10 8
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -2 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -2 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ -2 z^6-7 z^4-3 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 29, -4 } |
| Jones polynomial | [math]\displaystyle{ q^2-q+2-3 q^{-1} +4 q^{-2} -4 q^{-3} +4 q^{-4} -4 q^{-5} +3 q^{-6} -2 q^{-7} + q^{-8} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^4 a^6+3 z^2 a^6+a^6-z^6 a^4-4 z^4 a^4-3 z^2 a^4-z^6 a^2-5 z^4 a^2-7 z^2 a^2-3 a^2+z^4+4 z^2+3 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^2 a^{10}+2 z^3 a^9+3 z^4 a^8-2 z^2 a^8+4 z^5 a^7-7 z^3 a^7+2 z a^7+4 z^6 a^6-10 z^4 a^6+5 z^2 a^6-a^6+3 z^7 a^5-8 z^5 a^5+2 z^3 a^5+z a^5+2 z^8 a^4-6 z^6 a^4+z^4 a^4+3 z^2 a^4+z^9 a^3-3 z^7 a^3-z^5 a^3+5 z^3 a^3-z a^3+3 z^8 a^2-17 z^6 a^2+30 z^4 a^2-18 z^2 a^2+3 a^2+z^9 a-6 z^7 a+11 z^5 a-6 z^3 a+z^8-7 z^6+16 z^4-13 z^2+3 }[/math] |
| The A2 invariant | [math]\displaystyle{ q^{24}+q^{14}-q^{12}-q^8-q^6+1+ q^{-2} + q^{-4} + q^{-6} }[/math] |
| The G2 invariant | [math]\displaystyle{ q^{134}-q^{132}+q^{130}-q^{128}-q^{122}+2 q^{120}-2 q^{118}+2 q^{116}-2 q^{114}+q^{110}-q^{108}+3 q^{106}-3 q^{104}+2 q^{102}-q^{100}-q^{98}+2 q^{96}-3 q^{94}+3 q^{92}-2 q^{90}+q^{88}+q^{86}-q^{84}+2 q^{82}+q^{78}+q^{74}+2 q^{70}+q^{68}+q^{66}+q^{62}-q^{60}-3 q^{54}+3 q^{52}-5 q^{50}+3 q^{48}-q^{46}-5 q^{44}+5 q^{42}-7 q^{40}+2 q^{38}-2 q^{36}-2 q^{34}+2 q^{32}-3 q^{30}+3 q^{28}-q^{26}-2 q^{20}+q^{18}+q^{16}-q^{14}+2 q^{12}-3 q^{10}+3 q^8+q^6-3 q^4+6 q^2-6+4 q^{-2} + q^{-4} -3 q^{-6} +6 q^{-8} -4 q^{-10} +4 q^{-12} +2 q^{-18} -2 q^{-20} +2 q^{-22} + q^{-26} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 10_8.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | [math]\displaystyle{ q^{17}-q^{15}+q^{13}-q^{11}+q^3-q+ q^{-1} + q^{-5} }[/math] |
| 2 | [math]\displaystyle{ q^{46}-q^{44}+q^{40}-2 q^{38}+q^{36}+q^{34}-q^{32}-q^{26}+q^{22}-q^{20}+2 q^{16}+q^{14}-q^{12}+q^8-q^6-q^4+2 q^2- q^{-2} +2 q^{-4} -2 q^{-8} + q^{-10} + q^{-12} - q^{-14} + q^{-18} }[/math] |
| 3 | [math]\displaystyle{ q^{87}-q^{85}+q^{73}-q^{71}-q^{69}-q^{67}+q^{65}+q^{63}-2 q^{59}+4 q^{55}+2 q^{53}-3 q^{51}-2 q^{49}+q^{47}+2 q^{45}+q^{43}-q^{41}-2 q^{39}-q^{37}+3 q^{35}+q^{33}-3 q^{31}-3 q^{29}+q^{27}+q^{25}-q^{23}-q^{21}+q^{19}+q^{17}+q^{15}+q^{11}+2 q^9+q^7-q^5-q^3+q+2 q^{-1} -3 q^{-5} -2 q^{-7} +2 q^{-9} +3 q^{-11} -3 q^{-15} - q^{-17} +3 q^{-19} +2 q^{-21} - q^{-23} -3 q^{-25} +2 q^{-29} + q^{-31} - q^{-33} - q^{-35} + q^{-39} }[/math] |
