10 148

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-3 t^2+7 t-9+7 t^{-1} -3 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+3 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 31, -2 }
Jones polynomial	[math]\displaystyle{ -1+3 q^{-1} -4 q^{-2} +6 q^{-3} -5 q^{-4} +5 q^{-5} -4 q^{-6} +2 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^6-3 z^2 a^6-3 a^6+z^6 a^4+5 z^4 a^4+9 z^2 a^4+5 a^4-z^4 a^2-2 z^2 a^2-a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+2 z a^9+2 z^6 a^8-5 z^4 a^8+2 z^2 a^8+2 z^7 a^7-4 z^5 a^7+z^3 a^7-z a^7+z^8 a^6-z^6 a^6+2 z^4 a^6-6 z^2 a^6+3 a^6+3 z^7 a^5-7 z^5 a^5+9 z^3 a^5-5 z a^5+z^8 a^4-3 z^6 a^4+10 z^4 a^4-11 z^2 a^4+5 a^4+z^7 a^3-2 z^5 a^3+6 z^3 a^3-3 z a^3+3 z^4 a^2-3 z^2 a^2+a^2+z^3 a-z a }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}-2 q^{20}-q^{18}+q^{16}+3 q^{12}+q^{10}+2 q^8+q^6-q^4+q^2-1 }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+3 q^{120}-q^{118}-3 q^{116}+9 q^{114}-13 q^{112}+15 q^{110}-11 q^{108}-q^{106}+12 q^{104}-22 q^{102}+25 q^{100}-18 q^{98}+3 q^{96}+10 q^{94}-23 q^{92}+20 q^{90}-11 q^{88}-7 q^{86}+17 q^{84}-21 q^{82}+10 q^{80}+5 q^{78}-21 q^{76}+28 q^{74}-25 q^{72}+12 q^{70}+4 q^{68}-20 q^{66}+31 q^{64}-29 q^{62}+22 q^{60}-5 q^{58}-8 q^{56}+22 q^{54}-24 q^{52}+20 q^{50}-5 q^{48}-6 q^{46}+19 q^{44}-16 q^{42}+7 q^{40}+12 q^{38}-21 q^{36}+25 q^{34}-14 q^{32}-2 q^{30}+17 q^{28}-24 q^{26}+24 q^{24}-13 q^{22}+2 q^{20}+7 q^{18}-13 q^{16}+10 q^{14}-7 q^{12}+3 q^{10}-2 q^6-q^2+1- q^{-2} + q^{-4} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_148.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{17}+q^{15}-2 q^{13}+q^{11}+q^7+2 q^5-q^3+2 q- q^{-1} }[/math]
2	[math]\displaystyle{ q^{48}-q^{46}-q^{44}+4 q^{42}-q^{40}-6 q^{38}+5 q^{36}+3 q^{34}-8 q^{32}+q^{30}+5 q^{28}-5 q^{26}-2 q^{24}+4 q^{22}-3 q^{18}+q^{16}+7 q^{14}-4 q^{12}-3 q^{10}+9 q^8-2 q^6-5 q^4+5 q^2+1-2 q^{-2} }[/math]
3	[math]\displaystyle{ -q^{93}+q^{91}+q^{89}-q^{87}-3 q^{85}+q^{83}+7 q^{81}+q^{79}-11 q^{77}-7 q^{75}+12 q^{73}+18 q^{71}-8 q^{69}-24 q^{67}-2 q^{65}+26 q^{63}+14 q^{61}-25 q^{59}-21 q^{57}+17 q^{55}+26 q^{53}-11 q^{51}-26 q^{49}+3 q^{47}+24 q^{45}-q^{43}-18 q^{41}-6 q^{39}+13 q^{37}+9 q^{35}-9 q^{33}-17 q^{31}+3 q^{29}+24 q^{27}+5 q^{25}-25 q^{23}-13 q^{21}+29 q^{19}+21 q^{17}-21 q^{15}-24 q^{13}+12 q^{11}+24 q^9-q^7-18 q^5-3 q^3+8 q+7 q^{-1} -2 q^{-3} -4 q^{-5} - q^{-7} + q^{-11} }[/math]
