10 19

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-7 t^2+11 t-11+11 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+5 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 51, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+2 q^3-3 q^2+6 q-7+8 q^{-1} -8 q^{-2} +7 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^2 z^6+z^6-a^4 z^4+3 a^2 z^4-z^4 a^{-2} +4 z^4-2 a^4 z^2+a^2 z^2-3 z^2 a^{-2} +5 z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+2 z^8 a^{-2} +5 z^8+5 a^3 z^7+3 a z^7-z^7 a^{-1} +z^7 a^{-3} +6 a^4 z^6-3 a^2 z^6-10 z^6 a^{-2} -19 z^6+5 a^5 z^5-7 a^3 z^5-15 a z^5-8 z^5 a^{-1} -5 z^5 a^{-3} +3 a^6 z^4-8 a^4 z^4-4 a^2 z^4+16 z^4 a^{-2} +23 z^4+a^7 z^3-4 a^5 z^3+11 a z^3+13 z^3 a^{-1} +7 z^3 a^{-3} -a^6 z^2+3 a^4 z^2-9 z^2 a^{-2} -13 z^2+a^5 z+a^3 z-2 a z-4 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{18}+q^{16}+q^{10}-2 q^8+q^6-q^4+q^2+2+2 q^{-4} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+2 q^{92}-q^{90}-2 q^{88}+8 q^{86}-11 q^{84}+14 q^{82}-13 q^{80}+7 q^{78}+q^{76}-11 q^{74}+21 q^{72}-26 q^{70}+25 q^{68}-19 q^{66}+3 q^{64}+12 q^{62}-23 q^{60}+30 q^{58}-27 q^{56}+18 q^{54}-5 q^{52}-10 q^{50}+20 q^{48}-21 q^{46}+13 q^{44}-q^{42}-12 q^{40}+18 q^{38}-13 q^{36}+q^{34}+17 q^{32}-33 q^{30}+38 q^{28}-29 q^{26}+2 q^{24}+27 q^{22}-49 q^{20}+57 q^{18}-43 q^{16}+17 q^{14}+13 q^{12}-36 q^{10}+47 q^8-42 q^6+21 q^4+4 q^2-20+28 q^{-2} -18 q^{-4} +7 q^{-6} +14 q^{-8} -24 q^{-10} +24 q^{-12} -15 q^{-14} -3 q^{-16} +28 q^{-18} -39 q^{-20} +40 q^{-22} -24 q^{-24} + q^{-26} +23 q^{-28} -38 q^{-30} +38 q^{-32} -29 q^{-34} +9 q^{-36} +7 q^{-38} -20 q^{-40} +22 q^{-42} -17 q^{-44} +9 q^{-46} - q^{-48} -4 q^{-50} +4 q^{-52} -5 q^{-54} +3 q^{-56} - q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_19.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+2 q^{11}-2 q^9+2 q^7-q^5+q- q^{-1} +3 q^{-3} - q^{-5} + q^{-7} - q^{-9} }[/math]
2	[math]\displaystyle{ q^{36}-2 q^{34}+4 q^{30}-6 q^{28}+q^{26}+7 q^{24}-9 q^{22}+q^{20}+8 q^{18}-7 q^{16}-q^{14}+7 q^{12}-q^{10}-5 q^8+2 q^6+5 q^4-5 q^2-4+9 q^{-2} - q^{-4} -8 q^{-6} +8 q^{-8} +3 q^{-10} -8 q^{-12} +5 q^{-14} +4 q^{-16} -6 q^{-18} +3 q^{-22} -2 q^{-24} - q^{-26} + q^{-28} }[/math]
3	[math]\displaystyle{ -q^{69}+2 q^{67}-2 q^{63}+3 q^{59}+q^{57}-7 q^{55}+2 q^{53}+7 q^{51}-4 q^{49}-9 q^{47}+7 q^{45}+10 q^{43}-11 q^{41}-11 q^{39}+15 q^{37}+14 q^{35}-16 q^{33}-17 q^{31}+13 q^{29}+20 q^{27}-5 q^{25}-21 q^{23}-7 q^{21}+18 q^{19}+16 q^{17}-12 q^{15}-22 q^{13}+5 q^{11}+29 q^9+2 q^7-25 q^5-10 q^3+23 q+14 q^{-1} -19 q^{-3} -19 q^{-5} +14 q^{-7} +23 q^{-9} -8 q^{-11} -25 q^{-13} +2 q^{-15} +28 q^{-17} +5 q^{-19} -24 q^{-21} -14 q^{-23} +21 q^{-25} +18 q^{-27} -11 q^{-29} -21 q^{-31} +4 q^{-33} +17 q^{-35} +3 q^{-37} -12 q^{-39} -7 q^{-41} +6 q^{-43} +6 q^{-45} -2 q^{-47} -4 q^{-49} +2 q^{-53} + q^{-55} - q^{-57} }[/math]
