10 68

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-14 t+21-14 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 57, 0 }
Jones polynomial	[math]\displaystyle{ -q^3+3 q^2-5 q+8-9 q^{-1} +9 q^{-2} -8 q^{-3} +7 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+a^4+2 z^4 a^2+3 z^2 a^2+a^2+z^4-z^2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^5 z^9+a^3 z^9+2 a^6 z^8+6 a^4 z^8+4 a^2 z^8+a^7 z^7+a^5 z^7+7 a^3 z^7+7 a z^7-9 a^6 z^6-20 a^4 z^6-4 a^2 z^6+7 z^6-5 a^7 z^5-16 a^5 z^5-30 a^3 z^5-14 a z^5+5 z^5 a^{-1} +13 a^6 z^4+17 a^4 z^4-9 a^2 z^4+3 z^4 a^{-2} -10 z^4+8 a^7 z^3+23 a^5 z^3+27 a^3 z^3+8 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -7 a^6 z^2-5 a^4 z^2+7 a^2 z^2-z^2 a^{-2} +4 z^2-4 a^7 z-8 a^5 z-6 a^3 z-2 a z+a^6+a^4-a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{22}-2 q^{16}+2 q^{14}+q^{12}+2 q^8-q^6+q^4+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{108}-q^{106}+4 q^{104}-6 q^{102}+6 q^{100}-6 q^{98}-q^{96}+13 q^{94}-24 q^{92}+32 q^{90}-30 q^{88}+12 q^{86}+15 q^{84}-46 q^{82}+60 q^{80}-60 q^{78}+34 q^{76}+4 q^{74}-44 q^{72}+64 q^{70}-62 q^{68}+38 q^{66}+4 q^{64}-38 q^{62}+45 q^{60}-36 q^{58}+8 q^{56}+28 q^{54}-46 q^{52}+53 q^{50}-28 q^{48}-5 q^{46}+50 q^{44}-77 q^{42}+79 q^{40}-53 q^{38}+8 q^{36}+41 q^{34}-73 q^{32}+86 q^{30}-67 q^{28}+26 q^{26}+21 q^{24}-55 q^{22}+56 q^{20}-41 q^{18}+4 q^{16}+28 q^{14}-38 q^{12}+31 q^{10}-8 q^8-20 q^6+43 q^4-47 q^2+33-7 q^{-2} -20 q^{-4} +40 q^{-6} -43 q^{-8} +39 q^{-10} -21 q^{-12} +5 q^{-14} +12 q^{-16} -27 q^{-18} +28 q^{-20} -25 q^{-22} +17 q^{-24} -7 q^{-26} - q^{-28} +8 q^{-30} -13 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 10_68.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{15}+q^{13}-2 q^{11}+3 q^9-q^7+q^5-q+3 q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} }[/math]
2	[math]\displaystyle{ q^{44}-q^{42}-2 q^{40}+4 q^{38}-8 q^{34}+6 q^{32}+6 q^{30}-12 q^{28}+3 q^{26}+12 q^{24}-10 q^{22}-5 q^{20}+11 q^{18}-4 q^{16}-9 q^{14}+6 q^{12}+6 q^{10}-6 q^8-2 q^6+11 q^4-q^2-11+10 q^{-2} +4 q^{-4} -12 q^{-6} +6 q^{-8} +4 q^{-10} -6 q^{-12} +3 q^{-14} -2 q^{-18} + q^{-20} }[/math]
3	[math]\displaystyle{ -q^{87}+q^{85}+2 q^{83}-5 q^{79}-2 q^{77}+8 q^{75}+8 q^{73}-10 q^{71}-17 q^{69}+6 q^{67}+27 q^{65}+4 q^{63}-33 q^{61}-20 q^{59}+30 q^{57}+37 q^{55}-20 q^{53}-45 q^{51}+q^{49}+52 q^{47}+16 q^{45}-47 q^{43}-32 q^{41}+37 q^{39}+42 q^{37}-25 q^{35}-51 q^{33}+13 q^{31}+53 q^{29}-2 q^{27}-52 q^{25}-11 q^{23}+51 q^{21}+23 q^{19}-41 q^{17}-34 q^{15}+25 q^{13}+42 q^{11}-5 q^9-44 q^7-18 q^5+42 q^3+37 q-28 q^{-1} -45 q^{-3} +17 q^{-5} +47 q^{-7} -7 q^{-9} -37 q^{-11} - q^{-13} +25 q^{-15} + q^{-17} -14 q^{-19} - q^{-21} +8 q^{-23} -2 q^{-25} - q^{-27} - q^{-33} +2 q^{-37} - q^{-39} }[/math]
