10 53

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 6 t^2-18 t+25-18 t^{-1} +6 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 6 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 73, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} -3 q^{-3} +7 q^{-4} -9 q^{-5} +12 q^{-6} -12 q^{-7} +11 q^{-8} -9 q^{-9} +5 q^{-10} -3 q^{-11} + q^{-12} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^{12}-3 z^2 a^{10}-3 a^{10}+2 z^4 a^8+2 z^2 a^8+3 z^4 a^6+6 z^2 a^6+3 a^6+z^4 a^4+z^2 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{14}-3 z^4 a^{14}+2 z^2 a^{14}+3 z^7 a^{13}-10 z^5 a^{13}+10 z^3 a^{13}-3 z a^{13}+3 z^8 a^{12}-6 z^6 a^{12}+2 z^2 a^{12}+a^{12}+z^9 a^{11}+7 z^7 a^{11}-27 z^5 a^{11}+28 z^3 a^{11}-11 z a^{11}+7 z^8 a^{10}-13 z^6 a^{10}+6 z^4 a^{10}-5 z^2 a^{10}+3 a^{10}+z^9 a^9+10 z^7 a^9-26 z^5 a^9+21 z^3 a^9-7 z a^9+4 z^8 a^8-7 z^4 a^8+4 z^2 a^8+6 z^7 a^7-6 z^5 a^7+z^3 a^7+z a^7+6 z^6 a^6-9 z^4 a^6+8 z^2 a^6-3 a^6+3 z^5 a^5-2 z^3 a^5+z^4 a^4-z^2 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ q^{38}+q^{36}-2 q^{34}-q^{30}-4 q^{28}+q^{26}-q^{24}+q^{22}+2 q^{20}+4 q^{16}-q^{14}+q^{12}+2 q^{10}-2 q^8+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{190}-2 q^{188}+5 q^{186}-9 q^{184}+9 q^{182}-9 q^{180}-q^{178}+18 q^{176}-36 q^{174}+52 q^{172}-52 q^{170}+31 q^{168}+10 q^{166}-62 q^{164}+110 q^{162}-125 q^{160}+102 q^{158}-35 q^{156}-50 q^{154}+126 q^{152}-158 q^{150}+141 q^{148}-70 q^{146}-21 q^{144}+95 q^{142}-128 q^{140}+97 q^{138}-26 q^{136}-56 q^{134}+106 q^{132}-107 q^{130}+43 q^{128}+43 q^{126}-136 q^{124}+179 q^{122}-164 q^{120}+81 q^{118}+34 q^{116}-151 q^{114}+219 q^{112}-216 q^{110}+143 q^{108}-28 q^{106}-88 q^{104}+161 q^{102}-167 q^{100}+114 q^{98}-21 q^{96}-60 q^{94}+102 q^{92}-84 q^{90}+21 q^{88}+58 q^{86}-109 q^{84}+118 q^{82}-71 q^{80}-4 q^{78}+82 q^{76}-131 q^{74}+141 q^{72}-103 q^{70}+44 q^{68}+23 q^{66}-74 q^{64}+98 q^{62}-90 q^{60}+65 q^{58}-25 q^{56}-7 q^{54}+28 q^{52}-39 q^{50}+35 q^{48}-23 q^{46}+12 q^{44}-5 q^{40}+6 q^{38}-6 q^{36}+4 q^{34}-2 q^{32}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_53.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{25}-2 q^{23}+2 q^{21}-4 q^{19}+2 q^{17}-q^{15}+3 q^{11}-2 q^9+4 q^7-2 q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{70}-2 q^{68}-2 q^{66}+7 q^{64}-3 q^{62}-10 q^{60}+15 q^{58}+3 q^{56}-22 q^{54}+16 q^{52}+13 q^{50}-26 q^{48}+6 q^{46}+18 q^{44}-17 q^{42}-7 q^{40}+13 q^{38}+2 q^{36}-15 q^{34}-q^{32}+21 q^{30}-15 q^{28}-13 q^{26}+28 q^{24}-7 q^{22}-16 q^{20}+19 q^{18}-q^{16}-9 q^{14}+7 q^{12}+q^{10}-2 q^8+q^6 }[/math]
