9 24

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, 0 }
Jones polynomial	[math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+8-7 q^{-1} +7 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+z^4 a^{-2} -4 z^4-a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} -6 z^2-2 a^4+5 a^2+ a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+5 a z^7+3 z^7 a^{-1} +2 a^4 z^6+3 a^2 z^6+4 z^6 a^{-2} +5 z^6+a^5 z^5-2 a^3 z^5-7 a z^5-z^5 a^{-1} +3 z^5 a^{-3} -5 a^4 z^4-10 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -11 z^4-3 a^5 z^3-3 a^3 z^3+a z^3-3 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+10 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+2 a^5 z+3 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -2 a^4-5 a^2- a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}-q^{14}-q^{10}+3 q^8+2 q^6+q^4+2 q^2-2+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-3 q^{70}-3 q^{68}+9 q^{66}-16 q^{64}+19 q^{62}-19 q^{60}+8 q^{58}+7 q^{56}-26 q^{54}+41 q^{52}-47 q^{50}+34 q^{48}-11 q^{46}-23 q^{44}+45 q^{42}-53 q^{40}+46 q^{38}-16 q^{36}-11 q^{34}+34 q^{32}-36 q^{30}+22 q^{28}+10 q^{26}-32 q^{24}+44 q^{22}-27 q^{20}+2 q^{18}+37 q^{16}-60 q^{14}+71 q^{12}-55 q^{10}+21 q^8+20 q^6-60 q^4+77 q^2-69+40 q^{-2} -4 q^{-4} -32 q^{-6} +46 q^{-8} -43 q^{-10} +18 q^{-12} +8 q^{-14} -31 q^{-16} +33 q^{-18} -15 q^{-20} -14 q^{-22} +41 q^{-24} -50 q^{-26} +44 q^{-28} -20 q^{-30} -11 q^{-32} +35 q^{-34} -48 q^{-36} +47 q^{-38} -29 q^{-40} +9 q^{-42} +9 q^{-44} -21 q^{-46} +24 q^{-48} -19 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_24.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^9-2 q^7+3 q^5+q+ q^{-1} -2 q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+4 q^{26}-3 q^{24}-6 q^{22}+9 q^{20}-q^{18}-12 q^{16}+11 q^{14}+5 q^{12}-12 q^{10}+5 q^8+7 q^6-5 q^4-2 q^2+5+5 q^{-2} -10 q^{-4} +12 q^{-8} -11 q^{-10} -4 q^{-12} +13 q^{-14} -5 q^{-16} -5 q^{-18} +6 q^{-20} - q^{-22} -2 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{61}+q^{59}-q^{57}-3 q^{55}+3 q^{53}+7 q^{51}-3 q^{49}-13 q^{47}+q^{45}+21 q^{43}+5 q^{41}-30 q^{39}-17 q^{37}+34 q^{35}+30 q^{33}-30 q^{31}-48 q^{29}+26 q^{27}+54 q^{25}-11 q^{23}-59 q^{21}-q^{19}+55 q^{17}+12 q^{15}-46 q^{13}-18 q^{11}+34 q^9+25 q^7-16 q^5-30 q^3+3 q+32 q^{-1} +17 q^{-3} -36 q^{-5} -32 q^{-7} +32 q^{-9} +49 q^{-11} -27 q^{-13} -56 q^{-15} +16 q^{-17} +57 q^{-19} -3 q^{-21} -53 q^{-23} -7 q^{-25} +41 q^{-27} +13 q^{-29} -26 q^{-31} -14 q^{-33} +14 q^{-35} +11 q^{-37} -8 q^{-39} -5 q^{-41} +3 q^{-43} +3 q^{-45} - q^{-47} -2 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{104}-q^{102}-q^{100}+q^{98}+3 q^{94}-5 q^{92}-5 q^{90}+5 q^{88}+5 