9 30

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-12 t+17-12 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	{ 53, 0 }
Jones polynomial	[math]\displaystyle{ q^4-3 q^3+6 q^2-8 q+9-9 q^{-1} +8 q^{-2} -5 q^{-3} +3 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+5 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^4+4 a^2+2 a^{-2} -4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+7 a z^7+4 z^7 a^{-1} +3 a^4 z^6+8 a^2 z^6+5 z^6 a^{-2} +10 z^6+a^5 z^5-3 a^3 z^5-9 a z^5-2 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4-22 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -23 z^4-2 a^5 z^3-3 a^3 z^3-2 z^3 a^{-1} -3 z^3 a^{-3} +5 a^4 z^2+16 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +17 z^2+a^5 z+2 a^3 z+a z+z a^{-1} +z a^{-3} -a^4-4 a^2-2 a^{-2} -4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{12}-q^{10}+3 q^8+q^6+q^2-3+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-5 q^{70}-5 q^{68}+19 q^{66}-31 q^{64}+39 q^{62}-34 q^{60}+11 q^{58}+19 q^{56}-55 q^{54}+79 q^{52}-79 q^{50}+49 q^{48}-2 q^{46}-48 q^{44}+83 q^{42}-84 q^{40}+60 q^{38}-9 q^{36}-36 q^{34}+61 q^{32}-52 q^{30}+14 q^{28}+39 q^{26}-69 q^{24}+75 q^{22}-40 q^{20}-15 q^{18}+77 q^{16}-118 q^{14}+120 q^{12}-84 q^{10}+16 q^8+55 q^6-109 q^4+123 q^2-97+43 q^{-2} +15 q^{-4} -64 q^{-6} +72 q^{-8} -52 q^{-10} +6 q^{-12} +39 q^{-14} -60 q^{-16} +48 q^{-18} -8 q^{-20} -41 q^{-22} +79 q^{-24} -87 q^{-26} +65 q^{-28} -21 q^{-30} -29 q^{-32} +67 q^{-34} -77 q^{-36} +67 q^{-38} -34 q^{-40} +5 q^{-42} +19 q^{-44} -33 q^{-46} +32 q^{-48} -23 q^{-50} +13 q^{-52} -2 q^{-54} -4 q^{-56} +5 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_30.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-2 q^7+3 q^5-q^3+ q^{-1} -2 q^{-3} +3 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+7 q^{26}-3 q^{24}-9 q^{22}+14 q^{20}+q^{18}-18 q^{16}+13 q^{14}+8 q^{12}-17 q^{10}+4 q^8+10 q^6-6 q^4-6 q^2+6+9 q^{-2} -13 q^{-4} - q^{-6} +19 q^{-8} -13 q^{-10} -8 q^{-12} +17 q^{-14} -6 q^{-16} -8 q^{-18} +7 q^{-20} -2 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-7 q^{55}+3 q^{53}+16 q^{51}-2 q^{49}-26 q^{47}-7 q^{45}+39 q^{43}+23 q^{41}-47 q^{39}-45 q^{37}+48 q^{35}+68 q^{33}-36 q^{31}-91 q^{29}+19 q^{27}+98 q^{25}+4 q^{23}-96 q^{21}-26 q^{19}+87 q^{17}+40 q^{15}-65 q^{13}-50 q^{11}+41 q^9+55 q^7-12 q^5-57 q^3-15 q+55 q^{-1} +45 q^{-3} -48 q^{-5} -72 q^{-7} +37 q^{-9} +92 q^{-11} -20 q^{-13} -101 q^{-15} + q^{-17} +96 q^{-19} +20 q^{-21} -82 q^{-23} -32 q^{-25} +60 q^{-27} +33 q^{-29} -34 q^{-31} -30 q^{-33} +17 q^{-35} +21 q^{-37} -7 q^{-39} -10 q^{-41} + q^{-43} +4 q^{-45} -2 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+7 q^{94}-10 q^{92}-16 q^{90}+2 q^{88}+14 q^{86}+41 q^{84}-12 q^{82}-59 q^{80}-37 q^{78}+16 q^{76}+129 q^{74}+49 q^{72}-98 q^{70}-157 q^{68}-79 q^{66}+221 q^{64}+227 q^{62}-12 q^{60}-286 q^{58}-324 q^{56}+159 q^{54}+412 q^{52}+252 q^{50}-243 q^{48}-568 q^{46}-95 q^{44}+410 q^{42}+513 q^{40}-12 