9 37

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-11 t+19-11 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature	{ 45, 0 }
Jones polynomial	[math]\displaystyle{ q^4-2 q^3+5 q^2-7 q+7-8 q^{-1} +7 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4+z^4 a^2+z^2 a^2+2 a^2+z^4-z^2-2-2 z^2 a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+3 z^6 a^{-2} +5 z^6+a^5 z^5-6 a^3 z^5-13 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -8 a^4 z^4-17 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -13 z^4-2 a^5 z^3+3 a^3 z^3+13 a z^3+6 z^3 a^{-1} -2 z^3 a^{-3} +5 a^4 z^2+14 a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +12 z^2-2 a^3 z-7 a z-5 z a^{-1} -2 a^2+ a^{-4} -2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-q^{10}+3 q^8+q^6-3-2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-4 q^{70}-4 q^{68}+16 q^{66}-23 q^{64}+29 q^{62}-22 q^{60}+6 q^{58}+16 q^{56}-38 q^{54}+52 q^{52}-49 q^{50}+24 q^{48}+8 q^{46}-33 q^{44}+50 q^{42}-44 q^{40}+24 q^{38}+7 q^{36}-27 q^{34}+33 q^{32}-28 q^{30}-6 q^{28}+40 q^{26}-46 q^{24}+41 q^{22}-18 q^{20}-17 q^{18}+57 q^{16}-71 q^{14}+65 q^{12}-48 q^{10}+4 q^8+43 q^6-65 q^4+65 q^2-47+16 q^{-2} +18 q^{-4} -39 q^{-6} +32 q^{-8} -22 q^{-10} -7 q^{-12} +33 q^{-14} -38 q^{-16} +22 q^{-18} +6 q^{-20} -33 q^{-22} +50 q^{-24} -48 q^{-26} +29 q^{-28} -8 q^{-30} -20 q^{-32} +38 q^{-34} -40 q^{-36} +34 q^{-38} -14 q^{-40} +2 q^{-42} +9 q^{-44} -15 q^{-46} +15 q^{-48} -12 q^{-50} +8 q^{-52} -2 q^{-54} - q^{-56} +3 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_37.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-q^7+3 q^5-q^3-q-2 q^{-3} +3 q^{-5} - q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+6 q^{26}-2 q^{24}-7 q^{22}+10 q^{20}+3 q^{18}-12 q^{16}+8 q^{14}+6 q^{12}-13 q^{10}+6 q^6-3 q^4-4 q^2+5+9 q^{-2} -8 q^{-4} -2 q^{-6} +13 q^{-8} -9 q^{-10} -7 q^{-12} +11 q^{-14} -3 q^{-16} -5 q^{-18} +4 q^{-20} - q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-6 q^{55}+2 q^{53}+13 q^{51}-q^{49}-19 q^{47}-7 q^{45}+26 q^{43}+19 q^{41}-25 q^{39}-34 q^{37}+22 q^{35}+46 q^{33}-8 q^{31}-56 q^{29}-5 q^{27}+54 q^{25}+15 q^{23}-51 q^{21}-30 q^{19}+45 q^{17}+32 q^{15}-26 q^{13}-33 q^{11}+18 q^9+34 q^7+3 q^5-32 q^3-18 q+26 q^{-1} +32 q^{-3} -20 q^{-5} -51 q^{-7} +10 q^{-9} +56 q^{-11} +4 q^{-13} -58 q^{-15} -11 q^{-17} +49 q^{-19} +24 q^{-21} -38 q^{-23} -25 q^{-25} +25 q^{-27} +21 q^{-29} -10 q^{-31} -16 q^{-33} +4 q^{-35} +8 q^{-37} -2 q^{-39} -4 q^{-41} + q^{-43} + q^{-45} - q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+6 q^{94}-9 q^{92}-13 q^{90}+2 q^{88}+10 q^{86}+31 q^{84}-8 q^{82}-41 q^{80}-26 q^{78}+5 q^{76}+82 q^{74}+38 q^{72}-48 q^{70}-91 q^{68}-69 q^{66}+104 q^{64}+137 q^{62}+39 q^{60}-122 q^{58}-208 q^{56}+11 q^{54}+186 q^{52}+197 q^{50}-30 q^{48}-287 q^{46}-149 q^{44}+110 q^{42}+288 q^{40}+120 q^{38}-235 q^{36}-241 q^{34}-11 q^{32}+262 q^{30}+200 q^{28}-122 