5	[math]\displaystyle{ -q^{225}+q^{223}+q^{221}-q^{219}-q^{213}+q^{211}+4 q^{209}-q^{207}-6 q^{205}-5 q^{203}-3 q^{201}+7 q^{199}+19 q^{197}+18 q^{195}-7 q^{193}-34 q^{191}-43 q^{189}-21 q^{187}+34 q^{185}+82 q^{183}+80 q^{181}+3 q^{179}-100 q^{177}-154 q^{175}-100 q^{173}+51 q^{171}+208 q^{169}+238 q^{167}+73 q^{165}-179 q^{163}-345 q^{161}-273 q^{159}+37 q^{157}+376 q^{155}+471 q^{153}+190 q^{151}-277 q^{149}-584 q^{147}-454 q^{145}+64 q^{143}+584 q^{141}+659 q^{139}+189 q^{137}-452 q^{135}-754 q^{133}-435 q^{131}+258 q^{129}+735 q^{127}+587 q^{125}-53 q^{123}-619 q^{121}-641 q^{119}-108 q^{117}+477 q^{115}+596 q^{113}+203 q^{111}-328 q^{109}-510 q^{107}-222 q^{105}+215 q^{103}+398 q^{101}+212 q^{99}-134 q^{97}-312 q^{95}-182 q^{93}+88 q^{91}+240 q^{89}+173 q^{87}-44 q^{85}-217 q^{83}-197 q^{81}-5 q^{79}+201 q^{77}+257 q^{75}+91 q^{73}-190 q^{71}-356 q^{69}-219 q^{67}+148 q^{65}+448 q^{63}+388 q^{61}-43 q^{59}-516 q^{57}-583 q^{55}-110 q^{53}+505 q^{51}+733 q^{49}+329 q^{47}-397 q^{45}-807 q^{43}-535 q^{41}+200 q^{39}+754 q^{37}+673 q^{35}+51 q^{33}-576 q^{31}-695 q^{29}-268 q^{27}+327 q^{25}+595 q^{23}+381 q^{21}-72 q^{19}-393 q^{17}-381 q^{15}-104 q^{13}+193 q^{11}+278 q^9+162 q^7-23 q^5-150 q^3-143 q-48 q^{-1} +48 q^{-3} +78 q^{-5} +54 q^{-7} +7 q^{-9} -27 q^{-11} -33 q^{-13} -15 q^{-15} +4 q^{-17} +8 q^{-19} +8 q^{-21} +3 q^{-23} -2 q^{-25} -2 q^{-27} }[/math]
6	[math]\displaystyle{ q^{312}-q^{310}-q^{308}+q^{306}-2 q^{300}+3 q^{298}-4 q^{294}+3 q^{292}+4 q^{290}+4 q^{288}-5 q^{286}-q^{284}-9 q^{282}-18 q^{280}-2 q^{278}+17 q^{276}+36 q^{274}+24 q^{272}+23 q^{270}-24 q^{268}-81 q^{266}-87 q^{264}-47 q^{262}+47 q^{260}+115 q^{258}+195 q^{256}+142 q^{254}-27 q^{252}-218 q^{250}-336 q^{248}-285 q^{246}-92 q^{244}+298 q^{242}+561 q^{240}+567 q^{238}+236 q^{236}-296 q^{234}-790 q^{232}-996 q^{230}-545 q^{228}+252 q^{226}+1081 q^{224}+1421 q^{222}+1031 q^{220}-71 q^{218}-1389 q^{216}-1985 q^{214}-1561 q^{212}-115 q^{210}+1569 q^{208}+2619 q^{206}+2168 q^{204}+301 q^{202}-1848 q^{200}-3159 q^{198}-2641 q^{196}-518 q^{194}+2188 q^{192}+3668 q^{190}+2940 q^{188}+438 q^{186}-2487 q^{184}-3955 q^{182}-3058 q^{180}-123 q^{178}+2840 q^{176}+4015 q^{174}+2701 q^{172}-322 q^{170}-3053 q^{168}-3818 