| 4 | [math]\displaystyle{ q^{140}-q^{138}-q^{132}+2 q^{130}-q^{128}+q^{126}-q^{124}-2 q^{122}+q^{120}-2 q^{118}+3 q^{116}+2 q^{114}-2 q^{112}-2 q^{110}-2 q^{108}+6 q^{106}+5 q^{104}-2 q^{102}-6 q^{100}-7 q^{98}+6 q^{96}+9 q^{94}+2 q^{92}-6 q^{90}-11 q^{88}+q^{86}+8 q^{84}+6 q^{82}-q^{80}-10 q^{78}-5 q^{76}+2 q^{74}+6 q^{72}+8 q^{70}-q^{68}-7 q^{66}-7 q^{64}+9 q^{60}+7 q^{58}-q^{56}-8 q^{54}-6 q^{52}+4 q^{50}+8 q^{48}+2 q^{46}-4 q^{44}-4 q^{42}+2 q^{40}+3 q^{38}-2 q^{36}-4 q^{34}-2 q^{32}+3 q^{30}+5 q^{28}-2 q^{26}-5 q^{24}-3 q^{22}+2 q^{20}+7 q^{18}+2 q^{16}-q^{14}-4 q^{12}-5 q^{10}+3 q^8+4 q^6+5 q^4+q^2-7-3 q^{-2} +6 q^{-6} +7 q^{-8} -2 q^{-10} -4 q^{-12} -6 q^{-14} +7 q^{-18} +4 q^{-20} + q^{-22} -6 q^{-24} -6 q^{-26} +3 q^{-30} +7 q^{-32} + q^{-34} -4 q^{-36} -4 q^{-38} -3 q^{-40} +4 q^{-42} +4 q^{-44} +2 q^{-46} - q^{-48} -5 q^{-50} - q^{-52} + q^{-54} +2 q^{-56} +2 q^{-58} - q^{-60} - q^{-62} - q^{-64} + q^{-68} }[/math] |
| 5 | [math]\displaystyle{ q^{205}-q^{203}-q^{197}+q^{195}+q^{193}-q^{189}-q^{187}-2 q^{185}+2 q^{181}+3 q^{179}+q^{177}-2 q^{175}-4 q^{173}-2 q^{171}+5 q^{169}+8 q^{167}+2 q^{165}-8 q^{163}-11 q^{161}-3 q^{159}+10 q^{157}+16 q^{155}+3 q^{153}-15 q^{151}-16 q^{149}-4 q^{147}+13 q^{145}+19 q^{143}+5 q^{141}-15 q^{139}-20 q^{137}-5 q^{135}+14 q^{133}+21 q^{131}+8 q^{129}-12 q^{127}-24 q^{125}-12 q^{123}+11 q^{121}+22 q^{119}+15 q^{117}-3 q^{115}-16 q^{113}-18 q^{111}-6 q^{109}+10 q^{107}+16 q^{105}+13 q^{103}+3 q^{101}-11 q^{99}-19 q^{97}-15 q^{95}+4 q^{93}+17 q^{91}+20 q^{89}+8 q^{87}-11 q^{85}-20 q^{83}-11 q^{81}+5 q^{79}+14 q^{77}+10 q^{75}-8 q^{71}-7 q^{69}+q^{67}+3 q^{65}-5 q^{61}-7 q^{59}+q^{57}+11 q^{55}+10 q^{53}-13 q^{49}-16 q^{47}-3 q^{45}+15 q^{43}+20 q^{41}+10 q^{39}-9 q^{37}-20 q^{35}-14 q^{33}+4 q^{31}+17 q^{29}+16 q^{27}+3 q^{25}-14 q^{23}-17 q^{21}-7 q^{19}+7 q^{17}+14 q^{15}+10 q^{13}+q^{11}-9 q^9-11 q^7-4 q^5+3 q^3+8 q+10 q^{-1} +5 q^{-3} -4 q^{-5} -9 q^{-7} -9 q^{-9} -4 q^{-11} +4 q^{-13} +12 q^{-15} +11 q^{-17} + q^{-19} -8 q^{-21} -13 q^{-23} -9 q^{-25} + q^{-27} +12 q^{-29} +14 q^{-31} +6 q^{-33} -4 q^{-35} -12 q^{-37} -12 q^{-39} -4 q^{-41} +7 q^{-43} +12 q^{-45} +9 q^{-47} +2 q^{-49} -7 q^{-51} -11 q^{-53} -8 q^{-55} +7 q^{-59} +9 q^{-61} +6 q^{-63} - q^{-65} -6 q^{-67} -8 q^{-69} -4 q^{-71} +2 q^{-73} +5 q^{-75} +6 q^{-77} +3 q^{-79} -2 q^{-81} -5 q^{-83} -3 q^{-85} - q^{-87} + q^{-89} +3 q^{-91} +2 q^{-93} - q^{-97} - q^{-99} - q^{-101} + q^{-105} }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{24}+q^{14}-q^{12}-q^8-q^6+1+ q^{-2} + q^{-4} + q^{-6} }[/math] |