q^{26}-225 q^{24}-87 q^{22}+164 q^{20}+187 q^{18}-17 q^{16}-167 q^{14}-127 q^{12}+61 q^{10}+153 q^8+85 q^6-91 q^4-160 q^2-60+115 q^{-2} +209 q^{-4} +16 q^{-6} -183 q^{-8} -204 q^{-10} +30 q^{-12} +292 q^{-14} +149 q^{-16} -128 q^{-18} -296 q^{-20} -105 q^{-22} +253 q^{-24} +229 q^{-26} +7 q^{-28} -247 q^{-30} -196 q^{-32} +108 q^{-34} +185 q^{-36} +105 q^{-38} -106 q^{-40} -159 q^{-42} -5 q^{-44} +68 q^{-46} +92 q^{-48} -3 q^{-50} -63 q^{-52} -22 q^{-54} +34 q^{-58} +9 q^{-60} -13 q^{-62} -2 q^{-64} -6 q^{-66} +6 q^{-68} + q^{-70} -3 q^{-72} +2 q^{-74} -2 q^{-76} + q^{-78} - q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+q^{143}+9 q^{141}+13 q^{139}-2 q^{137}-20 q^{135}-22 q^{133}-7 q^{131}+24 q^{129}+49 q^{127}+36 q^{125}-30 q^{123}-87 q^{121}-77 q^{119}+q^{117}+113 q^{115}+163 q^{113}+70 q^{111}-122 q^{109}-251 q^{107}-195 q^{105}+42 q^{103}+322 q^{101}+387 q^{99}+126 q^{97}-304 q^{95}-569 q^{93}-409 q^{91}+149 q^{89}+686 q^{87}+733 q^{85}+168 q^{83}-652 q^{81}-1042 q^{79}-583 q^{77}+432 q^{75}+1201 q^{73}+1037 q^{71}-51 q^{69}-1202 q^{67}-1392 q^{65}-396 q^{63}+997 q^{61}+1595 q^{59}+828 q^{57}-676 q^{55}-1618 q^{53}-1126 q^{51}+330 q^{49}+1462 q^{47}+1286 q^{45}+2 q^{43}-1231 q^{41}-1298 q^{39}-210 q^{37}+938 q^{35}+1177 q^{33}+372 q^{31}-702 q^{29}-1041 q^{27}-424 q^{25}+476 q^{23}+875 q^{21}+485 q^{19}-293 q^{17}-756 q^{15}-533 q^{13}+112 q^{11}+662 q^9+652 q^7+95 q^5-573 q^3-803 q-357 q^{-1} +462 q^{-3} +977 q^{-5} +682 q^{-7} -284 q^{-9} -1125 q^{-11} -1033 q^{-13} +10 q^{-15} +1166 q^{-17} +1377 q^{-19} +345 q^{-21} -1089 q^{-23} -1605 q^{-25} -733 q^{-27} +812 q^{-29} +1696 q^{-31} +1087 q^{-33} -457 q^{-35} -1549 q^{-37} -1305 q^{-39} +15 q^{-41} +1246 q^{-43} +1358 q^{-45} +342 q^{-47} -825 q^{-49} -1191 q^{-51} -576 q^{-53} +399 q^{-55} +907 q^{-57} +640 q^{-59} -75 q^{-61} -579 q^{-63} -543 q^{-65} -127 q^{-67} +285 q^{-69} +390 q^{-71} +185 q^{-73} -99 q^{-75} -223 q^{-77} -155 q^{-79} -2 q^{-81} +105 q^{-83} +96 q^{-85} +29 q^{-87} -36 q^{-89} -49 q^{-91} -20 q^{-93} +9 q^{-95} +16 q^{-97} +11 q^{-99} +4 q^{-101} -8 q^{-103} -5 q^{-105} +2 q^{-107} - q^{-109} +3 q^{-113} - q^{-115} -2 q^{-117} + q^{-119} - q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-q^{10}+3 q^8+q^6-3-2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+42 q^{36}-70 q^{34}+100 q^{32}-136 q^{30}+169 q^{28}-186 q^{26}+186 q^{24}-156 q^{22}+117 q^{20}-40 q^{18}-44 q^{16}+138 q^{14}-229 q^{12}+284 q^{10}-348 q^8+352 q^6-343 q^4+300 q^2-214+146 q^{-2} -42 q^{-4} -30 q^{-6} +104 q^{-8} -154 q^{-10} +166 q^{-12} -170 q^{-14} +152 q^{-16} -132 q^{-18} +99 q^{-20} -72 q^{-22} +50 q^{-24} -32 q^{-26} +21 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{40}-2 q^{38}+3 q^{34}+q^{32}-6 q^{30}+8 q^{26}+5 q^{24}-6 q^{22}-2 q^{20}+8 q^{18}+4 q^{16}-9 q^{14}-5 q^{12}+2 q^{10}-5 q^8-4 q^6+2 q^2+2+8 q^{-2} +6 q^{-4} -3 q^{-6} - q^{-8} +8 q^{-10} -11 q^{-14} +7 