10 39

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+8 t^2-13 t+15-13 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-4 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 61, -4 }
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +5 q^{-2} -7 q^{-3} +9 q^{-4} -10 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8+2 z^2 a^8-z^6 a^6-3 z^4 a^6-2 z^2 a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-4 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-3 z^5 a^9-z^3 a^9+2 z a^9+3 z^8 a^8-2 z^6 a^8+z^2 a^8+z^9 a^7+4 z^7 a^7-9 z^5 a^7+4 z^3 a^7+5 z^8 a^6-10 z^6 a^6+5 z^4 a^6-z^2 a^6+z^9 a^5+2 z^7 a^5-9 z^5 a^5+5 z^3 a^5-z a^5+2 z^8 a^4-3 z^6 a^4-4 z^4 a^4+5 z^2 a^4-a^4+2 z^7 a^3-6 z^5 a^3+4 z^3 a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{30}-q^{28}-2 q^{22}+2 q^{20}-q^{18}+q^{16}-2 q^{12}+2 q^{10}-q^8+2 q^6+q^4+1 }[/math]
The G2 invariant	[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+12 q^{148}-17 q^{146}+22 q^{144}-22 q^{142}+12 q^{140}+3 q^{138}-21 q^{136}+38 q^{134}-49 q^{132}+50 q^{130}-37 q^{128}+11 q^{126}+26 q^{124}-56 q^{122}+77 q^{120}-70 q^{118}+44 q^{116}-7 q^{114}-37 q^{112}+60 q^{110}-59 q^{108}+33 q^{106}+9 q^{104}-43 q^{102}+49 q^{100}-29 q^{98}-16 q^{96}+61 q^{94}-92 q^{92}+84 q^{90}-45 q^{88}-16 q^{86}+83 q^{84}-119 q^{82}+121 q^{80}-83 q^{78}+20 q^{76}+43 q^{74}-88 q^{72}+100 q^{70}-77 q^{68}+32 q^{66}+22 q^{64}-56 q^{62}+58 q^{60}-32 q^{58}-13 q^{56}+50 q^{54}-69 q^{52}+51 q^{50}-12 q^{48}-38 q^{46}+82 q^{44}-92 q^{42}+72 q^{40}-28 q^{38}-23 q^{36}+59 q^{34}-72 q^{32}+65 q^{30}-37 q^{28}+8 q^{26}+18 q^{24}-30 q^{22}+31 q^{20}-21 q^{18}+12 q^{16}-q^{14}-4 q^{12}+6 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_39.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{21}-2 q^{19}+2 q^{17}-3 q^{15}+2 q^{13}-q^9+2 q^7-2 q^5+3 q^3-q+ q^{-1} }[/math]
2	[math]\displaystyle{ q^{58}-2 q^{56}-q^{54}+5 q^{52}-5 q^{50}-q^{48}+12 q^{46}-11 q^{44}-6 q^{42}+18 q^{40}-10 q^{38}-10 q^{36}+15 q^{34}-9 q^{30}+q^{28}+8 q^{26}-4 q^{24}-11 q^{22}+12 q^{20}+4 q^{18}-17 q^{16}+10 q^{14}+11 q^{12}-15 q^{10}+2 q^8+11 q^6-7 q^4-2 q^2+5- q^{-2} - q^{-4} + q^{-6} }[/math]
3	[math]\displaystyle{ q^{111}-2 q^{109}-q^{107}+2 q^{105}+3 q^{103}-2 q^{101}-4 q^{99}+6 q^{97}+q^{95}-11 q^{93}-3 q^{91}+22 q^{89}+5 q^{87}-35 q^{85}-12 q^{83}+50 q^{81}+25 q^{79}-59 q^{77}-43 q^{75}+60 q^{73}+58 q^{71}-51 q^{69}-65 q^{67}+29 q^{65}+69 q^{63}-7 q^{61}-58 q^{59}-16 q^{57}+43 q^{55}+36 q^{53}-27 q^{51}-52 q^{49}+10 q^{47}+62 q^{45}+6 q^{43}-69 q^{41}-23 q^{39}+68 q^{37}+42 q^{35}-64 q^{33}-56 q^{31}+49 q^{29}+67 q^{27}-29 q^{25}-71 q^{23}+7 q^{21}+66 q^{19}+12 q^{17}-49 q^{15}-23 q^{13}+32 q^{11}+29 q^9-17 q^7-22 q^5+4 q^3+16 q+ q^{-1} -8 q^{-3} -2 q^{-5} +4 q^{-7} + q^{-9} - q^{-11} - q^{-13} + q^{-15} }[/math]