3	[math]\displaystyle{ q^{135}-2 q^{133}-2 q^{131}+3 q^{129}+7 q^{127}-3 q^{125}-16 q^{123}+q^{121}+27 q^{119}+10 q^{117}-40 q^{115}-30 q^{113}+46 q^{111}+60 q^{109}-44 q^{107}-90 q^{105}+21 q^{103}+119 q^{101}+8 q^{99}-129 q^{97}-48 q^{95}+127 q^{93}+85 q^{91}-110 q^{89}-108 q^{87}+82 q^{85}+127 q^{83}-52 q^{81}-128 q^{79}+17 q^{77}+122 q^{75}+14 q^{73}-99 q^{71}-54 q^{69}+76 q^{67}+82 q^{65}-37 q^{63}-114 q^{61}-6 q^{59}+126 q^{57}+48 q^{55}-129 q^{53}-84 q^{51}+113 q^{49}+104 q^{47}-85 q^{45}-109 q^{43}+55 q^{41}+96 q^{39}-27 q^{37}-75 q^{35}+15 q^{33}+48 q^{31}-q^{29}-31 q^{27}+2 q^{25}+19 q^{23}-8 q^{19}+4 q^{15}+q^{13}-2 q^{11}+q^9 }[/math]
4	[math]\displaystyle{ q^{220}-2 q^{218}-2 q^{216}+3 q^{214}+3 q^{212}+7 q^{210}-10 q^{208}-16 q^{206}+2 q^{204}+13 q^{202}+42 q^{200}-9 q^{198}-59 q^{196}-41 q^{194}+6 q^{192}+132 q^{190}+68 q^{188}-84 q^{186}-169 q^{184}-129 q^{182}+203 q^{180}+278 q^{178}+71 q^{176}-267 q^{174}-451 q^{172}+32 q^{170}+441 q^{168}+469 q^{166}-57 q^{164}-717 q^{162}-427 q^{160}+243 q^{158}+802 q^{156}+466 q^{154}-584 q^{152}-825 q^{150}-277 q^{148}+744 q^{146}+911 q^{144}-127 q^{142}-854 q^{140}-719 q^{138}+391 q^{136}+998 q^{134}+294 q^{132}-616 q^{130}-860 q^{128}+45 q^{126}+819 q^{124}+518 q^{122}-321 q^{120}-786 q^{118}-233 q^{116}+527 q^{114}+647 q^{112}+13 q^{110}-608 q^{108}-528 q^{106}+109 q^{104}+708 q^{102}+452 q^{100}-257 q^{98}-787 q^{96}-458 q^{94}+558 q^{92}+842 q^{90}+278 q^{88}-759 q^{86}-935 q^{84}+131 q^{82}+867 q^{80}+734 q^{78}-377 q^{76}-979 q^{74}-280 q^{72}+488 q^{70}+773 q^{68}+38 q^{66}-611 q^{64}-360 q^{62}+83 q^{60}+465 q^{58}+164 q^{56}-227 q^{54}-184 q^{52}-62 q^{50}+174 q^{48}+91 q^{46}-60 q^{44}-44 q^{42}-40 q^{40}+54 q^{38}+24 q^{36}-21 q^{34}-3 q^{32}-13 q^{30}+17 q^{28}+6 q^{26}-7 q^{24}+q^{22}-3 q^{20}+4 q^{18}+q^{16}-2 q^{14}+q^{12} }[/math]
5	[math]\displaystyle{ q^{325}-2 q^{323}-2 q^{321}+3 q^{319}+3 q^{317}+3 q^{315}-10 q^{311}-16 q^{309}+2 q^{307}+23 q^{305}+28 q^{303}+14 q^{301}-29 q^{299}-71 q^{297}-55 q^{295}+36 q^{293}+126 q^{291}+131 q^{289}+15 q^{287}-181 q^{285}-286 q^{283}-145 q^{281}+200 q^{279}+474 q^{277}+405 q^{275}-62 q^{273}-653 q^{271}-825 q^{269}-287 q^{267}+671 q^{265}+1295 q^{263}+937 q^{261}-356 q^{259}-1657 q^{257}-1805 q^{255}-420 q^{253}+1639 q^{251}+2700 q^{249}+1625 q^{247}-1020 q^{245}-3254 q^{243}-3107 q^{241}-277 q^{239}+3219 q^{237}+4435 q^{235}+2074 q^{233}-2312 q^{231}-5276 