9 19

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-10 t+17-10 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 41, 0 }
Jones polynomial	[math]\displaystyle{ q^4-2 q^3+4 q^2-6 q+7-7 q^{-1} +6 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4+z^4 a^2+z^2 a^2+a^2+z^4-2 z^2 a^{-2} - a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +3 a^4 z^6+3 a^2 z^6+2 z^6 a^{-2} +2 z^6+a^5 z^5-7 a^3 z^5-11 a z^5-z^5 a^{-1} +2 z^5 a^{-3} -8 a^4 z^4-11 a^2 z^4+z^4 a^{-4} -4 z^4-2 a^5 z^3+4 a^3 z^3+10 a z^3+z^3 a^{-1} -3 z^3 a^{-3} +4 a^4 z^2+8 a^2 z^2-3 z^2 a^{-2} -2 z^2 a^{-4} +3 z^2-a^3 z-3 a z-z a^{-1} +z a^{-3} -a^2+ a^{-2} + a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-q^{10}+2 q^8+q^2-1+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-3 q^{70}-5 q^{68}+16 q^{66}-21 q^{64}+24 q^{62}-18 q^{60}+2 q^{58}+18 q^{56}-34 q^{54}+40 q^{52}-31 q^{50}+13 q^{48}+11 q^{46}-28 q^{44}+33 q^{42}-24 q^{40}+8 q^{38}+10 q^{36}-23 q^{34}+20 q^{32}-6 q^{30}-13 q^{28}+30 q^{26}-34 q^{24}+27 q^{22}-6 q^{20}-20 q^{18}+41 q^{16}-51 q^{14}+48 q^{12}-25 q^{10}-3 q^8+30 q^6-44 q^4+44 q^2-27+4 q^{-2} +14 q^{-4} -24 q^{-6} +19 q^{-8} -4 q^{-10} -13 q^{-12} +23 q^{-14} -22 q^{-16} +7 q^{-18} +9 q^{-20} -26 q^{-22} +33 q^{-24} -29 q^{-26} +17 q^{-28} -2 q^{-30} -14 q^{-32} +22 q^{-34} -24 q^{-36} +21 q^{-38} -12 q^{-40} +4 q^{-42} +4 q^{-44} -9 q^{-46} +11 q^{-48} -9 q^{-50} +8 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_19.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-q^7+2 q^5-q^3+ q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+6 q^{26}-q^{24}-7 q^{22}+7 q^{20}+3 q^{18}-10 q^{16}+5 q^{14}+6 q^{12}-8 q^{10}+q^8+5 q^6-2 q^4-4 q^2+2+7 q^{-2} -7 q^{-4} -3 q^{-6} +11 q^{-8} -5 q^{-10} -5 q^{-12} +8 q^{-14} -2 q^{-16} -3 q^{-18} +3 q^{-20} - q^{-22} - q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-6 q^{55}+q^{53}+13 q^{51}+2 q^{49}-16 q^{47}-10 q^{45}+17 q^{43}+20 q^{41}-13 q^{39}-30 q^{37}+6 q^{35}+33 q^{33}+5 q^{31}-35 q^{29}-13 q^{27}+34 q^{25}+20 q^{23}-27 q^{21}-24 q^{19}+21 q^{17}+23 q^{15}-13 q^{13}-24 q^{11}+5 q^9+21 q^7+6 q^5-17 q^3-17 q+13 q^{-1} +29 q^{-3} -5 q^{-5} -34 q^{-7} -4 q^{-9} +37 q^{-11} +13 q^{-13} -35 q^{-15} -17 q^{-17} +25 q^{-19} +19 q^{-21} -17 q^{-23} -16 q^{-25} +10 q^{-27} +11 q^{-29} -5 q^{-31} -6 q^{-33} +3 q^{-35} +3 q^{-37} -2 q^{-39} - q^{-41} +2 q^{-43} - q^{-47} - q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+6 q^{94}-8 q^{92}-13 q^{90}-q^{88}+7 q^{86}+31 q^{84}+q^{82}-30 q^{80}-28 q^{78}-13 q^{76}+57 q^{74}+44 q^{72}-7 q^{70}-56 q^{68}-81 q^{66}+31 q^{64}+85 q^{62}+70 q^{60}-25 q^{58}-138 q^{56}-53 q^{54}+60 q^{52}+142 q^{50}+61 q^{48}-130 q^{46}-126 q^{44}-14 q^{42}+147 q^{40}+128 q^{38}-73 q^{36}-141 q^{34}-73 q^{32}+105 q^{30}+139 q^{28}-17 q^{26}-110 q^{24}-87 q^{22}+54 q^{20}+112 q^{18}+27 q^{16}-69 q^{14}-85 q^{12}+q^{10}+74 q^8+76 q^6-15 q^4-84 q^2-75+20 q^{-2} +131 q^{-4} +61 q^{-6} -60 q^{-8} -145 q^{-10} -61 q^{-12} +138 q^{-14} +132 q^{-16} +10 q^{-18} -155 q^{-20} -132 q^{-22} +79 q^{-24} +134 q^{-26} +76 q^{-28} -88 q^{-30} -131 q^{-32} +6 q^{-34} +69 q^{-36} +82 q^{-38} -16 q^{-40} -73 q^{-42} -17 q^{-44} +9 q^{-46} +43 q^{-48} +9 q^{-50} -23 q^{-52} -6 q^{-54} -9 q^{-56} +12 q^{-58} +5 q^{-60} -4 q^{-62} +4 q^{-64} -6 q^{-66} + q^{-68} - q^{-72} +4 q^{-74} - q^{-76} - q^{-80} - q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+q^{143}+8 q^{141}+13 q^{139}+q^{137}-17 q^{135}-22 q^{133}-13 q^{131}+13 q^{129}+40 q^{127}+44 q^{125}-3 q^{123}-57 q^{121}-72 q^{119}-39 q^{117}+43 q^{115}+114 q^{113}+106 q^{111}-123 q^{107}-173 q^{105}-104 q^{103}+78 q^{101}+233 q^{99}+226 q^{97}+30 q^{95}-227 q^{93}-348 q^{91}-200 q^{89}+147 q^{87}+423 q^{85}+394 q^{83}+14 q^{81}-429 q^{79}-553 q^{77}-225 q^{75}+338 q^{73}+664 q^{71}+439 q^{69}-191 q^{67}-687 q^{65}-607 q^{63}+4 q^{61}+633 q^{59}+715 q^{57}+168 q^{55}-531 q^{53}-746 q^{51}-292 q^{49}+397 q^{47}+709 q^{45}+376 q^{43}-277 q^{41}-635 q^{39}-394 q^{37}+172 q^{35}+533 q^{33}+392 q^{31}-91 q^{29}-442 q^{27}-362 q^{25}+22 q^{23}+351 q^{21}+348 q^{19}+48 q^{17}-267 q^{15}-349 q^{13}-139 q^{11}+184 q^9+362 q^7+257 q^5-75 q^3-379 q-403 q^{-1} -65 q^{-3} +381 q^{-5} +543 q^{-7} +243 q^{-9} -324 q^{-11} -667 q^{-13} -442 q^{-15} +217 q^{-17} +733 q^{-19} +619 q^{-21} -50 q^{-23} -705 q^{-25} -752 q^{-27} -143 q^{-29} +598 q^{-31} +799 q^{-33} +301 q^{-35} -412 q^{-37} -740 q^{-39} -423 q^{-41} +218 q^{-43} +610 q^{-45} +453 q^{-47} -48 q^{-49} -430 q^{-51} -413 q^{-53} -72 q^{-55} +261 q^{-57} +325 q^{-59} +124 q^{-61} -130 q^{-63} -221 q^{-65} -125 q^{-67} +41 q^{-69} +133 q^{-71} +103 q^{-73} +2 q^{-75} -71 q^{-77} -68 q^{-79} -17 q^{-81} +30 q^{-83} +40 q^{-85} +20 q^{-87} -9 q^{-89} -23 q^{-91} -14 q^{-93} + q^{-95} +7 q^{-97} +9 q^{-99} +6 q^{-101} -4 q^{-103} -6 q^{-105} - q^{-107} -2 q^{-109} + q^{-111} +4 q^{-113} + q^{-115} - q^{-117} - q^{-121} - q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-q^{10}+2 q^8+q^2-1+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+40 q^{36}-62 q^{34}+86 q^{32}-112 q^{30}+129 q^{28}-130 q^{26}+120 q^{24}-90 q^{22}+48 q^{20}+8 q^{18}-70 q^{16}+128 q^{14}-177 q^{12}+216 q^{10}-234 q^8+234 q^6-214 q^4+172 q^2-122+64 q^{-2} -9 q^{-4} -38 q^{-6} +78 q^{-8} -94 q^{-10} +104 q^{-12} -100 q^{-14} +92 q^{-16} -78 q^{-18} +62 q^{-20} -50 q^{-22} +36 q^{-24} -26 q^{-26} +17 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{40}-2 q^{38}+3 q^{34}+2 q^{32}-5 q^{30}-q^{28}+5 q^{26}+3 q^{24}-5 q^{22}-3 q^{20}+6 q^{18}+3 q^{16}-5 q^{14}-q^{12}+5 q^{10}-q^8-2 q^6+q^4-q^2-1+2 q^{-2} +2 q^{-4} -5 q^{-6} -2 q^{-8} +8 q^{-10} +2 q^{-12} -6 q^{-14} + q^{-16} +7 q^{-18} + q^{-20} -5 q^{-22} -2 q^{-24} +2 q^{-26} -2 q^{-30} + q^{-34} + q^{-36} }[/math]