4	[math]\displaystyle{ q^{180}-2 q^{178}-q^{176}+2 q^{174}+6 q^{170}-5 q^{168}-3 q^{166}+q^{164}-9 q^{162}+13 q^{160}-4 q^{158}+8 q^{156}+12 q^{154}-31 q^{152}-5 q^{150}-16 q^{148}+45 q^{146}+67 q^{144}-49 q^{142}-74 q^{140}-89 q^{138}+88 q^{136}+203 q^{134}+5 q^{132}-164 q^{130}-260 q^{128}+52 q^{126}+358 q^{124}+176 q^{122}-153 q^{120}-443 q^{118}-118 q^{116}+374 q^{114}+354 q^{112}+10 q^{110}-448 q^{108}-295 q^{106}+183 q^{104}+364 q^{102}+208 q^{100}-242 q^{98}-329 q^{96}-66 q^{94}+209 q^{92}+288 q^{90}+16 q^{88}-231 q^{86}-241 q^{84}+26 q^{82}+278 q^{80}+211 q^{78}-121 q^{76}-342 q^{74}-111 q^{72}+242 q^{70}+351 q^{68}-16 q^{66}-398 q^{64}-237 q^{62}+160 q^{60}+443 q^{58}+131 q^{56}-354 q^{54}-350 q^{52}-19 q^{50}+425 q^{48}+294 q^{46}-170 q^{44}-354 q^{42}-227 q^{40}+243 q^{38}+338 q^{36}+68 q^{34}-192 q^{32}-310 q^{30}+6 q^{28}+208 q^{26}+177 q^{24}+14 q^{22}-205 q^{20}-106 q^{18}+29 q^{16}+117 q^{14}+100 q^{12}-57 q^{10}-71 q^8-44 q^6+23 q^4+63 q^2+5-12 q^{-2} -27 q^{-4} -8 q^{-6} +18 q^{-8} +4 q^{-10} +3 q^{-12} -6 q^{-14} -4 q^{-16} +4 q^{-18} + q^{-22} - q^{-24} - q^{-26} + q^{-28} }[/math]
5	[math]\displaystyle{ q^{265}-2 q^{263}-q^{261}+2 q^{259}+3 q^{255}+3 q^{253}-4 q^{251}-8 q^{249}-2 q^{247}-2 q^{245}+6 q^{243}+17 q^{241}+10 q^{239}-7 q^{237}-24 q^{235}-25 q^{233}-10 q^{231}+29 q^{229}+65 q^{227}+41 q^{225}-39 q^{223}-106 q^{221}-105 q^{219}+3 q^{217}+180 q^{215}+238 q^{213}+48 q^{211}-253 q^{209}-416 q^{207}-206 q^{205}+308 q^{203}+686 q^{201}+470 q^{199}-304 q^{197}-988 q^{195}-870 q^{193}+161 q^{191}+1272 q^{189}+1388 q^{187}+167 q^{185}-1453 q^{183}-1947 q^{181}-666 q^{179}+1416 q^{177}+2419 q^{175}+1300 q^{173}-1116 q^{171}-2695 q^{169}-1924 q^{167}+591 q^{165}+2652 q^{163}+2394 q^{161}+72 q^{159}-2266 q^{157}-2625 q^{155}-730 q^{153}+1665 q^{151}+2513 q^{149}+1246 q^{147}-907 q^{145}-2153 q^{143}-1572 q^{141}+203 q^{139}+1628 q^{137}+1659 q^{135}+406 q^{133}-1062 q^{131}-1614 q^{129}-862 q^{127}+562 q^{125}+1497 q^{123}+1180 q^{121}-152 q^{119}-1395 q^{117}-1433 q^{115}-156 q^{113}+1347 q^{111}+1665 q^{109}+412 q^{107}-1320 q^{105}-1908 q^{103}-689 q^{101}+1292 q^{99}+2165 q^{97}+1007 q^{95}-1184 q^{93}-2368 q^{91}-1399 q^{89}+920 q^{87}+2493 q^{85}+1813 q^{83}-513 q^{81}-2414 q^{79}-2189 q^{77}-51 q^{75}+2115 q^{73}+2429 q^{71}+670 q^{69}-1589 q^{67}-2440 q^{65}-1233 q^{63}+900 q^{61}+2164 