| 1,1 | [math]\displaystyle{ q^{68}-2 q^{66}+2 q^{64}-2 q^{62}+3 q^{60}-4 q^{58}+4 q^{56}-2 q^{54}+5 q^{52}-6 q^{50}+6 q^{48}-6 q^{46}+5 q^{44}-8 q^{42}+2 q^{40}-2 q^{38}+2 q^{34}-2 q^{32}+4 q^{30}-2 q^{28}+8 q^{26}-6 q^{24}+10 q^{22}-14 q^{20}+16 q^{18}-18 q^{16}+16 q^{14}-15 q^{12}+16 q^{10}-10 q^8+6 q^6-3 q^4-2 q^2+6-12 q^{-2} +13 q^{-4} -14 q^{-6} +14 q^{-8} -10 q^{-10} +8 q^{-12} -4 q^{-14} +4 q^{-16} + q^{-20} }[/math] |
| 2,0 | [math]\displaystyle{ q^{60}+q^{50}-q^{46}-q^{44}-2 q^{38}-q^{36}+q^{34}-2 q^{30}+2 q^{26}+q^{24}+2 q^{22}+2 q^{20}+2 q^{18}+q^{16}+q^{14}-2 q^{10}-q^8-q^4-2 q^2+ q^{-2} - q^{-6} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ q^{56}-q^{54}-q^{52}+2 q^{50}-2 q^{46}+q^{44}-2 q^{40}+q^{38}+q^{36}+q^{32}+2 q^{30}+q^{28}+q^{24}+q^{22}-q^{20}-2 q^{18}-2 q^{14}-2 q^{12}-2 q^8+q^4+1+2 q^{-2} + q^{-4} + q^{-6} +2 q^{-8} + q^{-12} }[/math] |
| 1,0,0 | [math]\displaystyle{ q^{31}+q^{27}-q^{25}+q^{23}+q^{19}-q^{13}-2 q^{11}-q^9-2 q^7+2 q+ q^{-1} +2 q^{-3} + q^{-5} + q^{-7} }[/math] |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{70}-q^{66}+q^{62}-q^{58}+q^{54}-2 q^{50}-q^{44}+q^{40}+2 q^{38}+2 q^{36}+3 q^{34}+2 q^{32}+q^{30}+q^{26}-q^{24}-3 q^{22}-q^{16}-2 q^{14}-q^{12}-2 q^{10}-3 q^8-3 q^6-2 q^4+1+3 q^{-2} +3 q^{-4} +4 q^{-6} +3 q^{-8} +2 q^{-10} + q^{-12} + q^{-14} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^{38}+q^{34}+q^{28}+q^{24}+q^{20}-q^{18}-q^{16}-2 q^{14}-2 q^{12}-2 q^{10}-2 q^8+2 q^2+2+2 q^{-2} +2 q^{-4} + q^{-6} + q^{-8} }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ q^{56}-q^{54}+q^{52}-2 q^{50}+2 q^{48}-2 q^{46}+3 q^{44}-2 q^{42}+2 q^{40}-q^{38}+q^{36}-q^{32}+2 q^{30}-3 q^{28}+4 q^{26}-5 q^{24}+5 q^{22}-5 q^{20}+4 q^{18}-4 q^{16}+2 q^{14}-2 q^{12}-2 q^6+3 q^4-2 q^2+3-2 q^{-2} +3 q^{-4} - q^{-6} +2 q^{-8} + q^{-12} }[/math] |
