9 41

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^2-12 t+19-12 t^{-1} +3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{7,t+1\} }[/math]
Determinant and Signature	{ 49, 0 }
Jones polynomial	[math]\displaystyle{ -q^3+3 q^2-5 q+8-8 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^6-3 z^2 a^4-3 a^4+2 z^4 a^2+4 z^2 a^2+3 a^2+z^4-z^2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^4 z^8+2 a^2 z^8+3 a^5 z^7+9 a^3 z^7+6 a z^7+a^6 z^6-a^4 z^6+5 a^2 z^6+7 z^6-10 a^5 z^5-26 a^3 z^5-11 a z^5+5 z^5 a^{-1} -3 a^6 z^4-12 a^4 z^4-23 a^2 z^4+3 z^4 a^{-2} -11 z^4+9 a^5 z^3+19 a^3 z^3+6 a z^3-3 z^3 a^{-1} +z^3 a^{-3} +3 a^6 z^2+13 a^4 z^2+17 a^2 z^2-z^2 a^{-2} +6 z^2-2 a^5 z-4 a^3 z-2 a z-a^6-3 a^4-3 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}-2 q^{16}-q^{12}-2 q^{10}+2 q^8+2 q^4+q^2+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-2 q^{92}+6 q^{90}-10 q^{88}+11 q^{86}-7 q^{84}-7 q^{82}+27 q^{80}-39 q^{78}+44 q^{76}-28 q^{74}-3 q^{72}+40 q^{70}-66 q^{68}+70 q^{66}-45 q^{64}+q^{62}+37 q^{60}-64 q^{58}+54 q^{56}-25 q^{54}-16 q^{52}+42 q^{50}-49 q^{48}+27 q^{46}+7 q^{44}-43 q^{42}+63 q^{40}-60 q^{38}+37 q^{36}+6 q^{34}-46 q^{32}+79 q^{30}-82 q^{28}+64 q^{26}-19 q^{24}-28 q^{22}+65 q^{20}-77 q^{18}+58 q^{16}-17 q^{14}-25 q^{12}+51 q^{10}-48 q^8+18 q^6+19 q^4-46 q^2+51-30 q^{-2} -3 q^{-4} +35 q^{-6} -51 q^{-8} +53 q^{-10} -32 q^{-12} +8 q^{-14} +16 q^{-16} -32 q^{-18} +33 q^{-20} -27 q^{-22} +18 q^{-24} -7 q^{-26} -2 q^{-28} +9 q^{-30} -14 q^{-32} +13 q^{-34} -10 q^{-36} +7 q^{-38} -2 q^{-40} - q^{-42} +2 q^{-44} -4 q^{-46} +3 q^{-48} -2 q^{-50} + q^{-52} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_41.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-2 q^{11}+2 q^9-2 q^7+q^5+3 q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} }[/math]
2	[math]\displaystyle{ q^{38}-2 q^{36}-3 q^{34}+7 q^{32}+q^{30}-11 q^{28}+7 q^{26}+9 q^{24}-13 q^{22}+12 q^{18}-8 q^{16}-6 q^{14}+9 q^{12}-8 q^8+2 q^6+9 q^4-5 q^2-8+14 q^{-2} + q^{-4} -13 q^{-6} +9 q^{-8} +4 q^{-10} -7 q^{-12} +3 q^{-14} -2 q^{-18} + q^{-20} }[/math]