q^{59}+1625 q^{57}-163 q^{55}-1663 q^{53}-1737 q^{51}-468 q^{49}+1006 q^{47}+1552 q^{45}+894 q^{43}-349 q^{41}-1165 q^{39}-1024 q^{37}-162 q^{35}+670 q^{33}+898 q^{31}+475 q^{29}-221 q^{27}-638 q^{25}-532 q^{23}-85 q^{21}+322 q^{19}+440 q^{17}+241 q^{15}-93 q^{13}-276 q^{11}-234 q^9-51 q^7+124 q^5+172 q^3+91 q-27 q^{-1} -92 q^{-3} -78 q^{-5} -12 q^{-7} +36 q^{-9} +43 q^{-11} +24 q^{-13} -9 q^{-15} -23 q^{-17} -12 q^{-19} +2 q^{-21} +4 q^{-23} +7 q^{-25} +3 q^{-27} -5 q^{-29} -2 q^{-31} +2 q^{-33} + q^{-39} - q^{-41} - q^{-43} + q^{-45} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{30}-q^{28}-2 q^{22}+2 q^{20}-q^{18}+q^{16}-2 q^{12}+2 q^{10}-q^8+2 q^6+q^4+1 }[/math]
1,1	[math]\displaystyle{ q^{84}-4 q^{82}+10 q^{80}-20 q^{78}+34 q^{76}-54 q^{74}+80 q^{72}-112 q^{70}+146 q^{68}-182 q^{66}+224 q^{64}-258 q^{62}+281 q^{60}-284 q^{58}+260 q^{56}-208 q^{54}+112 q^{52}+14 q^{50}-158 q^{48}+312 q^{46}-452 q^{44}+566 q^{42}-638 q^{40}+656 q^{38}-621 q^{36}+534 q^{34}-410 q^{32}+256 q^{30}-92 q^{28}-66 q^{26}+196 q^{24}-288 q^{22}+342 q^{20}-352 q^{18}+326 q^{16}-274 q^{14}+215 q^{12}-154 q^{10}+104 q^8-60 q^6+35 q^4-16 q^2+8-2 q^{-2} + q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{76}-q^{74}-q^{72}+q^{64}+4 q^{62}-2 q^{60}-4 q^{58}+3 q^{56}+5 q^{54}-7 q^{52}-5 q^{50}+7 q^{48}+3 q^{46}-6 q^{44}-2 q^{42}+8 q^{40}-3 q^{38}-6 q^{36}+4 q^{34}+q^{32}-5 q^{30}+2 q^{28}+5 q^{26}-5 q^{24}-3 q^{22}+7 q^{20}+3 q^{18}-7 q^{16}-q^{14}+8 q^{12}-4 q^8+q^6+4 q^4+q^2-1+ q^{-4} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{68}-2 q^{66}+4 q^{62}-6 q^{60}-q^{58}+10 q^{56}-8 q^{54}-5 q^{52}+16 q^{50}-7 q^{48}-8 q^{46}+13 q^{44}-4 q^{42}-7 q^{40}+4 q^{38}+3 q^{36}-3 q^{34}-6 q^{32}+7 q^{30}+3 q^{28}-13 q^{26}+6 q^{24}+8 q^{22}-14 q^{20}+4 q^{18}+8 q^{16}-9 q^{14}+5 q^{12}+5 q^{10}-3 q^8+3 q^6+2 q^4-q^2+1 }[/math]
1,0,0	[math]\displaystyle{ q^{39}-q^{37}+q^{35}-2 q^{33}+q^{31}-2 q^{29}+2 q^{27}-q^{25}+q^{23}-q^{19}-2 q^{15}+2 q^{13}-q^{11}+3 q^9+2 q^5+q }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{86}-q^{84}-2 q^{82}+3 q^{80}+2 q^{78}-6 q^{76}-2 q^{74}+7 q^{72}-8 q^{68}+2 q^{66}+10 q^{64}-4 q^{62}-7 q^{60}+9 q^{58}+4 q^{56}-9 q^{54}+3 q^{52}+8 q^{50}-7 q^{48}-6 q^{46}+7 q^{44}-q^{42}-12 q^{40}+q^{38}+10 q^{36}-6 q^{34}-8 q^{32}+8 q^{30}+3 q^{28}-7 q^{26}-q^{24}+5 q^{22}-q^{18}+3 q^{16}+4 q^{14}+q^{12}+2 q^{10}+3 q^8+q^6+q^2 }[/math]
