10 98

From Knot Atlas
Revision as of 18:50, 23 August 2005 by ScottKnotPageRobot (talk | contribs) (Resetting knot page to basic template.)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^6-3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]
Determinant and Signature { 81, -4 }
Jones polynomial [math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -9 q^{-3} +13 q^{-4} -14 q^{-5} +12 q^{-6} -11 q^{-7} +7 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+2 a^8-z^6 a^6-3 z^4 a^6-5 z^2 a^6-5 a^6-z^6 a^4-2 z^4 a^4+z^2 a^4+3 a^4+z^4 a^2+2 z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-2 z^3 a^{11}+6 z^6 a^{10}-7 z^4 a^{10}+4 z^2 a^{10}+8 z^7 a^9-14 z^5 a^9+14 z^3 a^9-6 z a^9+6 z^8 a^8-7 z^6 a^8+2 z^4 a^8+2 a^8+2 z^9 a^7+8 z^7 a^7-26 z^5 a^7+25 z^3 a^7-12 z a^7+10 z^8 a^6-23 z^6 a^6+17 z^4 a^6-10 z^2 a^6+5 a^6+2 z^9 a^5+3 z^7 a^5-17 z^5 a^5+14 z^3 a^5-6 z a^5+4 z^8 a^4-9 z^6 a^4+4 z^4 a^4-2 z^2 a^4+3 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2 }[/math]
The A2 invariant [math]\displaystyle{ q^{30}-q^{28}+2 q^{26}+2 q^{24}-3 q^{22}-5 q^{18}-q^{16}+q^{14}+5 q^{10}-q^8+2 q^6+q^4-q^2+1 }[/math]
The G2 invariant [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+6 q^{154}-5 q^{152}+10 q^{148}-21 q^{146}+31 q^{144}-39 q^{142}+33 q^{140}-18 q^{138}-12 q^{136}+58 q^{134}-94 q^{132}+117 q^{130}-107 q^{128}+56 q^{126}+17 q^{124}-112 q^{122}+185 q^{120}-203 q^{118}+154 q^{116}-41 q^{114}-84 q^{112}+183 q^{110}-192 q^{108}+134 q^{106}-19 q^{104}-110 q^{102}+175 q^{100}-143 q^{98}+34 q^{96}+116 q^{94}-230 q^{92}+256 q^{90}-161 q^{88}-7 q^{86}+163 q^{84}-297 q^{82}+322 q^{80}-242 q^{78}+78 q^{76}+92 q^{74}-232 q^{72}+288 q^{70}-230 q^{68}+90 q^{66}+43 q^{64}-158 q^{62}+191 q^{60}-129 q^{58}+8 q^{56}+128 q^{54}-194 q^{52}+181 q^{50}-68 q^{48}-83 q^{46}+204 q^{44}-244 q^{42}+197 q^{40}-81 q^{38}-51 q^{36}+157 q^{34}-188 q^{32}+162 q^{30}-84 q^{28}+3 q^{26}+51 q^{24}-78 q^{22}+69 q^{20}-44 q^{18}+20 q^{16}+2 q^{14}-11 q^{12}+13 q^{10}-10 q^8+6 q^6-2 q^4+q^2 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_98.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{21}-2 q^{19}+4 q^{17}-4 q^{15}+q^{13}-2 q^{11}-q^9+4 q^7-2 q^5+4 q^3-2 q+ q^{-1} }[/math]
2 [math]\displaystyle{ q^{58}-2 q^{56}+q^{54}+5 q^{52}-11 q^{50}+2 q^{48}+19 q^{46}-24 q^{44}-7 q^{42}+34 q^{40}-20 q^{38}-16 q^{36}+31 q^{34}-20 q^{30}+6 q^{28}+16 q^{26}-14 q^{24}-20 q^{22}+24 q^{20}+4 q^{18}-32 q^{16}+21 q^{14}+20 q^{12}-29 q^{10}+4 q^8+21 q^6-13 q^4-5 q^2+9- q^{-2} -2 q^{-4} + q^{-6} }[/math]