3	[math]\displaystyle{ q^{75}-2 q^{73}-3 q^{71}+2 q^{69}+10 q^{67}+4 q^{65}-18 q^{63}-16 q^{61}+16 q^{59}+34 q^{57}-4 q^{55}-47 q^{53}-17 q^{51}+46 q^{49}+41 q^{47}-34 q^{45}-60 q^{43}+15 q^{41}+66 q^{39}+9 q^{37}-62 q^{35}-28 q^{33}+55 q^{31}+39 q^{29}-43 q^{27}-46 q^{25}+32 q^{23}+50 q^{21}-19 q^{19}-53 q^{17}+7 q^{15}+49 q^{13}+11 q^{11}-47 q^9-33 q^7+32 q^5+57 q^3-11 q-66 q^{-1} -11 q^{-3} +66 q^{-5} +35 q^{-7} -56 q^{-9} -42 q^{-11} +34 q^{-13} +40 q^{-15} -15 q^{-17} -26 q^{-19} +5 q^{-21} +14 q^{-23} -4 q^{-25} -4 q^{-27} + q^{-31} - q^{-33} +2 q^{-37} - q^{-39} }[/math]
4	[math]\displaystyle{ q^{124}-2 q^{122}-3 q^{120}+2 q^{118}+5 q^{116}+13 q^{114}-3 q^{112}-23 q^{110}-24 q^{108}-8 q^{106}+55 q^{104}+57 q^{102}+3 q^{100}-72 q^{98}-121 q^{96}-2 q^{94}+116 q^{92}+159 q^{90}+43 q^{88}-191 q^{86}-202 q^{84}-47 q^{82}+217 q^{80}+294 q^{78}+9 q^{76}-254 q^{74}-328 q^{72}-12 q^{70}+352 q^{68}+302 q^{66}-22 q^{64}-398 q^{62}-304 q^{60}+142 q^{58}+393 q^{56}+244 q^{54}-237 q^{52}-398 q^{50}-86 q^{48}+293 q^{46}+335 q^{44}-67 q^{42}-340 q^{40}-178 q^{38}+193 q^{36}+310 q^{34}+9 q^{32}-275 q^{30}-205 q^{28}+130 q^{26}+287 q^{24}+96 q^{22}-210 q^{20}-278 q^{18}-8 q^{16}+244 q^{14}+274 q^{12}-17 q^{10}-326 q^8-273 q^6+42 q^4+409 q^2+310-163 q^{-2} -441 q^{-4} -287 q^{-6} +269 q^{-8} +479 q^{-10} +133 q^{-12} -301 q^{-14} -416 q^{-16} -3 q^{-18} +316 q^{-20} +233 q^{-22} -46 q^{-24} -251 q^{-26} -98 q^{-28} +88 q^{-30} +116 q^{-32} +38 q^{-34} -76 q^{-36} -41 q^{-38} +9 q^{-40} +20 q^{-42} +20 q^{-44} -17 q^{-46} -6 q^{-48} +2 q^{-50} +6 q^{-54} -3 q^{-56} + q^{-58} -2 q^{-62} + q^{-64} }[/math]
5	[math]\displaystyle{ q^{185}-2 q^{183}-3 q^{181}+2 q^{179}+5 q^{177}+8 q^{175}+6 q^{173}-8 q^{171}-31 q^{169}-30 q^{167}-2 q^{165}+44 q^{163}+84 q^{161}+72 q^{159}-17 q^{157}-143 q^{155}-190 q^{153}-106 q^{151}+98 q^{149}+309 q^{147}+346 q^{145}+105 q^{143}-294 q^{141}-567 q^{139}-486 q^{137}+2 q^{135}+617 q^{133}+899 q^{131}+537 q^{129}-314 q^{127}-1081 q^{125}-1177 q^{123}-371 q^{121}+857 q^{119}+1618 q^{117}+1232 q^{115}-157 q^{113}-1597 q^{111}-1983 q^{109}-858 q^{107}+1052 q^{105}+2327 q^{103}+1871 q^{101}-92 q^{99}-2114 q^{97}-2599 q^{95}-1026 q^{93}+1445 q^{91}+2851 q^{89}+1988 q^{87}-505 q^{85}-2620 q^{83}-2615 q^{81}-445 q^{79}+2075 q^{77}+2828 q^{75}+1183 q^{73}-1396 q^{71}-2689 q^{69}-1637 q^{67}+776 q^{65}+2358 q^{63}+1770 q^{61}-329 q^{59}-1961 q^{57}-1691 q^{55}+88 q^{53}+1632 q^{51}+1513 q^{49}-23 q^{47}-1421 q^{45}-1360 