4	[math]\displaystyle{ q^{144}-q^{142}-2 q^{140}+q^{136}+7 q^{134}-8 q^{130}-8 q^{128}-6 q^{126}+22 q^{124}+20 q^{122}-3 q^{120}-28 q^{118}-47 q^{116}+14 q^{114}+54 q^{112}+56 q^{110}-q^{108}-104 q^{106}-71 q^{104}+13 q^{102}+122 q^{100}+125 q^{98}-51 q^{96}-145 q^{94}-147 q^{92}+41 q^{90}+221 q^{88}+133 q^{86}-44 q^{84}-252 q^{82}-171 q^{80}+119 q^{78}+248 q^{76}+176 q^{74}-154 q^{72}-299 q^{70}-101 q^{68}+173 q^{66}+312 q^{64}+44 q^{62}-255 q^{60}-245 q^{58}+27 q^{56}+301 q^{54}+170 q^{52}-147 q^{50}-273 q^{48}-65 q^{46}+242 q^{44}+215 q^{42}-81 q^{40}-273 q^{38}-109 q^{36}+192 q^{34}+247 q^{32}-8 q^{30}-255 q^{28}-182 q^{26}+77 q^{24}+267 q^{22}+148 q^{20}-133 q^{18}-246 q^{16}-150 q^{14}+158 q^{12}+296 q^{10}+115 q^8-172 q^6-349 q^4-81 q^2+268+307 q^{-2} +33 q^{-4} -335 q^{-6} -243 q^{-8} +98 q^{-10} +279 q^{-12} +159 q^{-14} -170 q^{-16} -201 q^{-18} -22 q^{-20} +127 q^{-22} +128 q^{-24} -52 q^{-26} -87 q^{-28} -27 q^{-30} +30 q^{-32} +55 q^{-34} -17 q^{-36} -19 q^{-38} -5 q^{-40} +19 q^{-44} -8 q^{-46} -3 q^{-48} -3 q^{-52} +6 q^{-54} -2 q^{-56} + q^{-58} -2 q^{-62} + q^{-64} }[/math]
5	[math]\displaystyle{ -q^{215}+q^{213}+2 q^{211}-q^{207}-3 q^{205}-5 q^{203}+10 q^{199}+10 q^{197}+3 q^{195}-8 q^{193}-24 q^{191}-23 q^{189}+6 q^{187}+40 q^{185}+50 q^{183}+23 q^{181}-38 q^{179}-94 q^{177}-84 q^{175}+3 q^{173}+116 q^{171}+168 q^{169}+98 q^{167}-76 q^{165}-239 q^{163}-253 q^{161}-63 q^{159}+225 q^{157}+400 q^{155}+300 q^{153}-55 q^{151}-446 q^{149}-569 q^{147}-265 q^{145}+296 q^{143}+723 q^{141}+665 q^{139}+90 q^{137}-651 q^{135}-1000 q^{133}-610 q^{131}+291 q^{129}+1074 q^{127}+1134 q^{125}+316 q^{123}-842 q^{121}-1452 q^{119}-978 q^{117}+282 q^{115}+1434 q^{113}+1554 q^{111}+441 q^{109}-1088 q^{107}-1837 q^{105}-1162 q^{103}+473 q^{101}+1802 q^{99}+1708 q^{97}+222 q^{95}-1474 q^{93}-1985 q^{91}-855 q^{89}+983 q^{87}+1988 q^{85}+1313 q^{83}-464 q^{81}-1795 q^{79}-1541 q^{77}+40 q^{75}+1504 q^{73}+1573 q^{71}+238 q^{69}-1233 q^{67}-1481 q^{65}-353 q^{63}+1039 q^{61}+1358 q^{59}+358 q^{57}-944 q^{55}-1285 q^{53}-357 q^{51}+931 q^{49}+1305 q^{47}+412 q^{45}-894 q^{43}-1389 q^{41}-622 q^{39}+743 q^{37}+1488 q^{35}+965 q^{33}-394 q^{31}-1462 q^{29}-1378 q^{27}-197 q^{25}+1208 q^{23}+1742 q^{21}+923 q^{19}-683 q^{17}-1868 q^{15}-1658 q^{13}-83 q^{11}+1695 q^9+2208 q^7+912 q^5-1198 q^3-2404 q-1630 q^{-1} +521 q^{-3} +2236 q^{-5} +2054 q^{-7} +153 q^{-9} -1772 q^{-11} -2102 q^{-13} -671 q^{-15} +1186 q^{-17} +1855 q^{-19} +911 q^{-21} -647 q^{-23} -1424 q^{-25} -904 q^{-27} +256 q^{-29} +969 q^{-31} +737 q^{-33} -34 q^{-35} -596 q^{-37} -516 q^{-39} -35 q^{-41} +318 q^{-43} +308 q^{-45} +55 q^{-47} -163 q^{-49} -174 q^{-51} -25 q^{-53} +80 q^{-55} +71 q^{-57} +16 q^{-59} -32 q^{-61} -35 q^{-63} - q^{-65} +19 q^{-67} +11 q^{-69} -6 q^{-71} -8 q^{-73} - q^{-75} +5 q^{-79} +4 q^{-81} -3 q^{-83} -4 q^{-85} +2 q^{-87} - q^{-89} +2 q^{-93} - q^{-95} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{22}-2 q^{16}+2 q^{14}+q^{12}+2 q^8-q^6+q^4+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} }[/math]