3 [math]\displaystyle{ q^{111}-2 q^{109}+q^{107}+2 q^{105}-2 q^{103}-5 q^{101}+5 q^{99}+13 q^{97}-15 q^{95}-30 q^{93}+27 q^{91}+60 q^{89}-27 q^{87}-110 q^{85}+13 q^{83}+166 q^{81}+23 q^{79}-198 q^{77}-87 q^{75}+213 q^{73}+145 q^{71}-179 q^{69}-197 q^{67}+115 q^{65}+206 q^{63}-39 q^{61}-194 q^{59}-40 q^{57}+164 q^{55}+103 q^{53}-111 q^{51}-152 q^{49}+75 q^{47}+186 q^{45}-16 q^{43}-218 q^{41}-28 q^{39}+217 q^{37}+84 q^{35}-213 q^{33}-146 q^{31}+174 q^{29}+189 q^{27}-111 q^{25}-212 q^{23}+41 q^{21}+199 q^{19}+31 q^{17}-155 q^{15}-71 q^{13}+94 q^{11}+86 q^9-42 q^7-66 q^5+4 q^3+42 q+10 q^{-1} -19 q^{-3} -8 q^{-5} +6 q^{-7} +4 q^{-9} - q^{-11} -2 q^{-13} + q^{-15} }[/math]
4 [math]\displaystyle{ q^{180}-2 q^{178}+q^{176}+2 q^{174}-5 q^{172}+4 q^{170}-2 q^{168}+5 q^{166}+2 q^{164}-27 q^{162}+10 q^{160}+16 q^{158}+42 q^{156}+6 q^{154}-128 q^{152}-54 q^{150}+70 q^{148}+250 q^{146}+141 q^{144}-353 q^{142}-426 q^{140}-73 q^{138}+675 q^{136}+776 q^{134}-314 q^{132}-1120 q^{130}-884 q^{128}+743 q^{126}+1767 q^{124}+546 q^{122}-1303 q^{120}-2018 q^{118}-161 q^{116}+2024 q^{114}+1719 q^{112}-374 q^{110}-2249 q^{108}-1343 q^{106}+1040 q^{104}+1980 q^{102}+854 q^{100}-1289 q^{98}-1704 q^{96}-275 q^{94}+1237 q^{92}+1411 q^{90}-77 q^{88}-1307 q^{86}-1085 q^{84}+381 q^{82}+1433 q^{80}+736 q^{78}-864 q^{76}-1504 q^{74}-192 q^{72}+1394 q^{70}+1357 q^{68}-486 q^{66}-1871 q^{64}-819 q^{62}+1198 q^{60}+1976 q^{58}+222 q^{56}-1893 q^{54}-1621 q^{52}+376 q^{50}+2145 q^{48}+1267 q^{46}-1044 q^{44}-1950 q^{42}-877 q^{40}+1316 q^{38}+1772 q^{36}+314 q^{34}-1199 q^{32}-1498 q^{30}-13 q^{28}+1127 q^{26}+986 q^{24}-8 q^{22}-967 q^{20}-634 q^{18}+119 q^{16}+606 q^{14}+465 q^{12}-173 q^{10}-361 q^8-235 q^6+74 q^4+245 q^2+83-35 q^{-2} -108 q^{-4} -49 q^{-6} +40 q^{-8} +28 q^{-10} +18 q^{-12} -13 q^{-14} -14 q^{-16} +3 q^{-18} + q^{-20} +4 q^{-22} - q^{-24} -2 q^{-26} + q^{-28} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{30}-q^{28}+2 q^{26}+2 q^{24}-3 q^{22}-5 q^{18}-q^{16}+q^{14}+5 q^{10}-q^8+2 q^6+q^4-q^2+1 }[/math]
1,1 [math]\displaystyle{ q^{84}-4 q^{82}+10 q^{80}-20 q^{78}+40 q^{76}-72 q^{74}+114 q^{72}-174 q^{70}+266 q^{68}-376 q^{66}+494 q^{64}-632 q^{62}+766 q^{60}-860 q^{58}+866 q^{56}-786 q^{54}+593 q^{52}-272 q^{50}-140 q^{48}+616 q^{46}-1070 q^{44}+1472 q^{42}-1744 q^{40}+1876 q^{38}-1845 q^{36}+1644 q^{34}-1308 q^{32}+856 q^{30}-396 q^{28}-82 q^{26}+494 q^{24}-800 q^{22}+968 q^{20}-996 q^{18}+934 q^{16}-780 q^{14}+598 q^{12}-416 q^{10}+268 q^8-150 q^6+80 q^4-36 q^2+14-4 q^{-2} + q^{-4} }[/math]