4	[math]\displaystyle{ q^{112}-2 q^{110}+2 q^{106}-2 q^{104}+3 q^{102}-5 q^{100}+2 q^{98}+5 q^{96}-6 q^{94}+8 q^{92}-10 q^{90}+2 q^{88}+6 q^{86}-13 q^{84}+18 q^{82}-4 q^{80}-11 q^{76}-29 q^{74}+40 q^{72}+26 q^{70}+5 q^{68}-44 q^{66}-69 q^{64}+48 q^{62}+75 q^{60}+42 q^{58}-62 q^{56}-122 q^{54}+11 q^{52}+89 q^{50}+96 q^{48}-14 q^{46}-123 q^{44}-56 q^{42}+26 q^{40}+100 q^{38}+64 q^{36}-44 q^{34}-80 q^{32}-66 q^{30}+36 q^{28}+99 q^{26}+51 q^{24}-48 q^{22}-106 q^{20}-32 q^{18}+77 q^{16}+97 q^{14}-12 q^{12}-96 q^{10}-57 q^8+52 q^6+103 q^4-77-68 q^{-2} +29 q^{-4} +106 q^{-6} +20 q^{-8} -57 q^{-10} -92 q^{-12} -12 q^{-14} +101 q^{-16} +59 q^{-18} -4 q^{-20} -97 q^{-22} -75 q^{-24} +51 q^{-26} +80 q^{-28} +70 q^{-30} -49 q^{-32} -105 q^{-34} -28 q^{-36} +37 q^{-38} +102 q^{-40} +29 q^{-42} -62 q^{-44} -67 q^{-46} -34 q^{-48} +61 q^{-50} +62 q^{-52} +8 q^{-54} -36 q^{-56} -59 q^{-58} +2 q^{-60} +32 q^{-62} +31 q^{-64} +8 q^{-66} -31 q^{-68} -17 q^{-70} -2 q^{-72} +14 q^{-74} +16 q^{-76} -5 q^{-78} -6 q^{-80} -6 q^{-82} +6 q^{-86} + q^{-88} -2 q^{-92} - q^{-94} + q^{-96} }[/math]
5	[math]\displaystyle{ -q^{165}+2 q^{163}-2 q^{159}+2 q^{157}-q^{155}-q^{153}+2 q^{151}-4 q^{147}+2 q^{143}+2 q^{141}+4 q^{139}+2 q^{137}-4 q^{135}-14 q^{133}-7 q^{131}+10 q^{129}+19 q^{127}+27 q^{125}-51 q^{121}-59 q^{119}-4 q^{117}+76 q^{115}+110 q^{113}+37 q^{111}-115 q^{109}-192 q^{107}-77 q^{105}+152 q^{103}+280 q^{101}+152 q^{99}-166 q^{97}-387 q^{95}-260 q^{93}+150 q^{91}+477 q^{89}+387 q^{87}-77 q^{85}-517 q^{83}-524 q^{81}-49 q^{79}+497 q^{77}+622 q^{75}+207 q^{73}-382 q^{71}-648 q^{69}-370 q^{67}+195 q^{65}+578 q^{63}+484 q^{61}+21 q^{59}-412 q^{57}-513 q^{55}-235 q^{53}+195 q^{51}+457 q^{49}+380 q^{47}+28 q^{45}-330 q^{43}-440 q^{41}-214 q^{39}+188 q^{37}+435 q^{35}+323 q^{33}-68 q^{31}-385 q^{29}-358 q^{27}-17 q^{25}+336 q^{23}+362 q^{21}+34 q^{19}-310 q^{17}-338 q^{15}-39 q^{13}+309 q^{11}+349 q^9+28 q^7-329 q^5-372 q^3-47 q+346 q^{-1} +422 q^{-3} +103 q^{-5} -326 q^{-7} -471 q^{-9} -194 q^{-11} +263 q^{-13} +494 q^{-15} +305 q^{-17} -144 q^{-19} -470 q^{-21} -407 q^{-23} -13 q^{-25} +381 q^{-27} +463 q^{-29} +180 q^{-31} -231 q^{-33} -447 q^{-35} -325 q^{-37} +42 q^{-39} +363 q^{-41} +395 q^{-43} +144 q^{-45} -197 q^{-47} -391 q^{-49} -290 q^{-51} +24 q^{-53} +294 q^{-55} +339 q^{-57} +153 q^{-59} -141 q^{-61} -317 q^{-63} -250 q^{-65} -14 q^{-67} +204 q^{-69} +271 q^{-71} +141 q^{-73} -78 q^{-75} -216 q^{-77} -191 q^{-79} -40 q^{-81} +119 q^{-83} +177 q^{-85} +107 q^{-87} -24 q^{-89} -118 q^{-91} -119 q^{-93} -38 q^{-95} +50 q^{-97} +88 q^{-99} +63 q^{-101} + q^{-103} -48 q^{-105} -56 q^{-107} -22 q^{-109} +14 q^{-111} +32 q^{-113} +27 q^{-115} +4 q^{-117} -16 q^{-119} -17 q^{-121} -6 q^{-123} +2 q^{-125} +8 q^{-127} +8 q^{-129} -4 q^{-133} -3 q^{-135} - q^{-137} +2 q^{-141} + q^{-143} - q^{-145} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{18}+q^{16}+q^{10}-2 