A3 Invariants.

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

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-3 q^{70}-5 q^{68}+16 q^{66}-21 q^{64}+24 q^{62}-18 q^{60}+2 q^{58}+18 q^{56}-34 q^{54}+40 q^{52}-31 q^{50}+13 q^{48}+11 q^{46}-28 q^{44}+33 q^{42}-24 q^{40}+8 q^{38}+10 q^{36}-23 q^{34}+20 q^{32}-6 q^{30}-13 q^{28}+30 q^{26}-34 q^{24}+27 q^{22}-6 q^{20}-20 q^{18}+41 q^{16}-51 q^{14}+48 q^{12}-25 q^{10}-3 q^8+30 q^6-44 q^4+44 q^2-27+4 q^{-2} +14 q^{-4} -24 q^{-6} +19 q^{-8} -4 q^{-10} -13 q^{-12} +23 q^{-14} -22 q^{-16} +7 q^{-18} +9 q^{-20} -26 q^{-22} +33 q^{-24} -29 q^{-26} +17 q^{-28} -2 q^{-30} -14 q^{-32} +22 q^{-34} -24 q^{-36} +21 q^{-38} -12 q^{-40} +4 q^{-42} +4 q^{-44} -9 q^{-46} +11 q^{-48} -9 q^{-50} +8 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/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 19"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-10 t+17-10 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 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, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^4+z^4 a^2+z^2 a^2+a^2+z^4-2 z^2 a^{-2} - a^{-2} + a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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