q^{166}-2022 q^{164}+898 q^{162}+3084 q^{160}+3170 q^{158}+1206 q^{156}-1361 q^{154}-2857 q^{152}-2277 q^{150}-351 q^{148}+1604 q^{146}+2265 q^{144}+1365 q^{142}-303 q^{140}-1563 q^{138}-1537 q^{136}-543 q^{134}+689 q^{132}+1242 q^{130}+882 q^{128}-34 q^{126}-822 q^{124}-902 q^{122}-366 q^{120}+416 q^{118}+821 q^{116}+667 q^{114}+20 q^{112}-669 q^{110}-933 q^{108}-573 q^{106}+272 q^{104}+984 q^{102}+1182 q^{100}+553 q^{98}-563 q^{96}-1538 q^{94}-1584 q^{92}-480 q^{90}+1074 q^{88}+2227 q^{86}+1977 q^{84}+342 q^{82}-1865 q^{80}-3036 q^{78}-2226 q^{76}+108 q^{74}+2696 q^{72}+3684 q^{70}+2333 q^{68}-781 q^{66}-3493 q^{64}-4023 q^{62}-2045 q^{60}+1366 q^{58}+3933 q^{56}+4082 q^{54}+1550 q^{52}-1814 q^{50}-3942 q^{48}-3658 q^{46}-1135 q^{44}+1901 q^{42}+3640 q^{40}+3001 q^{38}+757 q^{36}-1674 q^{34}-2929 q^{32}-2368 q^{30}-556 q^{28}+1334 q^{26}+2124 q^{24}+1705 q^{22}+458 q^{20}-832 q^{18}-1437 q^{16}-1168 q^{14}-341 q^{12}+420 q^{10}+829 q^8+745 q^6+300 q^4-182 q^2-431-401 q^{-2} -216 q^{-4} +24 q^{-6} +187 q^{-8} +209 q^{-10} +120 q^{-12} +16 q^{-14} -54 q^{-16} -83 q^{-18} -63 q^{-20} -17 q^{-22} +13 q^{-24} +21 q^{-26} +18 q^{-28} +11 q^{-30} +2 q^{-32} -6 q^{-34} -3 q^{-36} - q^{-38} - q^{-40} - q^{-42} + q^{-46} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{24}-2 q^{20}-q^{18}+q^{16}+3 q^{12}+q^{10}+2 q^8+q^6-q^4+q^2-1 }[/math]
1,1	[math]\displaystyle{ q^{68}-2 q^{66}+6 q^{64}-12 q^{62}+21 q^{60}-32 q^{58}+48 q^{56}-60 q^{54}+64 q^{52}-66 q^{50}+54 q^{48}-30 q^{46}-3 q^{44}+38 q^{42}-70 q^{40}+98 q^{38}-118 q^{36}+118 q^{34}-120 q^{32}+96 q^{30}-77 q^{28}+44 q^{26}-8 q^{24}-16 q^{22}+54 q^{20}-60 q^{18}+72 q^{16}-60 q^{14}+53 q^{12}-36 q^{10}+20 q^8-12 q^6+6 q^4-2 q^2-2 q^{-2} + q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{62}+2 q^{56}+2 q^{54}-q^{52}-q^{50}+q^{46}-6 q^{44}-5 q^{42}-2 q^{38}-2 q^{36}-q^{34}+3 q^{32}+q^{28}+3 q^{26}+3 q^{24}+4 q^{20}+4 q^{18}-2 q^{16}+3 q^{12}+q^{10}-2 q^8-q^6+3 q^4-2 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{54}-q^{52}+q^{50}+2 q^{48}-3 q^{46}+2 q^{44}+2 q^{42}-6 q^{40}+3 q^{38}-8 q^{34}-2 q^{32}-2 q^{30}-4 q^{28}-q^{26}+4 q^{24}+6 q^{22}+5 q^{20}+3 q^{18}+9 q^{16}-q^{14}-3 q^{12}+5 q^{10}-4 q^8-4 q^6+3 q^4-q^2-1+ q^{-2} }[/math]
1,0,0	[math]\displaystyle{ -q^{31}-3 q^{27}-q^{25}-2 q^{23}+q^{21}+q^{19}+3 q^{17}+3 q^{15}+2 q^{13}+2 q^{11}+q^7-2 q^5+q^3-q }[/math]
