10 125

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-2 t^2+2 t-1+2 t^{-1} -2 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+4 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 11, 2 }
Jones polynomial	[math]\displaystyle{ -q^4+q^3-q^2+2 q-1+2 q^{-1} - q^{-2} + q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6-a^2 z^4-z^4 a^{-2} +6 z^4-4 a^2 z^2-4 z^2 a^{-2} +11 z^2-3 a^2-3 a^{-2} +7 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+a^3 z^7+2 a z^7+z^7 a^{-1} -6 a^2 z^6-6 z^6-6 a^3 z^5-11 a z^5-5 z^5 a^{-1} +11 a^2 z^4+2 z^4 a^{-2} +13 z^4+10 a^3 z^3+17 a z^3+8 z^3 a^{-1} +z^3 a^{-3} -8 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -15 z^2-4 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{12}-q^{10}-q^8+q^4+2 q^2+3+2 q^{-2} + q^{-4} - q^{-8} - q^{-10} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{60}+q^{56}-q^{54}-q^{48}-2 q^{44}-q^{40}-2 q^{38}-q^{36}-q^{34}-2 q^{32}-2 q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^6+2 q^4+3 q^2+1+2 q^{-2} +2 q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} - q^{-24} + q^{-26} - q^{-28} - q^{-30} - q^{-34} - q^{-36} - q^{-38} -2 q^{-40} - q^{-44} - q^{-46} - q^{-50} - q^{-56} + q^{-72} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_125.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^9+q^3+q+ q^{-1} + q^{-3} - q^{-9} }[/math]
2	[math]\displaystyle{ q^{28}-q^{24}-q^{18}-q^{16}+q^6+q^4+q^2+2+ q^{-2} + q^{-6} - q^{-12} - q^{-18} - q^{-20} + q^{-24} }[/math]
3	[math]\displaystyle{ -q^{57}+q^{53}+q^{51}-q^{47}+q^{43}+q^{41}-q^{37}-q^{35}-q^{27}-q^{25}-q^{23}-q^{17}-q^{15}+q^{13}+2 q^{11}+2 q^9+q^5+2 q^3+q+ q^{-3} + q^{-5} + q^{-13} - q^{-17} -2 q^{-19} - q^{-21} + q^{-23} +2 q^{-25} - q^{-27} -2 q^{-29} - q^{-31} +2 q^{-33} - q^{-37} + q^{-41} + q^{-43} - q^{-45} }[/math]
5	[math]\displaystyle{ -q^{145}+q^{141}+q^{139}+q^{137}-q^{133}-2 q^{131}-q^{129}+q^{125}+2 q^{123}+2 q^{121}-2 q^{117}-2 q^{115}-2 q^{113}-q^{111}+q^{109}+2 q^{107}+2 q^{105}+q^{103}-2 q^{99}-2 q^{97}-q^{95}+q^{91}+2 q^{89}+2 q^{87}+q^{85}-2 q^{81}-2 q^{79}-q^{77}+q^{75}+4 q^{73}+4 q^{71}+2 q^{69}-q^{67}-4 q^{65}-5 q^{63}-2 q^{61}+2 q^{59}+4 q^{57}+3 q^{55}-2 q^{53}-5 q^{51}-6 q^{49}-3 q^{47}+2 q^{45}+4 q^{43}+3 q^{41}-4 q^{37}-4 q^{35}-2 q^{33}+2 q^{31}+4 q^{29}+4 q^{27}+q^{25}-2 q^{23}-3 q^{21}+3 q^{17}+5 q^{15}+3 q^{13}-q^{11}-3 q^9-2 q^7+2 q^5+4 q^3+3 q-3 q^{-3} -2 q^{-5} + q^{-7} +3 q^{-9} +2 q^{-11} - q^{-13} - q^{-15} + q^{-19} -2 q^{-23} -3 q^{-25} - q^{-27} + q^{-29} +3 q^{-31} +2 q^{-33} + q^{-35} - q^{-37} -3 q^{-39} -3 q^{-41} - q^{-43} +2 q^{-45} +5 q^{-47} +5 q^{-49} + q^{-51} -5 q^{-53} -7 q^{-55} -5 q^{-57} + q^{-59} +6 q^{-61} +8 q^{-63} +2 q^{-65} -5 q^{-67} -7 q^{-69} -5 q^{-71} + q^{-73} +4 q^{-75} +3 q^{-77} + q^{-79} - q^{-81} - q^{-83} + q^{-89} + q^{-91} +2 q^{-93} - q^{-97} -2 q^{-99} - q^{-101} + q^{-107} }[/math]