2,0 [math]\displaystyle{ q^{76}-q^{74}+q^{72}+2 q^{70}-q^{68}-4 q^{66}+2 q^{64}+7 q^{62}-6 q^{60}-13 q^{58}+q^{56}+8 q^{54}-11 q^{52}-5 q^{50}+19 q^{48}+18 q^{46}-2 q^{44}+10 q^{40}-8 q^{38}-15 q^{36}-q^{34}-6 q^{32}-14 q^{30}+3 q^{28}+10 q^{26}-6 q^{24}-6 q^{22}+14 q^{20}+9 q^{18}-10 q^{16}-4 q^{14}+13 q^{12}+3 q^{10}-7 q^8-q^6+5 q^4+3 q^2-2- q^{-2} + q^{-4} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{68}-2 q^{66}+6 q^{62}-7 q^{60}-4 q^{58}+17 q^{56}-13 q^{54}-13 q^{52}+24 q^{50}-14 q^{48}-14 q^{46}+27 q^{44}-5 q^{40}+16 q^{38}+4 q^{36}-8 q^{34}-17 q^{32}-q^{28}-26 q^{26}+10 q^{24}+19 q^{22}-19 q^{20}+11 q^{18}+18 q^{16}-17 q^{14}+8 q^{12}+8 q^{10}-8 q^8+4 q^6+2 q^4-2 q^2+1 }[/math]
1,0,0 [math]\displaystyle{ q^{39}-q^{37}+3 q^{35}+3 q^{31}-3 q^{29}-6 q^{25}-3 q^{23}-2 q^{21}+3 q^{17}+q^{15}+5 q^{13}-q^{11}+3 q^9-q^7+2 q^5-q^3+q }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{86}-q^{84}-2 q^{82}+5 q^{80}+4 q^{78}-8 q^{76}-2 q^{74}+12 q^{72}-q^{70}-20 q^{68}-3 q^{66}+10 q^{64}-15 q^{62}-17 q^{60}+22 q^{58}+22 q^{56}-3 q^{54}+21 q^{52}+33 q^{50}-3 q^{48}-18 q^{46}+4 q^{44}-15 q^{42}-43 q^{40}-15 q^{38}+10 q^{36}-16 q^{34}-15 q^{32}+25 q^{30}+16 q^{28}-8 q^{26}+4 q^{24}+18 q^{22}+q^{20}-6 q^{18}+6 q^{16}+7 q^{14}-3 q^{12}-q^{10}+4 q^8-q^4+q^2 }[/math]
1,0,0,0 [math]\displaystyle{ q^{48}-q^{46}+3 q^{44}+q^{42}+q^{40}+3 q^{38}-3 q^{36}-6 q^{32}-4 q^{30}-4 q^{28}-2 q^{26}+2 q^{22}+4 q^{20}+q^{18}+5 q^{16}-q^{14}+3 q^{12}+2 q^6-q^4+q^2 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{68}-2 q^{66}+4 q^{64}-8 q^{62}+13 q^{60}-18 q^{58}+25 q^{56}-29 q^{54}+31 q^{52}-28 q^{50}+22 q^{48}-12 q^{46}-q^{44}+16 q^{42}-33 q^{40}+44 q^{38}-56 q^{36}+58 q^{34}-59 q^{32}+50 q^{30}-39 q^{28}+24 q^{26}-8 q^{24}-5 q^{22}+19 q^{20}-25 q^{18}+32 q^{16}-29 q^{14}+28 q^{12}-22 q^{10}+16 q^8-10 q^6+6 q^4-2 q^2+1 }[/math]
1,0 [math]\displaystyle{ q^{110}-2 q^{106}-2 q^{104}+2 q^{102}+7 q^{100}+2 q^{98}-10 q^{96}-11 q^{94}+5 q^{92}+21 q^{90}+6 q^{88}-23 q^{86}-22 q^{84}+12 q^{82}+30 q^{80}+q^{78}-31 q^{76}-15 q^{74}+24 q^{72}+25 q^{70}-10 q^{68}-22 q^{66}+8 q^{64}+27 q^{62}+6 q^{60}-22 q^{58}-8 q^{56}+16 q^{54}+8 q^{52}-21 q^{50}-19 q^{48}+10 q^{46}+16 q^{44}-13 q^{42}-29 q^{40}+2 