q^{86}+13 q^{84}-13 q^{82}-24 q^{80}+18 q^{76}+51 q^{74}-7 q^{72}-62 q^{70}-48 q^{68}+6 q^{66}+121 q^{64}+61 q^{62}-67 q^{60}-138 q^{58}-92 q^{56}+148 q^{54}+182 q^{52}+34 q^{50}-177 q^{48}-244 q^{46}+62 q^{44}+244 q^{42}+183 q^{40}-104 q^{38}-320 q^{36}-78 q^{34}+190 q^{32}+264 q^{30}+10 q^{28}-275 q^{26}-155 q^{24}+89 q^{22}+236 q^{20}+81 q^{18}-164 q^{16}-163 q^{14}-5 q^{12}+160 q^{10}+121 q^8-41 q^6-149 q^4-95 q^2+70+162 q^{-2} +100 q^{-4} -130 q^{-6} -197 q^{-8} -46 q^{-10} +180 q^{-12} +245 q^{-14} -61 q^{-16} -257 q^{-18} -180 q^{-20} +124 q^{-22} +327 q^{-24} +56 q^{-26} -203 q^{-28} -251 q^{-30} -3 q^{-32} +269 q^{-34} +135 q^{-36} -65 q^{-38} -202 q^{-40} -92 q^{-42} +129 q^{-44} +108 q^{-46} +32 q^{-48} -86 q^{-50} -80 q^{-52} +28 q^{-54} +36 q^{-56} +36 q^{-58} -18 q^{-60} -31 q^{-62} +6 q^{-64} +3 q^{-66} +12 q^{-68} -3 q^{-70} -8 q^{-72} +3 q^{-74} +3 q^{-78} - q^{-80} -2 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ -q^{155}+q^{153}+q^{151}-q^{149}-q^{143}+3 q^{141}+4 q^{139}-5 q^{137}-8 q^{135}-3 q^{133}+3 q^{131}+15 q^{129}+18 q^{127}-4 q^{125}-33 q^{123}-37 q^{121}-7 q^{119}+45 q^{117}+80 q^{115}+44 q^{113}-51 q^{111}-137 q^{109}-117 q^{107}+22 q^{105}+191 q^{103}+236 q^{101}+75 q^{99}-208 q^{97}-385 q^{95}-252 q^{93}+141 q^{91}+507 q^{89}+504 q^{87}+49 q^{85}-542 q^{83}-782 q^{81}-365 q^{79}+452 q^{77}+991 q^{75}+743 q^{73}-182 q^{71}-1079 q^{69}-1137 q^{67}-171 q^{65}+1004 q^{63}+1392 q^{61}+604 q^{59}-770 q^{57}-1536 q^{55}-963 q^{53}+466 q^{51}+1481 q^{49}+1212 q^{47}-122 q^{45}-1326 q^{43}-1310 q^{41}-155 q^{39}+1078 q^{37}+1280 q^{35}+345 q^{33}-813 q^{31}-1161 q^{29}-460 q^{27}+570 q^{25}+999 q^{23}+517 q^{21}-356 q^{19}-836 q^{17}-557 q^{15}+171 q^{13}+691 q^{11}+615 q^9+34 q^7-577 q^5-700 q^3-242 q+443 q^{-1} +820 q^{-3} +521 q^{-5} -300 q^{-7} -942 q^{-9} -808 q^{-11} +70 q^{-13} +1011 q^{-15} +1140 q^{-17} +203 q^{-19} -1002 q^{-21} -1395 q^{-23} -550 q^{-25} +857 q^{-27} +1563 q^{-29} +895 q^{-31} -596 q^{-33} -1567 q^{-35} -1165 q^{-37} +236 q^{-39} +1391 q^{-41} +1310 q^{-43} +137 q^{-45} -1074 q^{-47} -1286 q^{-49} -423 q^{-51} +677 q^{-53} +1100 q^{-55} +588 q^{-57} -303 q^{-59} -818 q^{-61} -603 q^{-63} +32 q^{-65} +508 q^{-67} +498 q^{-69} +118 q^{-71} -248 q^{-73} -349 q^{-75} -160 q^{-77} +94 q^{-79} +196 q^{-81} +126 q^{-83} -7 q^{-85} -90 q^{-87} -85 q^{-89} -12 q^{-91} +40 q^{-93} +36 q^{-95} +11 q^{-97} -11 q^{-99} -15 q^{-101} -8 q^{-103} +6 q^{-105} +8 q^{-107} -2 q^{-109} -3 q^{-111} +3 q^{-119} - q^{-121} -2 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}-q^{14}-q^{10}+3 q^8+2 q^6+q^4+2 q^2-2+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+6 