6	[math]\displaystyle{ q^{204}-q^{200}-q^{198}-q^{196}+2 q^{190}+2 q^{188}+q^{186}-q^{182}-2 q^{180}-3 q^{178}-q^{176}+2 q^{172}+3 q^{170}+3 q^{168}+2 q^{166}-q^{164}-2 q^{162}-3 q^{160}-3 q^{158}-2 q^{156}+2 q^{152}+3 q^{150}+3 q^{148}+2 q^{146}-2 q^{142}-3 q^{140}-3 q^{138}-2 q^{136}-q^{134}+q^{132}+2 q^{130}+4 q^{128}+3 q^{126}+q^{124}-q^{122}-4 q^{120}-5 q^{118}-5 q^{116}-q^{114}+3 q^{112}+6 q^{110}+7 q^{108}+6 q^{106}+q^{104}-5 q^{102}-7 q^{100}-7 q^{98}-2 q^{96}+4 q^{94}+9 q^{92}+9 q^{90}+5 q^{88}-q^{86}-8 q^{84}-10 q^{82}-7 q^{80}-q^{78}+4 q^{76}+8 q^{74}+8 q^{72}+2 q^{70}-4 q^{68}-8 q^{66}-9 q^{64}-7 q^{62}+6 q^{58}+8 q^{56}+5 q^{54}-6 q^{50}-10 q^{48}-7 q^{46}-q^{44}+7 q^{42}+10 q^{40}+9 q^{38}+2 q^{36}-6 q^{34}-9 q^{32}-6 q^{30}+q^{28}+7 q^{26}+11 q^{24}+6 q^{22}-2 q^{20}-7 q^{18}-8 q^{16}-2 q^{14}+4 q^{12}+9 q^{10}+6 q^8-q^6-5 q^4-6 q^2-1+4 q^{-2} +7 q^{-4} +5 q^{-6} - q^{-8} -4 q^{-10} -5 q^{-12} -2 q^{-14} + q^{-16} +4 q^{-18} +2 q^{-20} - q^{-22} -2 q^{-24} - q^{-26} +2 q^{-28} + q^{-30} -2 q^{-34} -3 q^{-36} - q^{-38} + q^{-40} +5 q^{-42} +5 q^{-44} +3 q^{-46} - q^{-48} -4 q^{-50} -5 q^{-52} -6 q^{-54} -3 q^{-56} +2 q^{-58} +7 q^{-60} +9 q^{-62} +8 q^{-64} + q^{-66} -7 q^{-68} -14 q^{-70} -13 q^{-72} -6 q^{-74} +5 q^{-76} +16 q^{-78} +15 q^{-80} +8 q^{-82} -5 q^{-84} -15 q^{-86} -18 q^{-88} -9 q^{-90} +6 q^{-92} +14 q^{-94} +14 q^{-96} +5 q^{-98} -4 q^{-100} -12 q^{-102} -9 q^{-104} +6 q^{-108} +8 q^{-110} +5 q^{-112} + q^{-114} -4 q^{-116} -5 q^{-118} -2 q^{-120} + q^{-122} +2 q^{-124} +2 q^{-126} + q^{-128} -2 q^{-130} -4 q^{-132} -2 q^{-134} + q^{-138} +2 q^{-140} +2 q^{-142} + q^{-144} - q^{-146} - q^{-148} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{12}-q^{10}-q^8+q^4+2 q^2+3+2 q^{-2} + q^{-4} - q^{-8} - q^{-10} - q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{36}+2 q^{32}-2 q^{30}+2 q^{28}-2 q^{26}+2 q^{24}-2 q^{22}-4 q^{20}-2 q^{18}-6 q^{16}-5 q^{12}+2 q^{10}+6 q^6+5 q^4+8 q^2+8+6 q^{-2} +5 q^{-4} -2 q^{-6} -2 q^{-8} -4 q^{-10} -5 q^{-12} -2 q^{-14} -2 q^{-16} +2 q^{-20} -2 q^{-26} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{34}+q^{32}+q^{30}-q^{24}-2 q^{22}-3 q^{20}-3 q^{18}-2 q^{16}-2 q^{14}-2 q^{12}-q^{10}+q^8+3 q^6+5 q^4+6 q^2+8+5 q^{-2} +4 q^{-4} - q^{-8} -3 q^{-10} -2 q^{-12} -3 q^{-14} -3 q^{-16} -2 q^{-18} - q^{-20} - q^{-24} + q^{-26} + q^{-28} + q^{-30} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{26}+q^{22}-q^{18}-2 q^{16}-3 q^{14}-4 q^{12}-5 q^{10}-2 q^8+5 q^4+7 q^2+10+8 q^{-2} +5 q^{-4} + q^{-6} -2 q^{-8} -4 q^{-10} -4 q^{-12} -3 q^{-14} -2 q^{-16} - q^{-18} + q^{-30} }[/math]
