^{38}+12 q^{36}-10 q^{34}-5 q^{32}+16 q^{30}-8 q^{28}-8 q^{26}+14 q^{24}-4 q^{22}-8 q^{20}+4 q^{18}+3 q^{16}-q^{14}-5 q^{12}+11 q^{10}+5 q^8-14 q^6+6 q^4+6 q^2-17+3 q^{-2} +7 q^{-4} -10 q^{-6} +5 q^{-8} +5 q^{-10} -3 q^{-12} +3 q^{-14} +2 q^{-16} - q^{-18} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ q^{29}+q^{25}-q^{23}+2 q^{21}-2 q^{19}+2 q^{17}-q^{15}+q^{13}-q^{11}-2 q^5+q^3-2 q+ q^{-1} -2 q^{-3} +2 q^{-5} +2 q^{-9} + q^{-11} + q^{-13} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-2 q^{46}+4 q^{44}-7 q^{42}+10 q^{40}-13 q^{38}+16 q^{36}-16 q^{34}+17 q^{32}-14 q^{30}+10 q^{28}-2 q^{26}-6 q^{24}+14 q^{22}-22 q^{20}+28 q^{18}-33 q^{16}+33 q^{14}-31 q^{12}+25 q^{10}-19 q^8+10 q^6-2 q^4-6 q^2+11-15 q^{-2} +17 q^{-4} -16 q^{-6} +15 q^{-8} -11 q^{-10} +9 q^{-12} -5 q^{-14} +4 q^{-16} - q^{-18} + q^{-20} }[/math]
1,0	[math]\displaystyle{ q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+6 q^{68}+q^{66}-8 q^{64}-7 q^{62}+6 q^{60}+14 q^{58}+q^{56}-15 q^{54}-11 q^{52}+10 q^{50}+17 q^{48}-q^{46}-17 q^{44}-6 q^{42}+13 q^{40}+11 q^{38}-9 q^{36}-13 q^{34}+4 q^{32}+12 q^{30}-q^{28}-12 q^{26}-q^{24}+11 q^{22}+5 q^{20}-9 q^{18}-4 q^{16}+11 q^{14}+10 q^{12}-9 q^{10}-15 q^8+3 q^6+17 q^4+4 q^2-17-15 q^{-2} +8 q^{-4} +17 q^{-6} -14 q^{-10} -7 q^{-12} +9 q^{-14} +9 q^{-16} - q^{-18} -6 q^{-20} - q^{-22} +4 q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+5 q^{106}-3 q^{104}-2 q^{102}+12 q^{100}-20 q^{98}+28 q^{96}-29 q^{94}+16 q^{92}+q^{90}-25 q^{88}+50 q^{86}-63 q^{84}+64 q^{82}-42 q^{80}+5 q^{78}+38 q^{76}-70 q^{74}+85 q^{72}-76 q^{70}+41 q^{68}+4 q^{66}-46 q^{64}+69 q^{62}-55 q^{60}+21 q^{58}+25 q^{56}-55 q^{54}+52 q^{52}-24 q^{50}-32 q^{48}+83 q^{46}-109 q^{44}+97 q^{42}-42 q^{40}-34 q^{38}+103 q^{36}-137 q^{34}+128 q^{32}-82 q^{30}+9 q^{28}+56 q^{26}-94 q^{24}+105 q^{22}-71 q^{20}+17 q^{18}+34 q^{16}-62 q^{14}+53 q^{12}-22 q^{10}-28 q^8+64 q^6-75 q^4+52 q^2-5-52 q^{-2} +93 q^{-4} -99 q^{-6} +70 q^{-8} -25 q^{-10} -28 q^{-12} +64 q^{-14} -74 q^{-16} +66 q^{-18} -35 q^{-20} +6 q^{-22} +19 q^{-24} -30 q^{-26} +30 q^{-28} -21 q^{-30} +12 q^{-32} - q^{-34} -4 q^{-36} +6 q^{-38} -5 q^{-40} +4 q^{-42} - q^{-44} + q^{-46} }[/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 29"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-7 t^2+15 t-17+15 t^{-1} -7 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 63, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-2 q^2+5 q-7+9 q^{-1} -11 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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