1,0,1	[math]\displaystyle{ q^{88}-2 q^{86}+5 q^{84}-5 q^{82}+2 q^{80}+7 q^{78}-19 q^{76}+28 q^{74}-23 q^{72}+5 q^{70}+22 q^{68}-51 q^{66}+60 q^{64}-51 q^{62}+22 q^{60}+20 q^{58}-47 q^{56}+68 q^{54}-52 q^{52}+31 q^{50}-14 q^{48}-8 q^{46}-19 q^{44}-4 q^{42}-14 q^{40}-33 q^{38}+53 q^{36}-67 q^{34}+76 q^{32}-21 q^{30}+10 q^{28}+56 q^{26}-50 q^{24}+59 q^{22}-26 q^{20}-q^{18}+16 q^{16}-22 q^{14}+7 q^{12}+q^{10}-7 q^8+4 q^6-4 q^2+3-2 q^{-2} + q^{-4} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{68}+q^{64}+3 q^{62}+5 q^{56}+q^{54}-5 q^{52}-2 q^{50}-q^{48}-9 q^{46}-13 q^{44}-6 q^{42}-5 q^{40}-10 q^{38}-q^{36}+10 q^{34}+5 q^{32}+9 q^{30}+18 q^{28}+11 q^{26}+4 q^{24}+6 q^{22}+4 q^{20}-5 q^{18}-7 q^{16}-q^{12}-5 q^{10}-q^8+3 q^6-q^4+1 }[/math]
1,0,0,0	[math]\displaystyle{ -q^{38}-3 q^{34}-2 q^{32}-2 q^{30}-2 q^{28}+q^{26}+q^{24}+4 q^{22}+3 q^{20}+4 q^{18}+2 q^{16}+2 q^{14}-2 q^6+q^4-q^2 }[/math]

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{74}-q^{72}+2 q^{70}-2 q^{68}+4 q^{66}-4 q^{64}+4 q^{62}-5 q^{60}+5 q^{58}-4 q^{56}+2 q^{54}-2 q^{52}-q^{50}-q^{48}-8 q^{46}+q^{44}-10 q^{42}+3 q^{40}-11 q^{38}+8 q^{36}-5 q^{34}+13 q^{32}+11 q^{28}+3 q^{26}+8 q^{24}+4 q^{22}-q^{20}+2 q^{18}-5 q^{16}+4 q^{14}-6 q^{12}+2 q^{10}-6 q^8+4 q^6-2 q^4+q^2-1+ q^{-2} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+3 q^{120}-q^{118}-3 q^{116}+9 q^{114}-13 q^{112}+15 q^{110}-11 q^{108}-q^{106}+12 q^{104}-22 q^{102}+25 q^{100}-18 q^{98}+3 q^{96}+10 q^{94}-23 q^{92}+20 q^{90}-11 q^{88}-7 q^{86}+17 q^{84}-21 q^{82}+10 q^{80}+5 q^{78}-21 q^{76}+28 q^{74}-25 q^{72}+12 q^{70}+4 q^{68}-20 q^{66}+31 q^{64}-29 q^{62}+22 q^{60}-5 q^{58}-8 q^{56}+22 q^{54}-24 q^{52}+20 q^{50}-5 q^{48}-6 q^{46}+19 q^{44}-16 q^{42}+7 q^{40}+12 q^{38}-21 q^{36}+25 q^{34}-14 q^{32}-2 q^{30}+17 q^{28}-24 q^{26}+24 q^{24}-13 q^{22}+2 q^{20}+7 q^{18}-13 q^{16}+10 q^{14}-7 q^{12}+3 q^{10}-2 q^6-q^2+1- q^{-2} + q^{-4} }[/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["10 148"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-3 t^2+7 t-9+7 t^{-1} -3 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+3 z^4+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]=

{ 31, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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