q^{43}+34 q^{41}+1345 q^{39}+1316 q^{37}+17 q^{35}-1317 q^{33}-1448 q^{31}-230 q^{29}+1238 q^{27}+1676 q^{25}+701 q^{23}-931 q^{21}-1927 q^{19}-1397 q^{17}+334 q^{15}+1957 q^{13}+2167 q^{11}+614 q^9-1637 q^7-2789 q^5-1726 q^3+888 q+2982 q^{-1} +2752 q^{-3} +198 q^{-5} -2647 q^{-7} -3387 q^{-9} -1306 q^{-11} +1844 q^{-13} +3402 q^{-15} +2149 q^{-17} -804 q^{-19} -2892 q^{-21} -2461 q^{-23} -109 q^{-25} +2034 q^{-27} +2259 q^{-29} +678 q^{-31} -1154 q^{-33} -1723 q^{-35} -843 q^{-37} +492 q^{-39} +1118 q^{-41} +702 q^{-43} -119 q^{-45} -603 q^{-47} -478 q^{-49} -30 q^{-51} +297 q^{-53} +265 q^{-55} +36 q^{-57} -118 q^{-59} -124 q^{-61} -33 q^{-63} +51 q^{-65} +55 q^{-67} +9 q^{-69} -23 q^{-71} -17 q^{-73} + q^{-75} +6 q^{-77} +8 q^{-79} +2 q^{-81} -6 q^{-83} -4 q^{-85} +3 q^{-87} - q^{-89} +2 q^{-93} - q^{-95} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}-2 q^{16}-q^{12}-2 q^{10}+2 q^8+2 q^4+q^2+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{52}-4 q^{50}+14 q^{48}-36 q^{46}+66 q^{44}-110 q^{42}+156 q^{40}-196 q^{38}+224 q^{36}-216 q^{34}+182 q^{32}-112 q^{30}+22 q^{28}+80 q^{26}-188 q^{24}+272 q^{22}-339 q^{20}+376 q^{18}-378 q^{16}+346 q^{14}-280 q^{12}+190 q^{10}-96 q^8-4 q^6+84 q^4-134 q^2+168-158 q^{-2} +149 q^{-4} -122 q^{-6} +98 q^{-8} -78 q^{-10} +56 q^{-12} -46 q^{-14} +38 q^{-16} -28 q^{-18} +18 q^{-20} -12 q^{-22} +8 q^{-24} -4 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{52}+q^{50}-q^{48}-5 q^{46}-3 q^{44}+4 q^{42}+5 q^{40}-2 q^{38}-3 q^{36}+6 q^{34}+8 q^{32}-3 q^{30}-8 q^{28}+2 q^{26}+2 q^{24}-4 q^{22}-7 q^{20}+3 q^{18}+4 q^{16}-3 q^{14}+q^{12}-q^8+q^6+6 q^4-3 q^2-3+7 q^{-2} +8 q^{-4} -5 q^{-6} -8 q^{-8} +7 q^{-10} +5 q^{-12} -5 q^{-14} -3 q^{-16} +2 q^{-18} +2 q^{-20} -2 q^{-22} - q^{-24} + q^{-26} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}-2 q^{38}+2 q^{36}+2 q^{34}-6 q^{32}+6 q^{30}-8 q^{26}+8 q^{24}-q^{22}-8 q^{20}+7 q^{18}+2 q^{16}-7 q^{14}+2 q^{12}+2 q^{10}-q^8-3 q^6+2 q^4+9 q^2-5+ q^{-2} +11 q^{-4} -9 q^{-6} -2 q^{-8} +8 q^{-10} -5 q^{-12} -3 q^{-14} +5 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} }[/math]