| 1,0 | [math]\displaystyle{ q^{90}-q^{86}-q^{84}+2 q^{80}+q^{78}-q^{76}-2 q^{74}-q^{72}+2 q^{70}+q^{68}-q^{66}-2 q^{64}+2 q^{60}+q^{58}-q^{56}-q^{54}+q^{52}+2 q^{50}+q^{48}-q^{46}-q^{44}+q^{42}+2 q^{40}-q^{36}+q^{32}-q^{30}-2 q^{28}+2 q^{24}-3 q^{20}-2 q^{18}+q^{16}+2 q^{14}-3 q^{10}-q^8+2 q^6+2 q^4-2+2 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-12} +2 q^{-14} + q^{-22} }[/math] |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{78}-q^{76}-q^{72}+2 q^{70}-q^{68}+q^{66}-2 q^{64}+2 q^{62}-2 q^{60}+q^{58}-2 q^{56}+q^{54}-q^{52}+q^{50}+q^{48}+q^{46}+2 q^{44}+3 q^{40}-q^{38}+4 q^{36}-3 q^{34}+4 q^{32}-4 q^{30}+4 q^{28}-4 q^{26}+2 q^{24}-5 q^{22}-3 q^{18}-q^{16}-3 q^{14}-2 q^{12}-2 q^8+2 q^6-q^4+4 q^2+4 q^{-2} +4 q^{-6} +2 q^{-10} + q^{-14} }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{134}-q^{132}+q^{130}-q^{128}-q^{122}+2 q^{120}-2 q^{118}+2 q^{116}-2 q^{114}+q^{110}-q^{108}+3 q^{106}-3 q^{104}+2 q^{102}-q^{100}-q^{98}+2 q^{96}-3 q^{94}+3 q^{92}-2 q^{90}+q^{88}+q^{86}-q^{84}+2 q^{82}+q^{78}+q^{74}+2 q^{70}+q^{68}+q^{66}+q^{62}-q^{60}-3 q^{54}+3 q^{52}-5 q^{50}+3 q^{48}-q^{46}-5 q^{44}+5 q^{42}-7 q^{40}+2 q^{38}-2 q^{36}-2 q^{34}+2 q^{32}-3 q^{30}+3 q^{28}-q^{26}-2 q^{20}+q^{18}+q^{16}-q^{14}+2 q^{12}-3 q^{10}+3 q^8+q^6-3 q^4+6 q^2-6+4 q^{-2} + q^{-4} -3 q^{-6} +6 q^{-8} -4 q^{-10} +4 q^{-12} +2 q^{-18} -2 q^{-20} +2 q^{-22} + q^{-26} }[/math] |
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 8"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ -2 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -2 t^{-3} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -2 z^6-7 z^4-3 z^2+1 }[/math] |
In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 29, -4 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ q^2-q+2-3 q^{-1} +4 q^{-2} -4 q^{-3} +4 q^{-4} -4 q^{-5} +3 q^{-6} -2 q^{-7} + q^{-8} }[/math] |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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[math]\displaystyle{ z^4 a^6+3 z^2 a^6+a^6-z^6 a^4-4 z^4 a^4-3 z^2 a^4-z^6 a^2-5 z^4 a^2-7 z^2 a^2-3 a^2+z^4+4 z^2+3 }[/math] |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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[math]\displaystyle{ z^2 a^{10}+2 z^3 a^9+3 z^4 a^8-2 z^2 a^8+4 z^5 a^7-7 z^3 a^7+2 z a^7+4 z^6 a^6-10 z^4 a^6+5 z^2 a^6-a^6+3 z^7 a^5-8 z^5 a^5+2 z^3 a^5+z a^5+2 z^8 a^4-6 z^6 a^4+z^4 a^4+3 z^2 a^4+z^9 a^3-3 z^7 a^3-z^5 a^3+5 z^3 a^3-z a^3+3 z^8 a^2-17 z^6 a^2+30 z^4 a^2-18 z^2 a^2+3 a^2+z^9 a-6 z^7 a+11 z^5 a-6 z^3 a+z^8-7 z^6+16 z^4-13 z^2+3 }[/math] |
