10 11

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -4 t^2+11 t-13+11 t^{-1} -4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -4 z^4-5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 43, -2 }
Jones polynomial	[math]\displaystyle{ q^3-q^2+3 q-5+6 q^{-1} -7 q^{-2} +7 q^{-3} -6 q^{-4} +4 q^{-5} -2 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-z^2 a^4-2 z^4 a^2-4 z^2 a^2-a^2-z^4-2 z^2-1+z^2 a^{-2} +2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+2 a^4 z^8+3 a^2 z^8+z^8+3 a^5 z^7-2 a^3 z^7-4 a z^7+z^7 a^{-1} +3 a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -2 z^6+2 a^7 z^5-7 a^5 z^5+5 a^3 z^5+11 a z^5-3 z^5 a^{-1} +a^8 z^4-6 a^6 z^4+5 a^4 z^4+16 a^2 z^4-5 z^4 a^{-2} -z^4-3 a^7 z^3+9 a^5 z^3-5 a^3 z^3-16 a z^3+z^3 a^{-1} -2 a^8 z^2+5 a^6 z^2-12 a^2 z^2+7 z^2 a^{-2} +2 z^2-2 a^5 z+2 a^3 z+5 a z+z a^{-1} -a^6+a^2-2 a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}+2 q^{16}-q^{14}-q^8+q^6-2 q^4-1- q^{-2} +2 q^{-4} + q^{-6} + q^{-8} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+2 q^{106}-q^{104}-2 q^{102}+6 q^{100}-8 q^{98}+10 q^{96}-9 q^{94}+5 q^{92}+2 q^{90}-11 q^{88}+19 q^{86}-21 q^{84}+18 q^{82}-12 q^{80}-q^{78}+15 q^{76}-23 q^{74}+29 q^{72}-21 q^{70}+9 q^{68}+5 q^{66}-16 q^{64}+19 q^{62}-13 q^{60}+2 q^{58}+14 q^{56}-19 q^{54}+16 q^{52}+q^{50}-19 q^{48}+33 q^{46}-37 q^{44}+25 q^{42}-7 q^{40}-18 q^{38}+36 q^{36}-42 q^{34}+36 q^{32}-20 q^{30}-5 q^{28}+19 q^{26}-29 q^{24}+25 q^{22}-17 q^{20}+q^{18}+13 q^{16}-18 q^{14}+15 q^{12}-q^{10}-15 q^8+24 q^6-24 q^4+10 q^2+5-21 q^{-2} +31 q^{-4} -28 q^{-6} +19 q^{-8} -3 q^{-10} -14 q^{-12} +20 q^{-14} -19 q^{-16} +16 q^{-18} -8 q^{-20} + q^{-22} +5 q^{-24} -6 q^{-26} +9 q^{-28} -6 q^{-30} +4 q^{-32} +2 q^{-38} -2 q^{-40} +2 q^{-42} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_11.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-q^{13}+2 q^{11}-2 q^9+q^7-q^3+q-2 q^{-1} +2 q^{-3} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{42}-q^{40}+3 q^{36}-3 q^{34}-2 q^{32}+6 q^{30}-5 q^{28}-5 q^{26}+10 q^{24}-3 q^{22}-6 q^{20}+6 q^{18}+q^{16}-3 q^{14}-q^{12}+5 q^{10}-6 q^6+6 q^4+4 q^2-8+3 q^{-2} +6 q^{-4} -7 q^{-6} - q^{-8} +4 q^{-10} -3 q^{-12} - q^{-14} +2 q^{-16} + q^{-22} }[/math]