1,0,0	[math]\displaystyle{ q^{27}+q^{25}+q^{23}-2 q^{21}-3 q^{17}-q^{15}-2 q^{13}+2 q^{11}+q^9+2 q^7+2 q^5+q^3+q- q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} + q^{-11} - q^{-13} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{40}-2 q^{38}+6 q^{36}-8 q^{34}+10 q^{32}-12 q^{30}+12 q^{28}-12 q^{26}+8 q^{24}-5 q^{22}-2 q^{20}+7 q^{18}-14 q^{16}+19 q^{14}-22 q^{12}+24 q^{10}-21 q^8+19 q^6-12 q^4+9 q^2-1-3 q^{-2} +9 q^{-4} -11 q^{-6} +12 q^{-8} -12 q^{-10} +9 q^{-12} -7 q^{-14} +5 q^{-16} -3 q^{-18} +2 q^{-20} - q^{-22} }[/math]
1,0	[math]\displaystyle{ q^{66}-2 q^{62}-2 q^{60}+4 q^{58}+5 q^{56}-4 q^{54}-8 q^{52}+11 q^{48}+5 q^{46}-11 q^{44}-10 q^{42}+7 q^{40}+13 q^{38}-13 q^{34}-5 q^{32}+9 q^{30}+8 q^{28}-5 q^{26}-8 q^{24}+2 q^{22}+7 q^{20}-q^{18}-9 q^{16}-q^{14}+9 q^{12}+3 q^{10}-9 q^8-5 q^6+8 q^4+11 q^2-4-10 q^{-2} +2 q^{-4} +14 q^{-6} +3 q^{-8} -10 q^{-10} -8 q^{-12} +5 q^{-14} +10 q^{-16} -7 q^{-20} -5 q^{-22} +2 q^{-24} +5 q^{-26} + q^{-28} -2 q^{-30} -2 q^{-32} + q^{-36} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{94}-2 q^{92}+6 q^{90}-10 q^{88}+11 q^{86}-7 q^{84}-7 q^{82}+27 q^{80}-39 q^{78}+44 q^{76}-28 q^{74}-3 q^{72}+40 q^{70}-66 q^{68}+70 q^{66}-45 q^{64}+q^{62}+37 q^{60}-64 q^{58}+54 q^{56}-25 q^{54}-16 q^{52}+42 q^{50}-49 q^{48}+27 q^{46}+7 q^{44}-43 q^{42}+63 q^{40}-60 q^{38}+37 q^{36}+6 q^{34}-46 q^{32}+79 q^{30}-82 q^{28}+64 q^{26}-19 q^{24}-28 q^{22}+65 q^{20}-77 q^{18}+58 q^{16}-17 q^{14}-25 q^{12}+51 q^{10}-48 q^8+18 q^6+19 q^4-46 q^2+51-30 q^{-2} -3 q^{-4} +35 q^{-6} -51 q^{-8} +53 q^{-10} -32 q^{-12} +8 q^{-14} +16 q^{-16} -32 q^{-18} +33 q^{-20} -27 q^{-22} +18 q^{-24} -7 q^{-26} -2 q^{-28} +9 q^{-30} -14 q^{-32} +13 q^{-34} -10 q^{-36} +7 q^{-38} -2 q^{-40} - q^{-42} +2 q^{-44} -4 q^{-46} +3 q^{-48} -2 q^{-50} + q^{-52} }[/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 41"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 3 t^2-12 t+19-12 t^{-1} +3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 3 z^4+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{ \{7,t+1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^4 z^8+2 a^2 z^8+3 a^5 z^7+9 a^3 z^7+6 a z^7+a^6 z^6-a^4 z^6+5 a^2 z^6+7 z^6-10 a^5 z^5-26 a^3 z^5-11 a z^5+5 z^5 a^{-1} -3 a^6 z^4-12 a^4 z^4-23 a^2 z^4+3 z^4 a^{-2} -11 z^4+9 a^5 z^3+19 a^3 z^3+6 a z^3-3 z^3 a^{-1} +z^3 a^{-3} +3 a^6 z^2+13 a^4 z^2+17 a^2 z^2-z^2 a^{-2} +6 z^2-2 a^5 z-4 a^3 z-2 a z-a^6-3 a^4-3 a^2 }[/math]