10 37

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-13 t+19-13 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 53, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+2 q^4-4 q^3+7 q^2-8 q+9-8 q^{-1} +7 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4-a^4+z^4 a^2+z^2 a^2+a^2+2 z^4+3 z^2+1+z^4 a^{-2} +z^2 a^{-2} + a^{-2} -z^2 a^{-4} - a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7+2 z^7 a^{-3} +2 a^4 z^6-3 a^2 z^6-3 z^6 a^{-2} +2 z^6 a^{-4} -10 z^6+a^5 z^5-2 a^3 z^5-2 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+2 a^2 z^4+2 z^4 a^{-2} -5 z^4 a^{-4} +14 z^4-3 a^5 z^3-3 a^3 z^3+a z^3+z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +3 a^4 z^2+3 z^2 a^{-4} -6 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} +2 z a^{-3} +2 z a^{-5} -a^4-a^2- a^{-2} - a^{-4} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}-2 q^{10}+2 q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} +2 q^{-8} -2 q^{-10} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-2 q^{70}-3 q^{68}+8 q^{66}-13 q^{64}+15 q^{62}-15 q^{60}+8 q^{58}+q^{56}-16 q^{54}+31 q^{52}-41 q^{50}+37 q^{48}-25 q^{46}-2 q^{44}+30 q^{42}-52 q^{40}+62 q^{38}-47 q^{36}+19 q^{34}+17 q^{32}-46 q^{30}+48 q^{28}-29 q^{26}+3 q^{24}+27 q^{22}-38 q^{20}+31 q^{18}+q^{16}-34 q^{14}+62 q^{12}-69 q^{10}+46 q^8-6 q^6-39 q^4+75 q^2-85+75 q^{-2} -39 q^{-4} -6 q^{-6} +46 q^{-8} -69 q^{-10} +62 q^{-12} -34 q^{-14} + q^{-16} +31 q^{-18} -38 q^{-20} +27 q^{-22} +3 q^{-24} -29 q^{-26} +48 q^{-28} -46 q^{-30} +17 q^{-32} +19 q^{-34} -47 q^{-36} +62 q^{-38} -52 q^{-40} +30 q^{-42} -2 q^{-44} -25 q^{-46} +37 q^{-48} -41 q^{-50} +31 q^{-52} -16 q^{-54} + q^{-56} +8 q^{-58} -15 q^{-60} +15 q^{-62} -13 q^{-64} +8 q^{-66} -3 q^{-68} -2 q^{-70} +3 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_37.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^9-2 q^7+3 q^5-q^3+q+ q^{-1} - q^{-3} +3 q^{-5} -2 q^{-7} + q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+3 q^{26}-3 q^{24}-3 q^{22}+9 q^{20}-4 q^{18}-10 q^{16}+13 q^{14}-14 q^{10}+9 q^8+5 q^6-8 q^4+2 q^2+7+2 q^{-2} -8 q^{-4} +5 q^{-6} +9 q^{-8} -14 q^{-10} +13 q^{-14} -10 q^{-16} -4 q^{-18} +9 q^{-20} -3 q^{-22} -3 q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{61}+q^{59}-2 q^{55}+q^{53}+2 q^{51}-3 q^{49}-4 q^{47}+6 q^{45}+9 q^{43}-8 q^{41}-18 q^{39}+8 q^{37}+31 q^{35}-2 q^{33}-41 q^{31}-15 q^{29}+50 q^{27}+27 q^{25}-48 q^{23}-44 