q^{229}-4075 q^{227}+718 q^{225}+5315 q^{223}+5765 q^{221}+1363 q^{219}-4494 q^{217}-6857 q^{215}-3466 q^{213}+3050 q^{211}+7131 q^{209}+5166 q^{207}-1268 q^{205}-6660 q^{203}-6272 q^{201}-417 q^{199}+5697 q^{197}+6653 q^{195}+1749 q^{193}-4489 q^{191}-6506 q^{189}-2623 q^{187}+3361 q^{185}+5972 q^{183}+3091 q^{181}-2390 q^{179}-5358 q^{177}-3285 q^{175}+1605 q^{173}+4734 q^{171}+3464 q^{169}-866 q^{167}-4244 q^{165}-3695 q^{163}+46 q^{161}+3673 q^{159}+4146 q^{157}+1058 q^{155}-3011 q^{153}-4639 q^{151}-2422 q^{149}+1949 q^{147}+5030 q^{145}+4063 q^{143}-504 q^{141}-5047 q^{139}-5644 q^{137}-1380 q^{135}+4479 q^{133}+6888 q^{131}+3444 q^{129}-3249 q^{127}-7456 q^{125}-5331 q^{123}+1495 q^{121}+7135 q^{119}+6656 q^{117}+473 q^{115}-5981 q^{113}-7138 q^{111}-2211 q^{109}+4252 q^{107}+6689 q^{105}+3369 q^{103}-2367 q^{101}-5509 q^{99}-3794 q^{97}+755 q^{95}+3985 q^{93}+3479 q^{91}+331 q^{89}-2443 q^{87}-2765 q^{85}-859 q^{83}+1285 q^{81}+1869 q^{79}+887 q^{77}-478 q^{75}-1110 q^{73}-712 q^{71}+106 q^{69}+571 q^{67}+438 q^{65}+38 q^{63}-245 q^{61}-235 q^{59}-51 q^{57}+95 q^{55}+110 q^{53}+25 q^{51}-38 q^{49}-35 q^{47}-6 q^{45}+15 q^{43}+14 q^{41}+5 q^{39}-14 q^{37}-5 q^{35}+8 q^{33}+4 q^{31}-q^{29}+2 q^{27}-2 q^{25}-3 q^{23}+4 q^{21}+q^{19}-2 q^{17}+q^{15} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{38}+q^{36}-2 q^{34}-q^{30}-4 q^{28}+q^{26}-q^{24}+q^{22}+2 q^{20}+4 q^{16}-q^{14}+q^{12}+2 q^{10}-2 q^8+q^6 }[/math]
2,0	[math]\displaystyle{ q^{96}+q^{94}-q^{92}-4 q^{90}-2 q^{88}+4 q^{86}+q^{84}-5 q^{82}-2 q^{80}+10 q^{78}+7 q^{76}-9 q^{74}-4 q^{72}+14 q^{70}+9 q^{68}-10 q^{66}-8 q^{64}+9 q^{62}+q^{60}-15 q^{58}-7 q^{56}+2 q^{54}-5 q^{52}-4 q^{50}+4 q^{48}-2 q^{46}-5 q^{44}+9 q^{42}+11 q^{40}-11 q^{38}-5 q^{36}+18 q^{34}+7 q^{32}-14 q^{30}-q^{28}+14 q^{26}+3 q^{24}-10 q^{22}+7 q^{18}-q^{16}-2 q^{14}+q^{12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{80}-2 q^{78}+q^{76}+2 q^{74}-8 q^{72}+5 q^{70}+4 q^{68}-13 q^{66}+13 q^{64}+11 q^{62}-16 q^{60}+15 q^{58}+9 q^{56}-22 q^{54}+3 q^{50}-14 q^{48}-8 q^{46}+2 q^{44}+7 q^{42}-8 q^{40}-3 q^{38}+22 q^{36}-9 q^{34}-8 q^{32}+24 q^{30}-5 q^{28}-10 q^{26}+17 q^{24}-2 q^{22}-7 q^{20}+6 q^{18}-2 q^{14}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ q^{51}+q^{49}+q^{47}-2 q^{45}-3 q^{41}-q^{39}-4 q^{37}+q^{35}-2 q^{33}+q^{31}+q^{29}+2 q^{27}+2 q^{25}+q^{23}+4 q^{21}-q^{19}+2 q^{17}-q^{15}+2 q^{13}-2 q^{11}+q^9 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+12 q^{72}-17 q^{70}+20 q^{68}-23 