q^{38}-602 q^{36}-351 q^{34}+218 q^{32}+576 q^{30}+213 q^{28}-431 q^{26}-436 q^{24}+446 q^{20}+311 q^{18}-184 q^{16}-383 q^{14}-165 q^{12}+245 q^{10}+330 q^8+66 q^6-278 q^4-306 q^2+7+324 q^{-2} +339 q^{-4} -128 q^{-6} -432 q^{-8} -272 q^{-10} +246 q^{-12} +574 q^{-14} +101 q^{-16} -431 q^{-18} -517 q^{-20} +37 q^{-22} +623 q^{-24} +326 q^{-26} -233 q^{-28} -559 q^{-30} -203 q^{-32} +420 q^{-34} +381 q^{-36} +25 q^{-38} -367 q^{-40} -279 q^{-42} +144 q^{-44} +234 q^{-46} +133 q^{-48} -125 q^{-50} -176 q^{-52} + q^{-54} +70 q^{-56} +88 q^{-58} -14 q^{-60} -59 q^{-62} -11 q^{-64} +4 q^{-66} +26 q^{-68} +3 q^{-70} -11 q^{-72} - q^{-74} -2 q^{-76} +4 q^{-78} -2 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}+10 q^{141}+16 q^{139}-2 q^{137}-23 q^{135}-29 q^{133}-13 q^{131}+32 q^{129}+71 q^{127}+52 q^{125}-43 q^{123}-132 q^{121}-124 q^{119}+8 q^{117}+199 q^{115}+275 q^{113}+97 q^{111}-254 q^{109}-473 q^{107}-316 q^{105}+194 q^{103}+700 q^{101}+693 q^{99}+26 q^{97}-853 q^{95}-1164 q^{93}-488 q^{91}+802 q^{89}+1646 q^{87}+1166 q^{85}-454 q^{83}-1975 q^{81}-1968 q^{79}-198 q^{77}+1994 q^{75}+2703 q^{73}+1111 q^{71}-1652 q^{69}-3227 q^{67}-2060 q^{65}+989 q^{63}+3361 q^{61}+2906 q^{59}-129 q^{57}-3148 q^{55}-3462 q^{53}-718 q^{51}+2634 q^{49}+3644 q^{47}+1441 q^{45}-1965 q^{43}-3518 q^{41}-1914 q^{39}+1292 q^{37}+3139 q^{35}+2125 q^{33}-659 q^{31}-2658 q^{29}-2166 q^{27}+164 q^{25}+2139 q^{23}+2091 q^{21}+265 q^{19}-1642 q^{17}-2014 q^{15}-655 q^{13}+1175 q^{11}+1966 q^9+1091 q^7-705 q^5-1948 q^3-1594 q+160 q^{-1} +1931 q^{-3} +2165 q^{-5} +487 q^{-7} -1821 q^{-9} -2732 q^{-11} -1254 q^{-13} +1532 q^{-15} +3201 q^{-17} +2084 q^{-19} -1027 q^{-21} -3429 q^{-23} -2860 q^{-25} +307 q^{-27} +3311 q^{-29} +3458 q^{-31} +531 q^{-33} -2851 q^{-35} -3692 q^{-37} -1334 q^{-39} +2073 q^{-41} +3538 q^{-43} +1939 q^{-45} -1174 q^{-47} -3018 q^{-49} -2189 q^{-51} +321 q^{-53} +2244 q^{-55} +2098 q^{-57} +317 q^{-59} -1438 q^{-61} -1736 q^{-63} -624 q^{-65} +727 q^{-67} +1234 q^{-69} +688 q^{-71} -244 q^{-73} -766 q^{-75} -564 q^{-77} +2 q^{-79} +393 q^{-81} +378 q^{-83} +89 q^{-85} -167 q^{-87} -223 q^{-89} -84 q^{-91} +63 q^{-93} +102 q^{-95} +54 q^{-97} -13 q^{-99} -41 q^{-101} -32 q^{-103} +3 q^{-105} +19 q^{-107} +8 q^{-109} - q^{-111} -2 q^{-113} -4 q^{-115} -2 q^{-117} +4 q^{-119} -2 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{12}-q^{10}+3 q^8+q^6+q^2-3+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+54 q^{36}-96 q^{34}+150 q^{32}-214 q^{30}+280 q^{28}-336 q^{26}+366 q^{24}-352 q^{22}+299 q^{20}-192 q^{18}+42 q^{16}+138 q^{14}-321 q^{12}+486 q^{10}-622 q^8+698 q^6-714 q^4+662 q^2-550+398 q^{-2} -213 q^{-4} +36 q^{-6} +132 q^{-8} -256 q^{-10} +334 q^{-12} -360 q^{-14} +344 q^{-16} -304 q^{-18} +239 q^{-20} -174 q^{-22} +118 q^{-24} -74 q^{-26} +42 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{42}-2 q^{38}-2 q^{36}+2 q^{34}+3 q^{32}-4 q^{30}-3 q^{28}+6 q^{26}+5 q^{24}-6 q^{22}-3 q^{20}+8 