q^{38}+32 q^{36}+15 q^{34}-25 q^{32}-24 q^{30}+16 q^{28}+34 q^{26}+2 q^{24}-25 q^{22}-13 q^{20}+17 q^{18}+17 q^{16}-4 q^{14}-12 q^{12}-2 q^{10}+7 q^8+4 q^6-2 q^4-2 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}+7 q^{86}-10 q^{84}+10 q^{82}-13 q^{80}+21 q^{78}-23 q^{76}+19 q^{74}-25 q^{72}+24 q^{70}-21 q^{68}+12 q^{66}-11 q^{64}+6 q^{62}+14 q^{60}-9 q^{58}+28 q^{56}-26 q^{54}+44 q^{52}-41 q^{50}+41 q^{48}-55 q^{46}+34 q^{44}-47 q^{42}+24 q^{40}-34 q^{38}+13 q^{36}-6 q^{34}+3 q^{32}+11 q^{30}-10 q^{28}+26 q^{26}-19 q^{24}+26 q^{22}-23 q^{20}+25 q^{18}-18 q^{16}+17 q^{14}-12 q^{12}+10 q^{10}-5 q^8+4 q^6-2 q^4+q^2 }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+6 q^{154}-5 q^{152}+10 q^{148}-21 q^{146}+31 q^{144}-39 q^{142}+33 q^{140}-18 q^{138}-12 q^{136}+58 q^{134}-94 q^{132}+117 q^{130}-107 q^{128}+56 q^{126}+17 q^{124}-112 q^{122}+185 q^{120}-203 q^{118}+154 q^{116}-41 q^{114}-84 q^{112}+183 q^{110}-192 q^{108}+134 q^{106}-19 q^{104}-110 q^{102}+175 q^{100}-143 q^{98}+34 q^{96}+116 q^{94}-230 q^{92}+256 q^{90}-161 q^{88}-7 q^{86}+163 q^{84}-297 q^{82}+322 q^{80}-242 q^{78}+78 q^{76}+92 q^{74}-232 q^{72}+288 q^{70}-230 q^{68}+90 q^{66}+43 q^{64}-158 q^{62}+191 q^{60}-129 q^{58}+8 q^{56}+128 q^{54}-194 q^{52}+181 q^{50}-68 q^{48}-83 q^{46}+204 q^{44}-244 q^{42}+197 q^{40}-81 q^{38}-51 q^{36}+157 q^{34}-188 q^{32}+162 q^{30}-84 q^{28}+3 q^{26}+51 q^{24}-78 q^{22}+69 q^{20}-44 q^{18}+20 q^{16}+2 q^{14}-11 q^{12}+13 q^{10}-10 q^8+6 q^6-2 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 98"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^6-3 z^4+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{ \left\{t^2-t+1\right\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 81, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -9 q^{-3} +13 q^{-4} -14 q^{-5} +12 q^{-6} -11 q^{-7} +7 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+2 a^8-z^6 a^6-3 z^4 a^6-5 z^2 a^6-5 a^6-z^6 a^4-2 z^4 a^4+z^2 a^4+3 a^4+z^4 a^2+2 z^2 a^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}-2 z^3 a^{11}+6 z^6 a^{10}-7 z^4 a^{10}+4 z^2 a^{10}+8 z^7 a^9-14 z^5 a^9+14 z^3 a^9-6 z a^9+6 z^8 a^8-7 z^6 a^8+2 z^4 a^8+2 a^8+2 z^9 a^7+8 z^7 a^7-26 z^5 a^7+25 z^3 a^7-12 z a^7+10 z^8 a^6-23 z^6 a^6+17 z^4 a^6-10 z^2 a^6+5 a^6+2 z^9 a^5+3 z^7 a^5-17 z^5 a^5+14 z^3 a^5-6 z a^5+4 z^8 a^4-9 z^6 a^4+4 z^4 a^4-2 z^2 a^4+3 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_98&oldid=4328"