q^{40}-12 q^{38}+25 q^{36}-42 q^{34}+66 q^{32}-100 q^{30}+130 q^{28}-168 q^{26}+186 q^{24}-196 q^{22}+177 q^{20}-132 q^{18}+72 q^{16}+26 q^{14}-114 q^{12}+216 q^{10}-294 q^8+352 q^6-385 q^4+376 q^2-338+268 q^{-2} -177 q^{-4} +80 q^{-6} +16 q^{-8} -96 q^{-10} +156 q^{-12} -186 q^{-14} +192 q^{-16} -178 q^{-18} +152 q^{-20} -120 q^{-22} +86 q^{-24} -58 q^{-26} +36 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{42}+q^{40}-q^{36}+q^{34}+q^{32}-4 q^{30}-6 q^{28}+q^{24}-4 q^{22}+8 q^{18}+7 q^{16}-3 q^{14}+2 q^{12}+5 q^{10}-3 q^8-3 q^6+3 q^4-4+2 q^{-2} +4 q^{-4} -4 q^{-6} -3 q^{-8} +6 q^{-10} +2 q^{-12} -6 q^{-14} + q^{-16} +5 q^{-18} - q^{-20} -3 q^{-22} +2 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}+q^{42}+q^{40}+q^{38}+q^{36}-2 q^{34}-5 q^{32}-4 q^{30}-4 q^{28}-11 q^{26}-7 q^{24}+6 q^{22}+5 q^{20}+2 q^{18}+14 q^{16}+20 q^{14}+6 q^{12}+5 q^8-q^6-16 q^4-6 q^2+1-9 q^{-2} -5 q^{-4} +10 q^{-6} +2 q^{-8} -5 q^{-10} +6 q^{-12} +8 q^{-14} -3 q^{-16} -5 q^{-18} +5 q^{-20} +2 q^{-22} -5 q^{-24} - q^{-26} +3 q^{-28} - q^{-30} - q^{-32} + q^{-34} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{24}-2 q^{22}-2 q^{20}-q^{16}+4 q^{14}+3 q^{12}+4 q^{10}+4 q^8+q^6+q^4-2 q^2-1-3 q^{-2} -2 q^{-6} + q^{-8} +2 q^{-14} - q^{-16} + q^{-18} }[/math]

B2 Invariants.

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

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{46}-q^{44}+2 q^{42}-2 q^{40}+4 q^{38}-6 q^{36}+4 q^{34}-9 q^{32}+5 q^{30}-12 q^{28}+4 q^{26}-8 q^{24}+7 q^{22}-q^{20}+4 q^{18}+9 q^{16}+3 q^{14}+15 q^{12}-8 q^{10}+14 q^8-13 q^6+14 q^4-19 q^2+9-17 q^{-2} +9 q^{-4} -8 q^{-6} +3 q^{-8} -3 q^{-10} + q^{-12} +7 q^{-14} -4 q^{-16} +7 q^{-18} -8 q^{-20} +10 q^{-22} -7 q^{-24} +6 q^{-26} -7 q^{-28} +5 q^{-30} -3 q^{-32} +2 q^{-34} -2 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-3 q^{70}-3 q^{68}+9 q^{66}-16 q^{64}+19 q^{62}-19 q^{60}+8 q^{58}+7 q^{56}-26 q^{54}+41 q^{52}-47 q^{50}+34 q^{48}-11 q^{46}-23 q^{44}+45 q^{42}-53 q^{40}+46 q^{38}-16 q^{36}-11 q^{34}+34 q^{32}-36 q^{30}+22 q^{28}+10 q^{26}-32 q^{24}+44 q^{22}-27 q^{20}+2 q^{18}+37 q^{16}-60 q^{14}+71 q^{12}-55 q^{10}+21 q^8+20 q^6-60 q^4+77 q^2-69+40 q^{-2} -4 q^{-4} -32 q^{-6} +46 q^{-8} -43 q^{-10} +18 q^{-12} +8 q^{-14} -31 q^{-16} +33 q^{-18} -15 q^{-20} -14 q^{-22} +41 q^{-24} -50 q^{-26} +44 q^{-28} -20 q^{-30} -11 q^{-32} +35 q^{-34} -48 q^{-36} +47 q^{-38} -29 q^{-40} +9 q^{-42} +9 q^{-44} -21 q^{-46} +24 q^{-48} -19 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 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 24"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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