q^{-18} -8 q^{-22} -3 q^{-24} +4 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} + q^{-36} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-2 q^{32}+2 q^{28}-6 q^{26}+4 q^{24}+6 q^{22}-6 q^{20}+7 q^{18}+8 q^{16}-11 q^{14}+2 q^{10}-9 q^8-2 q^6+4 q^4+6 q^2+1+ q^{-2} +8 q^{-4} -5 q^{-6} -10 q^{-8} +8 q^{-10} -4 q^{-12} -8 q^{-14} +9 q^{-16} -3 q^{-20} +4 q^{-22} + q^{-24} - q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}+q^{19}+q^{15}-q^{13}+3 q^{11}+q^9+2 q^7-2 q-3 q^{-1} -2 q^{-5} + q^{-7} +2 q^{-11} - q^{-13} + q^{-15} + q^{-17} + q^{-19} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-4 q^{30}+6 q^{28}-8 q^{26}+10 q^{24}-10 q^{22}+10 q^{20}-7 q^{18}+6 q^{16}+q^{14}-4 q^{12}+12 q^{10}-15 q^8+18 q^6-20 q^4+18 q^2-19+11 q^{-2} -8 q^{-4} + q^{-6} +2 q^{-8} -6 q^{-10} +10 q^{-12} -10 q^{-14} +11 q^{-16} -8 q^{-18} +7 q^{-20} -4 q^{-22} +3 q^{-24} - q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+2 q^{48}+4 q^{46}-2 q^{44}-7 q^{42}-2 q^{40}+9 q^{38}+8 q^{36}-4 q^{34}-9 q^{32}+2 q^{30}+12 q^{28}+7 q^{26}-9 q^{24}-9 q^{22}+2 q^{20}+7 q^{18}-4 q^{16}-10 q^{14}-2 q^{12}+6 q^{10}+2 q^8-6 q^6-q^4+8 q^2+9-3 q^{-2} -5 q^{-4} +4 q^{-6} +9 q^{-8} -2 q^{-10} -11 q^{-12} -5 q^{-14} +8 q^{-16} +7 q^{-18} -7 q^{-20} -11 q^{-22} +10 q^{-26} +4 q^{-28} -4 q^{-30} -5 q^{-32} + q^{-34} +4 q^{-36} +2 q^{-38} - q^{-40} - q^{-42} + q^{-46} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-4 q^{70}-4 q^{68}+16 q^{66}-23 q^{64}+29 q^{62}-22 q^{60}+6 q^{58}+16 q^{56}-38 q^{54}+52 q^{52}-49 q^{50}+24 q^{48}+8 q^{46}-33 q^{44}+50 q^{42}-44 q^{40}+24 q^{38}+7 q^{36}-27 q^{34}+33 q^{32}-28 q^{30}-6 q^{28}+40 q^{26}-46 q^{24}+41 q^{22}-18 q^{20}-17 q^{18}+57 q^{16}-71 q^{14}+65 q^{12}-48 q^{10}+4 q^8+43 q^6-65 q^4+65 q^2-47+16 q^{-2} +18 q^{-4} -39 q^{-6} +32 q^{-8} -22 q^{-10} -7 q^{-12} +33 q^{-14} -38 q^{-16} +22 q^{-18} +6 q^{-20} -33 q^{-22} +50 q^{-24} -48 q^{-26} +29 q^{-28} -8 q^{-30} -20 q^{-32} +38 q^{-34} -40 q^{-36} +34 q^{-38} -14 q^{-40} +2 q^{-42} +9 q^{-44} -15 q^{-46} +15 q^{-48} -12 q^{-50} +8 q^{-52} -2 q^{-54} - q^{-56} +3 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/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["9 37"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-11 t+19-11 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^4-3 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{ \{3,t+1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^4+z^4 a^2+z^2 a^2+2 a^2+z^4-z^2-2-2 z^2 a^{-2} + a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+3 z^6 a^{-2} +5 z^6+a^5 z^5-6 a^3 z^5-13 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -8 a^4 z^4-17 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -13 z^4-2 a^5 z^3+3 a^3 z^3+13 a z^3+6 z^3 a^{-1} -2 z^3 a^{-3} +5 a^4 z^2+14 a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +12 z^2-2 a^3 z-7 a z-5 z a^{-1} -2 a^2+ a^{-4} -2 }[/math]