q^{21}+39 q^{19}+52 q^{17}-26 q^{15}-53 q^{13}+14 q^{11}+47 q^9+2 q^7-36 q^5-13 q^3+26 q+26 q^{-1} -13 q^{-3} -36 q^{-5} +2 q^{-7} +47 q^{-9} +14 q^{-11} -53 q^{-13} -26 q^{-15} +52 q^{-17} +39 q^{-19} -44 q^{-21} -48 q^{-23} +27 q^{-25} +50 q^{-27} -15 q^{-29} -41 q^{-31} -2 q^{-33} +31 q^{-35} +8 q^{-37} -18 q^{-39} -8 q^{-41} +9 q^{-43} +6 q^{-45} -4 q^{-47} -3 q^{-49} +2 q^{-51} + q^{-53} -2 q^{-55} + q^{-59} + q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-q^{102}-q^{100}-q^{96}+4 q^{94}-q^{92}+2 q^{88}-6 q^{86}+3 q^{84}-6 q^{82}+2 q^{80}+15 q^{78}-3 q^{76}+q^{74}-29 q^{72}-14 q^{70}+33 q^{68}+31 q^{66}+35 q^{64}-59 q^{62}-82 q^{60}-2 q^{58}+69 q^{56}+143 q^{54}-15 q^{52}-156 q^{50}-133 q^{48}+16 q^{46}+258 q^{44}+133 q^{42}-128 q^{40}-261 q^{38}-137 q^{36}+248 q^{34}+268 q^{32}+12 q^{30}-264 q^{28}-261 q^{26}+126 q^{24}+273 q^{22}+127 q^{20}-158 q^{18}-256 q^{16}-4 q^{14}+174 q^{12}+161 q^{10}-35 q^8-175 q^6-91 q^4+67 q^2+159+67 q^{-2} -91 q^{-4} -175 q^{-6} -35 q^{-8} +161 q^{-10} +174 q^{-12} -4 q^{-14} -256 q^{-16} -158 q^{-18} +127 q^{-20} +273 q^{-22} +126 q^{-24} -261 q^{-26} -264 q^{-28} +12 q^{-30} +268 q^{-32} +248 q^{-34} -137 q^{-36} -261 q^{-38} -128 q^{-40} +133 q^{-42} +258 q^{-44} +16 q^{-46} -133 q^{-48} -156 q^{-50} -15 q^{-52} +143 q^{-54} +69 q^{-56} -2 q^{-58} -82 q^{-60} -59 q^{-62} +35 q^{-64} +31 q^{-66} +33 q^{-68} -14 q^{-70} -29 q^{-72} + q^{-74} -3 q^{-76} +15 q^{-78} +2 q^{-80} -6 q^{-82} +3 q^{-84} -6 q^{-86} +2 q^{-88} - q^{-92} +4 q^{-94} - q^{-96} - q^{-100} - q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+q^{153}+q^{151}+q^{147}-q^{145}-4 q^{143}-q^{141}+2 q^{139}+5 q^{135}+6 q^{133}-3 q^{131}-7 q^{129}-7 q^{127}-7 q^{125}+4 q^{123}+21 q^{121}+19 q^{119}-q^{117}-24 q^{115}-42 q^{113}-26 q^{111}+22 q^{109}+72 q^{107}+73 q^{105}+6 q^{103}-89 q^{101}-142 q^{99}-86 q^{97}+68 q^{95}+222 q^{93}+222 q^{91}+23 q^{89}-258 q^{87}-400 q^{85}-220 q^{83}+202 q^{81}+571 q^{79}+520 q^{77}-21 q^{75}-661 q^{73}-844 q^{71}-319 q^{69}+593 q^{67}+1155 q^{65}+750 q^{63}-381 q^{61}-1297 q^{59}-1189 q^{57}-17 q^{55}+1284 q^{53}+1545 q^{51}+447 q^{49}-1080 q^{47}-1710 q^{45}-867 q^{43}+751 q^{41}+1692 q^{39}+1151 q^{37}-378 q^{35}-1510 q^{33}-1271 q^{31}+39 q^{29}+1210 q^{27}+1247 q^{25}+223 q^{23}-881 q^{21}-1120 q^{19}-381 q^{17}+574 q^{15}+942 q^{13}+489 q^{11}-314 q^9-779 q^7-562 q^5+101 q^3+650 q+650 q^{-1} +101 q^{-3} -562 q^{-5} -779 