1,0,0,0	[math]\displaystyle{ q^{48}-q^{46}+q^{44}-q^{42}-q^{40}+q^{38}-2 q^{36}+2 q^{34}-q^{32}+q^{30}-q^{24}-q^{22}-2 q^{18}+2 q^{16}-q^{14}+3 q^{12}+q^{10}+q^8+2 q^6+q^2 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{68}-2 q^{66}+4 q^{64}-6 q^{62}+8 q^{60}-11 q^{58}+14 q^{56}-16 q^{54}+15 q^{52}-14 q^{50}+9 q^{48}-4 q^{46}-3 q^{44}+12 q^{42}-19 q^{40}+26 q^{38}-29 q^{36}+31 q^{34}-30 q^{32}+25 q^{30}-19 q^{28}+11 q^{26}-4 q^{24}-4 q^{22}+10 q^{20}-14 q^{18}+16 q^{16}-15 q^{14}+15 q^{12}-11 q^{10}+9 q^8-5 q^6+4 q^4-q^2+1 }[/math]
1,0	[math]\displaystyle{ q^{110}-2 q^{106}-2 q^{104}+2 q^{102}+5 q^{100}-7 q^{96}-6 q^{94}+5 q^{92}+12 q^{90}+2 q^{88}-13 q^{86}-10 q^{84}+8 q^{82}+17 q^{80}-16 q^{76}-7 q^{74}+12 q^{72}+11 q^{70}-8 q^{68}-12 q^{66}+3 q^{64}+12 q^{62}-11 q^{58}-3 q^{56}+9 q^{54}+4 q^{52}-9 q^{50}-6 q^{48}+8 q^{46}+9 q^{44}-7 q^{42}-14 q^{40}+2 q^{38}+17 q^{36}+6 q^{34}-15 q^{32}-14 q^{30}+8 q^{28}+17 q^{26}+q^{24}-13 q^{22}-7 q^{20}+9 q^{18}+9 q^{16}-q^{14}-6 q^{12}-q^{10}+4 q^8+3 q^6-q^4-q^2+ q^{-2} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{94}-2 q^{92}+2 q^{90}-3 q^{88}+5 q^{86}-7 q^{84}+6 q^{82}-8 q^{80}+12 q^{78}-12 q^{76}+10 q^{74}-11 q^{72}+14 q^{70}-8 q^{68}+5 q^{66}-4 q^{64}+7 q^{60}-11 q^{58}+12 q^{56}-19 q^{54}+22 q^{52}-22 q^{50}+23 q^{48}-25 q^{46}+22 q^{44}-18 q^{42}+16 q^{40}-14 q^{38}+7 q^{36}-3 q^{34}-q^{32}+3 q^{30}-9 q^{28}+11 q^{26}-12 q^{24}+12 q^{22}-12 q^{20}+14 q^{18}-8 q^{16}+10 q^{14}-5 q^{12}+7 q^{10}-2 q^8+3 q^6-q^4+q^2 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+12 q^{148}-17 q^{146}+22 q^{144}-22 q^{142}+12 q^{140}+3 q^{138}-21 q^{136}+38 q^{134}-49 q^{132}+50 q^{130}-37 q^{128}+11 q^{126}+26 q^{124}-56 q^{122}+77 q^{120}-70 q^{118}+44 q^{116}-7 q^{114}-37 q^{112}+60 q^{110}-59 q^{108}+33 q^{106}+9 q^{104}-43 q^{102}+49 q^{100}-29 q^{98}-16 q^{96}+61 q^{94}-92 q^{92}+84 q^{90}-45 q^{88}-16 q^{86}+83 q^{84}-119 q^{82}+121 q^{80}-83 q^{78}+20 q^{76}+43 q^{74}-88 q^{72}+100 q^{70}-77 q^{68}+32 q^{66}+22 q^{64}-56 q^{62}+58 q^{60}-32 q^{58}-13 q^{56}+50 q^{54}-69 q^{52}+51 q^{50}-12 q^{48}-38 q^{46}+82 q^{44}-92 q^{42}+72 q^{40}-28 q^{38}-23 q^{36}+59 q^{34}-72 q^{32}+65 q^{30}-37 q^{28}+8 q^{26}+18 q^{24}-30 q^{22}+31 q^{20}-21 q^{18}+12 q^{16}-q^{14}-4 q^{12}+6 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/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 39"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+8 t^2-13 t+15-13 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^6-4 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]=

{ 61, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 1-2 q^{-1} +5 q^{-2} -7 q^{-3} +9 q^{-4} -10 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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