3	[math]\displaystyle{ q^{81}-q^{79}+q^{75}+q^{73}-2 q^{71}-2 q^{69}+2 q^{67}+2 q^{65}-4 q^{63}-3 q^{61}+6 q^{59}+6 q^{57}-10 q^{55}-11 q^{53}+14 q^{51}+17 q^{49}-12 q^{47}-25 q^{45}+12 q^{43}+28 q^{41}-6 q^{39}-26 q^{37}-q^{35}+22 q^{33}+7 q^{31}-14 q^{29}-12 q^{27}+5 q^{25}+15 q^{23}+q^{21}-18 q^{19}-9 q^{17}+20 q^{15}+13 q^{13}-19 q^{11}-16 q^9+16 q^7+22 q^5-13 q^3-24 q+6 q^{-1} +26 q^{-3} +2 q^{-5} -21 q^{-7} -9 q^{-9} +19 q^{-11} +12 q^{-13} -10 q^{-15} -12 q^{-17} +4 q^{-19} +10 q^{-21} - q^{-23} -6 q^{-25} -2 q^{-27} +3 q^{-29} - q^{-33} - q^{-35} + q^{-37} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{132}-q^{130}+q^{126}-q^{124}+2 q^{122}-3 q^{120}-q^{118}+2 q^{116}-3 q^{114}+6 q^{112}-3 q^{110}-q^{108}+3 q^{106}-9 q^{104}+6 q^{102}-5 q^{100}+6 q^{98}+15 q^{96}-9 q^{94}-4 q^{92}-29 q^{90}+4 q^{88}+46 q^{86}+19 q^{84}-6 q^{82}-76 q^{80}-33 q^{78}+62 q^{76}+71 q^{74}+33 q^{72}-100 q^{70}-89 q^{68}+31 q^{66}+92 q^{64}+79 q^{62}-64 q^{60}-100 q^{58}-24 q^{56}+53 q^{54}+88 q^{52}+2 q^{50}-57 q^{48}-52 q^{46}-3 q^{44}+53 q^{42}+45 q^{40}-4 q^{38}-56 q^{36}-41 q^{34}+20 q^{32}+69 q^{30}+29 q^{28}-58 q^{26}-61 q^{24}+q^{22}+88 q^{20}+51 q^{18}-61 q^{16}-80 q^{14}-25 q^{12}+93 q^{10}+79 q^8-31 q^6-84 q^4-72 q^2+60+94 q^{-2} +25 q^{-4} -48 q^{-6} -98 q^{-8} -5 q^{-10} +60 q^{-12} +61 q^{-14} +16 q^{-16} -72 q^{-18} -44 q^{-20} +5 q^{-22} +44 q^{-24} +47 q^{-26} -18 q^{-28} -30 q^{-30} -24 q^{-32} +6 q^{-34} +31 q^{-36} +7 q^{-38} -3 q^{-40} -15 q^{-42} -8 q^{-44} +8 q^{-46} +3 q^{-48} +5 q^{-50} -3 q^{-52} -4 q^{-54} + q^{-56} -2 q^{-58} +2 q^{-60} - q^{-64} + q^{-66} - q^{-68} + q^{-76} }[/math]
5	[math]\displaystyle{ q^{195}-q^{193}+q^{189}-q^{187}+q^{183}-2 q^{181}-2 q^{179}+2 q^{177}+q^{175}+4 q^{171}-q^{169}-6 q^{167}-4 q^{165}-q^{163}+3 q^{161}+10 q^{159}+10 q^{157}-q^{155}-11 q^{153}-18 q^{151}-10 q^{149}+5 q^{147}+26 q^{145}+34 q^{143}+12 q^{141}-29 q^{139}-61 q^{137}-51 q^{135}+11 q^{133}+91 q^{131}+118 q^{129}+34 q^{127}-109 q^{125}-196 q^{123}-119 q^{121}+90 q^{119}+268 q^{117}+239 q^{115}-30 q^{113}-320 q^{111}-357 q^{109}-72 q^{107}+303 q^{105}+458 q^{103}+210 q^{101}-249 q^{99}-495 q^{97}-320 q^{95}+131 q^{93}+464 q^{91}+396 q^{89}+3 q^{87}-374 q^{85}-408 q^{83}-111 q^{81}+237 q^{79}+361 q^{77}+192 q^{75}-102 q^{73}-277 q^{71}-221 q^{69}-11 q^{67}+177 q^{65}+217 q^{63}+96 q^{61}-92 q^{59}-202 q^{57}-145 q^{55}+31 q^{53}+185 q^{51}+182 q^{49}+3 q^{47}-194 q^{45}-210 