2,0	[math]\displaystyle{ q^{58}-q^{54}-q^{52}+2 q^{50}+2 q^{48}-4 q^{46}-4 q^{44}+3 q^{42}+5 q^{40}-3 q^{38}-6 q^{36}+4 q^{34}+8 q^{32}-3 q^{30}-7 q^{28}+q^{26}+4 q^{24}-3 q^{22}-6 q^{20}+2 q^{18}+4 q^{16}-q^{14}+3 q^{12}+2 q^{10}-2 q^8+3 q^6+7 q^4-2 q^2-6+4 q^{-2} +8 q^{-4} -6 q^{-6} -9 q^{-8} +6 q^{-10} +6 q^{-12} -4 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^{46}-q^{44}+2 q^{42}+q^{40}-4 q^{38}+3 q^{36}-2 q^{34}-8 q^{32}+5 q^{30}-2 q^{28}-7 q^{26}+10 q^{24}-5 q^{20}+9 q^{18}+3 q^{16}-4 q^{14}+2 q^{12}+3 q^{10}-6 q^6+2 q^4+7 q^2-8+2 q^{-2} +10 q^{-4} -9 q^{-6} - 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^{29}-q^{25}-2 q^{21}+2 q^{19}+2 q^{15}+2 q^{11}+q^5+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^{46}+q^{44}-4 q^{42}+5 q^{40}-8 q^{38}+11 q^{36}-12 q^{34}+14 q^{32}-13 q^{30}+12 q^{28}-7 q^{26}+2 q^{24}+4 q^{22}-11 q^{20}+17 q^{18}-23 q^{16}+26 q^{14}-26 q^{12}+25 q^{10}-20 q^8+16 q^6-8 q^4+3 q^2+4-8 q^{-2} +12 q^{-4} -13 q^{-6} +13 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^{76}-q^{72}-q^{70}+3 q^{68}+3 q^{66}-3 q^{64}-6 q^{62}+8 q^{58}+3 q^{56}-10 q^{54}-10 q^{52}+5 q^{50}+13 q^{48}+q^{46}-15 q^{44}-7 q^{42}+11 q^{40}+13 q^{38}-4 q^{36}-13 q^{34}-q^{32}+11 q^{30}+5 q^{28}-7 q^{26}-4 q^{24}+7 q^{22}+6 q^{20}-5 q^{18}-7 q^{16}+5 q^{14}+10 q^{12}-2 q^{10}-12 q^8-2 q^6+11 q^4+7 q^2-9-10 q^{-2} +5 q^{-4} +14 q^{-6} + q^{-8} -11 q^{-10} -7 q^{-12} +6 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^{108}-q^{106}+4 q^{104}-6 q^{102}+6 q^{100}-6 q^{98}-q^{96}+13 q^{94}-24 q^{92}+32 q^{90}-30 q^{88}+12 q^{86}+15 q^{84}-46 q^{82}+60 q^{80}-60 q^{78}+34 q^{76}+4 q^{74}-44 q^{72}+64 q^{70}-62 q^{68}+38 q^{66}+4 q^{64}-38 q^{62}+45 q^{60}-36 q^{58}+8 q^{56}+28 q^{54}-46 q^{52}+53 q^{50}-28 q^{48}-5 q^{46}+50 q^{44}-77 q^{42}+79 q^{40}-53 q^{38}+8 q^{36}+41 q^{34}-73 q^{32}+86 q^{30}-67 q^{28}+26 q^{26}+21 q^{24}-55 q^{22}+56 q^{20}-41 q^{18}+4 q^{16}+28 q^{14}-38 q^{12}+31 q^{10}-8 q^8-20 q^6+43 q^4-47 q^2+33-7 q^{-2} -20 q^{-4} +40 q^{-6} -43 q^{-8} +39 q^{-10} -21 q^{-12} +5 q^{-14} +12 q^{-16} -27 q^{-18} +28 q^{-20} -25 q^{-22} +17 q^{-24} -7 q^{-26} - q^{-28} +8 q^{-30} -13 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]

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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]:=

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 68"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t^2-14 t+21-14 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 4 z^4+2 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]=

{ 57, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^3+3 q^2-5 q+8-9 q^{-1} +9 q^{-2} -8 q^{-3} +7 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+a^4+2 z^4 a^2+3 z^2 a^2+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{ a^5 z^9+a^3 z^9+2 a^6 z^8+6 a^4 z^8+4 a^2 z^8+a^7 z^7+a^5 z^7+7 a^3 z^7+7 a z^7-9 a^6 z^6-20 a^4 z^6-4 a^2 z^6+7 z^6-5 a^7 z^5-16 a^5 z^5-30 a^3 z^5-14 a z^5+5 z^5 a^{-1} +13 a^6 z^4+17 a^4 z^4-9 a^2 z^4+3 z^4 a^{-2} -10 z^4+8 a^7 z^3+23 a^5 z^3+27 a^3 z^3+8 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -7 a^6 z^2-5 a^4 z^2+7 a^2 z^2-z^2 a^{-2} +4 z^2-4 a^7 z-8 a^5 z-6 a^3 z-2 a z+a^6+a^4-a^2 }[/math]