1,0,0	[math]\displaystyle{ -q^{15}-q^{13}-2 q^{11}-q^9-q^7+q^5+3 q^3+4 q+4 q^{-1} +3 q^{-3} + q^{-5} - q^{-7} - q^{-9} -2 q^{-11} - q^{-13} - q^{-15} }[/math]
1,0,1	[math]\displaystyle{ q^{44}+2 q^{40}+q^{36}+q^{32}+q^{30}+q^{26}-4 q^{24}-2 q^{22}-9 q^{20}-7 q^{18}-14 q^{16}-11 q^{14}-11 q^{12}-5 q^{10}+5 q^8+11 q^6+24 q^4+22 q^2+30+18 q^{-2} +16 q^{-4} +3 q^{-6} -6 q^{-8} -11 q^{-10} -17 q^{-12} -13 q^{-14} -15 q^{-16} -5 q^{-18} -4 q^{-20} +4 q^{-22} +3 q^{-24} +5 q^{-26} +3 q^{-28} - q^{-30} -2 q^{-34} + q^{-48} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{32}+q^{30}+2 q^{28}+2 q^{26}+2 q^{24}-3 q^{20}-6 q^{18}-9 q^{16}-11 q^{14}-12 q^{12}-8 q^{10}-3 q^8+5 q^6+11 q^4+18 q^2+21+19 q^{-2} +14 q^{-4} +8 q^{-6} -6 q^{-10} -9 q^{-12} -10 q^{-14} -10 q^{-16} -7 q^{-18} -4 q^{-20} -2 q^{-22} - q^{-24} + q^{-26} +2 q^{-28} + q^{-30} + q^{-32} + q^{-34} + q^{-36} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{18}-q^{16}-2 q^{14}-2 q^{12}-2 q^{10}-q^8+q^6+3 q^4+5 q^2+5+5 q^{-2} +3 q^{-4} + q^{-6} - q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} - q^{-16} - q^{-18} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{26}-q^{22}-q^{18}-q^{14}+q^{10}+2 q^6+q^4+3 q^2+2 q^{-2} + q^{-4} + q^{-6} - q^{-14} - q^{-18} - q^{-30} }[/math]
1,0	[math]\displaystyle{ q^{44}+q^{36}-q^{32}-q^{30}-q^{26}-2 q^{24}-2 q^{22}-q^{20}-q^{18}-2 q^{16}-q^{14}-q^{12}+q^{10}+q^8+3 q^6+2 q^4+4 q^2+4+4 q^{-2} +2 q^{-4} +3 q^{-6} +2 q^{-8} + q^{-10} - q^{-12} - q^{-14} - q^{-16} - q^{-18} -2 q^{-20} -2 q^{-22} - q^{-24} - q^{-26} - q^{-28} - q^{-30} + q^{-48} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{34}+q^{30}+q^{26}-q^{24}-q^{22}-3 q^{20}-4 q^{18}-5 q^{16}-6 q^{14}-4 q^{12}-3 q^{10}+q^8+3 q^6+8 q^4+9 q^2+12+9 q^{-2} +8 q^{-4} +4 q^{-6} + q^{-8} -2 q^{-10} -4 q^{-12} -5 q^{-14} -5 q^{-16} -3 q^{-18} -3 q^{-20} - q^{-22} - q^{-24} + q^{-42} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{60}+q^{56}-q^{54}-q^{48}-2 q^{44}-q^{40}-2 q^{38}-q^{36}-q^{34}-2 q^{32}-2 q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^6+2 q^4+3 q^2+1+2 q^{-2} +2 q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} - q^{-24} + q^{-26} - q^{-28} - q^{-30} - q^{-34} - q^{-36} - q^{-38} -2 q^{-40} - q^{-44} - q^{-46} - q^{-50} - q^{-56} + q^{-72} }[/math]

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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["10 125"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

{ 11, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+a^3 z^7+2 a z^7+z^7 a^{-1} -6 a^2 z^6-6 z^6-6 a^3 z^5-11 a z^5-5 z^5 a^{-1} +11 a^2 z^4+2 z^4 a^{-2} +13 z^4+10 a^3 z^3+17 a z^3+8 z^3 a^{-1} +z^3 a^{-3} -8 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -15 z^2-4 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7 }[/math]