q^{18}+5 q^{16}-8 q^{14}-q^{12}+5 q^{10}-6 q^8-5 q^6+2 q^4-3+7 q^{-2} +9 q^{-4} -3 q^{-6} -2 q^{-8} +9 q^{-10} + q^{-12} -12 q^{-14} - q^{-16} +6 q^{-18} - q^{-20} -5 q^{-22} +4 q^{-26} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+7 q^{28}-10 q^{26}+13 q^{24}-14 q^{22}+15 q^{20}-12 q^{18}+9 q^{16}-q^{14}-4 q^{12}+13 q^{10}-19 q^8+25 q^6-28 q^4+27 q^2-26+19 q^{-2} -13 q^{-4} +5 q^{-6} +2 q^{-8} -8 q^{-10} +13 q^{-12} -14 q^{-14} +15 q^{-16} -13 q^{-18} +11 q^{-20} -7 q^{-22} +4 q^{-24} -2 q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+5 q^{46}-2 q^{44}-9 q^{42}-4 q^{40}+10 q^{38}+10 q^{36}-6 q^{34}-15 q^{32}-q^{30}+16 q^{28}+10 q^{26}-11 q^{24}-12 q^{22}+5 q^{20}+14 q^{18}-11 q^{14}-2 q^{12}+10 q^{10}+4 q^8-9 q^6-6 q^4+7 q^2+9-6 q^{-2} -11 q^{-4} +3 q^{-6} +12 q^{-8} -14 q^{-12} -6 q^{-14} +13 q^{-16} +13 q^{-18} -7 q^{-20} -15 q^{-22} +14 q^{-26} +7 q^{-28} -7 q^{-30} -9 q^{-32} + q^{-34} +6 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}-2 q^{44}+3 q^{42}-4 q^{40}+6 q^{38}-9 q^{36}+8 q^{34}-11 q^{32}+11 q^{30}-13 q^{28}+9 q^{26}-8 q^{24}+9 q^{22}-q^{20}+9 q^{16}-4 q^{14}+17 q^{12}-17 q^{10}+19 q^8-20 q^6+21 q^4-24 q^2+14-20 q^{-2} +12 q^{-4} -9 q^{-6} +2 q^{-8} -2 q^{-10} - q^{-12} +11 q^{-14} -7 q^{-16} +11 q^{-18} -11 q^{-20} +14 q^{-22} -9 q^{-24} +8 q^{-26} -9 q^{-28} +6 q^{-30} -3 q^{-32} +2 q^{-34} -2 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-5 q^{70}-5 q^{68}+19 q^{66}-31 q^{64}+39 q^{62}-34 q^{60}+11 q^{58}+19 q^{56}-55 q^{54}+79 q^{52}-79 q^{50}+49 q^{48}-2 q^{46}-48 q^{44}+83 q^{42}-84 q^{40}+60 q^{38}-9 q^{36}-36 q^{34}+61 q^{32}-52 q^{30}+14 q^{28}+39 q^{26}-69 q^{24}+75 q^{22}-40 q^{20}-15 q^{18}+77 q^{16}-118 q^{14}+120 q^{12}-84 q^{10}+16 q^8+55 q^6-109 q^4+123 q^2-97+43 q^{-2} +15 q^{-4} -64 q^{-6} +72 q^{-8} -52 q^{-10} +6 q^{-12} +39 q^{-14} -60 q^{-16} +48 q^{-18} -8 q^{-20} -41 q^{-22} +79 q^{-24} -87 q^{-26} +65 q^{-28} -21 q^{-30} -29 q^{-32} +67 q^{-34} -77 q^{-36} +67 q^{-38} -34 q^{-40} +5 q^{-42} +19 q^{-44} -33 q^{-46} +32 q^{-48} -23 q^{-50} +13 q^{-52} -2 q^{-54} -4 q^{-56} +5 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 30"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-12 t+17-12 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]=

{ 53, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-3 q^3+6 q^2-8 q+9-9 q^{-1} +8 q^{-2} -5 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^6+2 a^2 z^4+z^4 a^{-2} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^4+4 a^2+2 a^{-2} -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+7 a z^7+4 z^7 a^{-1} +3 a^4 z^6+8 a^2 z^6+5 z^6 a^{-2} +10 z^6+a^5 z^5-3 a^3 z^5-9 a z^5-2 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4-22 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -23 z^4-2 a^5 z^3-3 a^3 z^3-2 z^3 a^{-1} -3 z^3 a^{-3} +5 a^4 z^2+16 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +17 z^2+a^5 z+2 a^3 z+a z+z a^{-1} +z a^{-3} -a^4-4 a^2-2 a^{-2} -4 }[/math]