q^{-7} -314 q^{-9} +489 q^{-11} +942 q^{-13} +574 q^{-15} -381 q^{-17} -1120 q^{-19} -881 q^{-21} +223 q^{-23} +1247 q^{-25} +1210 q^{-27} +39 q^{-29} -1271 q^{-31} -1510 q^{-33} -378 q^{-35} +1151 q^{-37} +1692 q^{-39} +751 q^{-41} -867 q^{-43} -1710 q^{-45} -1080 q^{-47} +447 q^{-49} +1545 q^{-51} +1284 q^{-53} -17 q^{-55} -1189 q^{-57} -1297 q^{-59} -381 q^{-61} +750 q^{-63} +1155 q^{-65} +593 q^{-67} -319 q^{-69} -844 q^{-71} -661 q^{-73} -21 q^{-75} +520 q^{-77} +571 q^{-79} +202 q^{-81} -220 q^{-83} -400 q^{-85} -258 q^{-87} +23 q^{-89} +222 q^{-91} +222 q^{-93} +68 q^{-95} -86 q^{-97} -142 q^{-99} -89 q^{-101} +6 q^{-103} +73 q^{-105} +72 q^{-107} +22 q^{-109} -26 q^{-111} -42 q^{-113} -24 q^{-115} - q^{-117} +19 q^{-119} +21 q^{-121} +4 q^{-123} -7 q^{-125} -7 q^{-127} -7 q^{-129} -3 q^{-131} +6 q^{-133} +5 q^{-135} +2 q^{-139} - q^{-141} -4 q^{-143} - q^{-145} + q^{-147} + q^{-151} + q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}-2 q^{10}+2 q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} +2 q^{-8} -2 q^{-10} - q^{-16} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+6 q^{40}-12 q^{38}+21 q^{36}-34 q^{34}+50 q^{32}-74 q^{30}+102 q^{28}-134 q^{26}+166 q^{24}-194 q^{22}+209 q^{20}-206 q^{18}+172 q^{16}-114 q^{14}+27 q^{12}+70 q^{10}-180 q^8+282 q^6-359 q^4+420 q^2-426+420 q^{-2} -359 q^{-4} +282 q^{-6} -180 q^{-8} +70 q^{-10} +27 q^{-12} -114 q^{-14} +172 q^{-16} -206 q^{-18} +209 q^{-20} -194 q^{-22} +166 q^{-24} -134 q^{-26} +102 q^{-28} -74 q^{-30} +50 q^{-32} -34 q^{-34} +21 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{38}+2 q^{34}-4 q^{30}-q^{28}+6 q^{26}-8 q^{22}-2 q^{20}+7 q^{18}+2 q^{16}-11 q^{14}-q^{12}+6 q^{10}-2 q^8-3 q^6+5 q^4+6 q^2+2+6 q^{-2} +5 q^{-4} -3 q^{-6} -2 q^{-8} +6 q^{-10} - q^{-12} -11 q^{-14} +2 q^{-16} +7 q^{-18} -2 q^{-20} -8 q^{-22} +6 q^{-26} - q^{-28} -4 q^{-30} +2 q^{-34} - q^{-38} + q^{-42} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-q^{32}+q^{30}+2 q^{28}-4 q^{26}+3 q^{22}-10 q^{20}+9 q^{16}-11 q^{14}+2 q^{12}+11 q^{10}-6 q^8-q^6+7 q^4+q^2-2+ q^{-2} +7 q^{-4} - q^{-6} -6 q^{-8} +11 q^{-10} +2 q^{-12} -11 q^{-14} +9 q^{-16} -10 q^{-20} +3 q^{-22} -4 q^{-26} +2 q^{-28} + q^{-30} - q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}-2 q^{13}+2 q^{11}+2 q^7+2 q^3+2 q^{-3} +2 q^{-7} +2 q^{-11} -2 q^{-13} - q^{-17} - q^{-21} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+q^{32}-3 