q^8+q^6-q^4+q^2+2+2 q^{-4} - q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{52}-4 q^{50}+8 q^{48}-12 q^{46}+20 q^{44}-32 q^{42}+42 q^{40}-52 q^{38}+64 q^{36}-78 q^{34}+86 q^{32}-88 q^{30}+90 q^{28}-86 q^{26}+74 q^{24}-56 q^{22}+29 q^{20}+4 q^{18}-44 q^{16}+88 q^{14}-132 q^{12}+180 q^{10}-210 q^8+228 q^6-229 q^4+216 q^2-186+136 q^{-2} -85 q^{-4} +32 q^{-6} +24 q^{-8} -66 q^{-10} +107 q^{-12} -124 q^{-14} +136 q^{-16} -130 q^{-18} +111 q^{-20} -92 q^{-22} +66 q^{-24} -44 q^{-26} +25 q^{-28} -14 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{46}-q^{44}-q^{42}+q^{40}+q^{34}-4 q^{30}-2 q^{28}+4 q^{26}+q^{24}-5 q^{22}+3 q^{20}+7 q^{18}-q^{16}-3 q^{14}+3 q^{12}+2 q^{10}-4 q^8-2 q^6+q^4-2 q^2-2+4 q^{-2} + q^{-4} -2 q^{-6} +4 q^{-8} +5 q^{-10} -3 q^{-14} +3 q^{-16} +3 q^{-18} -2 q^{-20} -4 q^{-22} + q^{-26} - q^{-28} - q^{-30} + q^{-34} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{58}-2 q^{56}+q^{54}-2 q^{52}+5 q^{50}-5 q^{48}+4 q^{46}-6 q^{44}+9 q^{42}-7 q^{40}+6 q^{38}-8 q^{36}+7 q^{34}-4 q^{32}+3 q^{30}-2 q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+8 q^{20}-13 q^{18}+12 q^{16}-15 q^{14}+15 q^{12}-15 q^{10}+15 q^8-10 q^6+13 q^4-6 q^2+9+ q^{-2} +2 q^{-4} +3 q^{-6} -4 q^{-8} +7 q^{-10} -7 q^{-12} +6 q^{-14} -9 q^{-16} +7 q^{-18} -8 q^{-20} +5 q^{-22} -6 q^{-24} +4 q^{-26} -3 q^{-28} +2 q^{-30} - q^{-32} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+2 q^{92}-q^{90}-2 q^{88}+8 q^{86}-11 q^{84}+14 q^{82}-13 q^{80}+7 q^{78}+q^{76}-11 q^{74}+21 q^{72}-26 q^{70}+25 q^{68}-19 q^{66}+3 q^{64}+12 q^{62}-23 q^{60}+30 q^{58}-27 q^{56}+18 q^{54}-5 q^{52}-10 q^{50}+20 q^{48}-21 q^{46}+13 q^{44}-q^{42}-12 q^{40}+18 q^{38}-13 q^{36}+q^{34}+17 q^{32}-33 q^{30}+38 q^{28}-29 q^{26}+2 q^{24}+27 q^{22}-49 q^{20}+57 q^{18}-43 q^{16}+17 q^{14}+13 q^{12}-36 q^{10}+47 q^8-42 q^6+21 q^4+4 q^2-20+28 q^{-2} -18 q^{-4} +7 q^{-6} +14 q^{-8} -24 q^{-10} +24 q^{-12} -15 q^{-14} -3 q^{-16} +28 q^{-18} -39 q^{-20} +40 q^{-22} -24 q^{-24} + q^{-26} +23 q^{-28} -38 q^{-30} +38 q^{-32} -29 q^{-34} +9 q^{-36} +7 q^{-38} -20 q^{-40} +22 q^{-42} -17 q^{-44} +9 q^{-46} - q^{-48} -4 q^{-50} +4 q^{-52} -5 q^{-54} +3 q^{-56} - q^{-58} + q^{-60} }[/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 19"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-7 t^2+11 t-11+11 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+5 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]=

{ 51, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+2 z^8 a^{-2} +5 z^8+5 a^3 z^7+3 a z^7-z^7 a^{-1} +z^7 a^{-3} +6 a^4 z^6-3 a^2 z^6-10 z^6 a^{-2} -19 z^6+5 a^5 z^5-7 a^3 z^5-15 a z^5-8 z^5 a^{-1} -5 z^5 a^{-3} +3 a^6 z^4-8 a^4 z^4-4 a^2 z^4+16 z^4 a^{-2} +23 z^4+a^7 z^3-4 a^5 z^3+11 a z^3+13 z^3 a^{-1} +7 z^3 a^{-3} -a^6 z^2+3 a^4 z^2-9 z^2 a^{-2} -13 z^2+a^5 z+a^3 z-2 a z-4 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]