q^{43}-16 q^{41}+212 q^{39}+244 q^{37}+31 q^{35}-243 q^{33}-297 q^{31}-53 q^{29}+266 q^{27}+353 q^{25}+100 q^{23}-261 q^{21}-402 q^{19}-180 q^{17}+227 q^{15}+432 q^{13}+260 q^{11}-136 q^9-420 q^7-339 q^5+22 q^3+355 q+388 q^{-1} +111 q^{-3} -247 q^{-5} -380 q^{-7} -213 q^{-9} +97 q^{-11} +317 q^{-13} +283 q^{-15} +37 q^{-17} -208 q^{-19} -275 q^{-21} -146 q^{-23} +73 q^{-25} +227 q^{-27} +194 q^{-29} +29 q^{-31} -132 q^{-33} -183 q^{-35} -102 q^{-37} +38 q^{-39} +135 q^{-41} +118 q^{-43} +25 q^{-45} -66 q^{-47} -97 q^{-49} -60 q^{-51} +16 q^{-53} +63 q^{-55} +55 q^{-57} +16 q^{-59} -24 q^{-61} -41 q^{-63} -22 q^{-65} +4 q^{-67} +19 q^{-69} +20 q^{-71} +7 q^{-73} -9 q^{-75} -10 q^{-77} -6 q^{-79} -2 q^{-81} +6 q^{-83} +6 q^{-85} - q^{-89} - q^{-91} -4 q^{-93} +2 q^{-97} + q^{-103} - q^{-105} - q^{-107} + q^{-115} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}+2 q^{16}-q^{14}-q^8+q^6-2 q^4-1- q^{-2} +2 q^{-4} + q^{-6} + q^{-8} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{60}-2 q^{58}+4 q^{56}-8 q^{54}+15 q^{52}-20 q^{50}+28 q^{48}-40 q^{46}+54 q^{44}-60 q^{42}+68 q^{40}-80 q^{38}+80 q^{36}-70 q^{34}+52 q^{32}-34 q^{30}-3 q^{28}+44 q^{26}-86 q^{24}+120 q^{22}-150 q^{20}+170 q^{18}-168 q^{16}+166 q^{14}-137 q^{12}+110 q^{10}-66 q^8+30 q^6+7 q^4-48 q^2+72-84 q^{-2} +84 q^{-4} -86 q^{-6} +72 q^{-8} -56 q^{-10} +40 q^{-12} -32 q^{-14} +22 q^{-16} -12 q^{-18} +10 q^{-20} -4 q^{-22} +4 q^{-24} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{56}+q^{50}+2 q^{48}-3 q^{44}+3 q^{40}-3 q^{38}-5 q^{36}+q^{34}+4 q^{32}-2 q^{30}-5 q^{28}+3 q^{26}+2 q^{24}-3 q^{22}+q^{20}+4 q^{18}+3 q^{12}-2 q^8+3 q^6+6 q^4-q^2-3+4 q^{-2} + q^{-4} -5 q^{-6} -5 q^{-8} - q^{-10} -3 q^{-14} - q^{-16} +2 q^{-18} +3 q^{-20} + q^{-22} + q^{-24} + q^{-26} + q^{-28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-q^{46}+2 q^{42}-3 q^{40}+6 q^{36}-5 q^{34}-2 q^{32}+8 q^{30}-4 q^{28}-3 q^{26}+6 q^{24}-2 q^{22}-3 q^{20}+q^{18}+2 q^{16}-2 q^{12}+6 q^{10}+2 q^8-7 q^6+2 q^4+2 q^2-9+ q^{-2} +2 q^{-4} -4 q^{-6} +3 q^{-8} +2 q^{-10} +2 q^{-14} +2 q^{-16} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ q^{29}+q^{25}+2 q^{21}-q^{19}+q^{17}-q^{15}-q^{11}-2 q^5-2 q- q^{-3} +2 q^{-5} + q^{-7} +2 q^{-9} + q^{-11} + q^{-13} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-q^{46}+2 q^{44}-4 q^{42}+5 q^{40}-6 q^{38}+8 q^{36}-7 q^{34}+8 q^{32}-6 q^{30}+4 q^{28}+q^{26}-4 q^{24}+8 q^{22}-11 q^{20}+13 q^{18}-16 q^{16}+14 q^{14}-14 q^{12}+10 q^{10}-8 q^8+3 q^6-4 q^2+5-7 q^{-2} +8 q^{-4} -6 q^{-6} +7 q^{-8} -4 q^{-10} +4 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-20} }[/math]