q^{30}+4 q^{28}-6 q^{26}+8 q^{24}-11 q^{22}+12 q^{20}-12 q^{18}+11 q^{16}-7 q^{14}+4 q^{12}+3 q^{10}-8 q^8+15 q^6-19 q^4+23 q^2-24+23 q^{-2} -19 q^{-4} +15 q^{-6} -8 q^{-8} +3 q^{-10} +4 q^{-12} -7 q^{-14} +11 q^{-16} -12 q^{-18} +12 q^{-20} -11 q^{-22} +8 q^{-24} -6 q^{-26} +4 q^{-28} -3 q^{-30} + q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-q^{52}-q^{50}+2 q^{48}+3 q^{46}-q^{44}-5 q^{42}-3 q^{40}+4 q^{38}+7 q^{36}-3 q^{34}-12 q^{32}-5 q^{30}+10 q^{28}+11 q^{26}-6 q^{24}-13 q^{22}-q^{20}+13 q^{18}+6 q^{16}-8 q^{14}-7 q^{12}+6 q^{10}+8 q^8-2 q^6-7 q^4+2 q^2+9+2 q^{-2} -7 q^{-4} -2 q^{-6} +8 q^{-8} +6 q^{-10} -7 q^{-12} -8 q^{-14} +6 q^{-16} +13 q^{-18} - q^{-20} -13 q^{-22} -6 q^{-24} +11 q^{-26} +10 q^{-28} -5 q^{-30} -12 q^{-32} -3 q^{-34} +7 q^{-36} +4 q^{-38} -3 q^{-40} -5 q^{-42} - q^{-44} +3 q^{-46} +2 q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-2 q^{70}-3 q^{68}+8 q^{66}-13 q^{64}+15 q^{62}-15 q^{60}+8 q^{58}+q^{56}-16 q^{54}+31 q^{52}-41 q^{50}+37 q^{48}-25 q^{46}-2 q^{44}+30 q^{42}-52 q^{40}+62 q^{38}-47 q^{36}+19 q^{34}+17 q^{32}-46 q^{30}+48 q^{28}-29 q^{26}+3 q^{24}+27 q^{22}-38 q^{20}+31 q^{18}+q^{16}-34 q^{14}+62 q^{12}-69 q^{10}+46 q^8-6 q^6-39 q^4+75 q^2-85+75 q^{-2} -39 q^{-4} -6 q^{-6} +46 q^{-8} -69 q^{-10} +62 q^{-12} -34 q^{-14} + q^{-16} +31 q^{-18} -38 q^{-20} +27 q^{-22} +3 q^{-24} -29 q^{-26} +48 q^{-28} -46 q^{-30} +17 q^{-32} +19 q^{-34} -47 q^{-36} +62 q^{-38} -52 q^{-40} +30 q^{-42} -2 q^{-44} -25 q^{-46} +37 q^{-48} -41 q^{-50} +31 q^{-52} -16 q^{-54} + q^{-56} +8 q^{-58} -15 q^{-60} +15 q^{-62} -13 q^{-64} +8 q^{-66} -3 q^{-68} -2 q^{-70} +3 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/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 37"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t^2-13 t+19-13 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 4 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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 53, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7+2 z^7 a^{-3} +2 a^4 z^6-3 a^2 z^6-3 z^6 a^{-2} +2 z^6 a^{-4} -10 z^6+a^5 z^5-2 a^3 z^5-2 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+2 a^2 z^4+2 z^4 a^{-2} -5 z^4 a^{-4} +14 z^4-3 a^5 z^3-3 a^3 z^3+a z^3+z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +3 a^4 z^2+3 z^2 a^{-4} -6 z^2+2 a^5 z+2 a^3 z-a z-z a^{-1} +2 z a^{-3} +2 z a^{-5} -a^4-a^2- a^{-2} - a^{-4} +1 }[/math]