q^{66}+23 q^{64}-21 q^{62}+14 q^{60}-7 q^{58}-5 q^{56}+16 q^{54}-28 q^{52}+37 q^{50}-44 q^{48}+46 q^{46}-44 q^{44}+37 q^{42}-28 q^{40}+17 q^{38}-4 q^{36}-5 q^{34}+16 q^{32}-20 q^{30}+25 q^{28}-22 q^{26}+21 q^{24}-16 q^{22}+13 q^{20}-8 q^{18}+4 q^{16}-2 q^{14}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{130}-2 q^{126}-2 q^{124}+3 q^{122}+5 q^{120}-3 q^{118}-10 q^{116}-3 q^{114}+13 q^{112}+11 q^{110}-10 q^{108}-19 q^{106}+2 q^{104}+25 q^{102}+14 q^{100}-18 q^{98}-20 q^{96}+9 q^{94}+26 q^{92}+3 q^{90}-22 q^{88}-12 q^{86}+12 q^{84}+10 q^{82}-12 q^{80}-16 q^{78}+4 q^{76}+12 q^{74}-6 q^{72}-18 q^{70}+18 q^{66}+5 q^{64}-17 q^{62}-10 q^{60}+18 q^{58}+19 q^{56}-8 q^{54}-23 q^{52}+q^{50}+25 q^{48}+14 q^{46}-15 q^{44}-18 q^{42}+4 q^{40}+19 q^{38}+7 q^{36}-9 q^{34}-10 q^{32}+2 q^{30}+7 q^{28}+2 q^{26}-2 q^{24}-2 q^{22}+q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{190}-2 q^{188}+5 q^{186}-9 q^{184}+9 q^{182}-9 q^{180}-q^{178}+18 q^{176}-36 q^{174}+52 q^{172}-52 q^{170}+31 q^{168}+10 q^{166}-62 q^{164}+110 q^{162}-125 q^{160}+102 q^{158}-35 q^{156}-50 q^{154}+126 q^{152}-158 q^{150}+141 q^{148}-70 q^{146}-21 q^{144}+95 q^{142}-128 q^{140}+97 q^{138}-26 q^{136}-56 q^{134}+106 q^{132}-107 q^{130}+43 q^{128}+43 q^{126}-136 q^{124}+179 q^{122}-164 q^{120}+81 q^{118}+34 q^{116}-151 q^{114}+219 q^{112}-216 q^{110}+143 q^{108}-28 q^{106}-88 q^{104}+161 q^{102}-167 q^{100}+114 q^{98}-21 q^{96}-60 q^{94}+102 q^{92}-84 q^{90}+21 q^{88}+58 q^{86}-109 q^{84}+118 q^{82}-71 q^{80}-4 q^{78}+82 q^{76}-131 q^{74}+141 q^{72}-103 q^{70}+44 q^{68}+23 q^{66}-74 q^{64}+98 q^{62}-90 q^{60}+65 q^{58}-25 q^{56}-7 q^{54}+28 q^{52}-39 q^{50}+35 q^{48}-23 q^{46}+12 q^{44}-5 q^{40}+6 q^{38}-6 q^{36}+4 q^{34}-2 q^{32}+q^{30} }[/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 53"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 6 t^2-18 t+25-18 t^{-1} +6 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 6 z^4+6 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]=

{ 73, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{14}-3 z^4 a^{14}+2 z^2 a^{14}+3 z^7 a^{13}-10 z^5 a^{13}+10 z^3 a^{13}-3 z a^{13}+3 z^8 a^{12}-6 z^6 a^{12}+2 z^2 a^{12}+a^{12}+z^9 a^{11}+7 z^7 a^{11}-27 z^5 a^{11}+28 z^3 a^{11}-11 z a^{11}+7 z^8 a^{10}-13 z^6 a^{10}+6 z^4 a^{10}-5 z^2 a^{10}+3 a^{10}+z^9 a^9+10 z^7 a^9-26 z^5 a^9+21 z^3 a^9-7 z a^9+4 z^8 a^8-7 z^4 a^8+4 z^2 a^8+6 z^7 a^7-6 z^5 a^7+z^3 a^7+z a^7+6 z^6 a^6-9 z^4 a^6+8 z^2 a^6-3 a^6+3 z^5 a^5-2 z^3 a^5+z^4 a^4-z^2 a^4 }[/math]