1,0	[math]\displaystyle{ q^{78}-q^{74}-q^{72}+q^{70}+3 q^{68}-4 q^{64}-3 q^{62}+3 q^{60}+7 q^{58}-7 q^{54}-5 q^{52}+5 q^{50}+8 q^{48}-q^{46}-8 q^{44}-2 q^{42}+6 q^{40}+4 q^{38}-5 q^{36}-5 q^{34}+3 q^{32}+5 q^{30}-q^{28}-5 q^{26}+q^{24}+5 q^{22}+2 q^{20}-4 q^{18}-q^{16}+5 q^{14}+4 q^{12}-5 q^{10}-7 q^8+2 q^6+8 q^4+q^2-8-7 q^{-2} +4 q^{-4} +7 q^{-6} - q^{-8} -6 q^{-10} -3 q^{-12} +4 q^{-14} +3 q^{-16} -2 q^{-20} +2 q^{-24} +2 q^{-26} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+2 q^{106}-q^{104}-2 q^{102}+6 q^{100}-8 q^{98}+10 q^{96}-9 q^{94}+5 q^{92}+2 q^{90}-11 q^{88}+19 q^{86}-21 q^{84}+18 q^{82}-12 q^{80}-q^{78}+15 q^{76}-23 q^{74}+29 q^{72}-21 q^{70}+9 q^{68}+5 q^{66}-16 q^{64}+19 q^{62}-13 q^{60}+2 q^{58}+14 q^{56}-19 q^{54}+16 q^{52}+q^{50}-19 q^{48}+33 q^{46}-37 q^{44}+25 q^{42}-7 q^{40}-18 q^{38}+36 q^{36}-42 q^{34}+36 q^{32}-20 q^{30}-5 q^{28}+19 q^{26}-29 q^{24}+25 q^{22}-17 q^{20}+q^{18}+13 q^{16}-18 q^{14}+15 q^{12}-q^{10}-15 q^8+24 q^6-24 q^4+10 q^2+5-21 q^{-2} +31 q^{-4} -28 q^{-6} +19 q^{-8} -3 q^{-10} -14 q^{-12} +20 q^{-14} -19 q^{-16} +16 q^{-18} -8 q^{-20} + q^{-22} +5 q^{-24} -6 q^{-26} +9 q^{-28} -6 q^{-30} +4 q^{-32} +2 q^{-38} -2 q^{-40} +2 q^{-42} + q^{-46} }[/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["10 11"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -4 t^2+11 t-13+11 t^{-1} -4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -4 z^4-5 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]=

{ 43, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-z^2 a^4-2 z^4 a^2-4 z^2 a^2-a^2-z^4-2 z^2-1+z^2 a^{-2} +2 a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^9+a z^9+2 a^4 z^8+3 a^2 z^8+z^8+3 a^5 z^7-2 a^3 z^7-4 a z^7+z^7 a^{-1} +3 a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -2 z^6+2 a^7 z^5-7 a^5 z^5+5 a^3 z^5+11 a z^5-3 z^5 a^{-1} +a^8 z^4-6 a^6 z^4+5 a^4 z^4+16 a^2 z^4-5 z^4 a^{-2} -z^4-3 a^7 z^3+9 a^5 z^3-5 a^3 z^3-16 a z^3+z^3 a^{-1} -2 a^8 z^2+5 a^6 z^2-12 a^2 z^2+7 z^2 a^{-2} +2 z^2-2 a^5 z+2 a^3 z+5 a z+z a^{-1} -a^6+a^2-2 a^{-2} -1 }[/math]