10 140

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2-2 t+3-2 t^{-1} + t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature	{ 9, 0 }
Jones polynomial	[math]\displaystyle{ 1- q^{-1} + q^{-2} - q^{-3} +2 q^{-4} - q^{-5} + q^{-6} - q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^6-2 a^6+z^4 a^4+4 z^2 a^4+4 a^4-z^2 a^2-2 a^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^6 z^8+a^4 z^8+a^7 z^7+2 a^5 z^7+a^3 z^7-6 a^6 z^6-6 a^4 z^6-6 a^7 z^5-11 a^5 z^5-5 a^3 z^5+11 a^6 z^4+12 a^4 z^4+a^2 z^4+10 a^7 z^3+16 a^5 z^3+6 a^3 z^3-8 a^6 z^2-12 a^4 z^2-4 a^2 z^2-4 a^7 z-6 a^5 z-2 a^3 z+2 a^6+4 a^4+2 a^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{22}-q^{20}-q^{18}+2 q^{14}+2 q^{12}+2 q^{10}-q^6-q^4+1+ q^{-2} }[/math]
The G2 invariant	[math]\displaystyle{ q^{108}+q^{104}-q^{102}-q^{96}+q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{82}-4 q^{80}-4 q^{70}+q^{68}+3 q^{66}-q^{62}-q^{60}+3 q^{58}+5 q^{56}+2 q^{54}-q^{52}+q^{50}+3 q^{48}+4 q^{46}-2 q^{42}+q^{40}+3 q^{38}+q^{36}-q^{34}-2 q^{30}+q^{28}-3 q^{24}-q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-q^8+q^6+2+ q^{-4} + q^{-6} + q^{-10} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_140.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{15}+q^9+q^7+ q^{-1} }[/math]
2	[math]\displaystyle{ q^{44}-q^{40}-q^{34}-q^{32}+q^{28}+q^{26}+q^{22}-q^{18}-q^{14}-q^{12}+q^{10}+q^8+q^6+q^2+2- q^{-4} }[/math]
3	[math]\displaystyle{ -q^{87}+q^{83}+q^{81}-q^{77}+q^{73}+q^{71}-q^{67}-2 q^{65}-q^{63}+q^{59}-q^{55}+2 q^{51}+2 q^{49}-q^{45}+q^{41}-2 q^{37}-2 q^{35}-q^{29}+q^{25}+q^{23}+q^{17}+q^{15}-q^{11}-q^9+2 q^7+3 q^5-2 q+3 q^{-3} + q^{-5} - q^{-7} - q^{-9} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{22}-q^{20}-q^{18}+2 q^{14}+2 q^{12}+2 q^{10}-q^6-q^4+1+ q^{-2} }[/math]
1,1	[math]\displaystyle{ q^{60}+2 q^{56}-2 q^{54}+2 q^{52}-2 q^{50}-2 q^{46}-4 q^{44}-4 q^{40}+2 q^{38}+q^{36}+4 q^{34}+4 q^{32}+4 q^{30}+q^{28}-2 q^{26}-4 q^{24}-4 q^{22}-4 q^{20}-2 q^{18}+6 q^{14}+2 q^{12}+6 q^{10}+2 q^8+2 q^4-2 q^2+2-2 q^{-2} + q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{58}+q^{56}+q^{54}-q^{48}-2 q^{46}-4 q^{44}-4 q^{42}-3 q^{40}+4 q^{36}+4 q^{34}+6 q^{32}+4 q^{30}+3 q^{28}-2 q^{26}-4 q^{24}-5 q^{22}-5 q^{20}-3 q^{18}+4 q^{14}+4 q^{12}+5 q^{10}+2 q^8+1- q^{-2} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{46}+q^{42}-q^{38}-2 q^{36}-2 q^{34}-2 q^{32}-3 q^{30}+2 q^{24}+2 q^{22}+4 q^{20}+3 q^{18}+2 q^{16}+2 q^{14}-q^{10}-2 q^8-q^6-q^4+2+ q^{-2} + q^{-4} }[/math]
1,0,0	[math]\displaystyle{ -q^{29}-q^{27}-2 q^{25}-q^{23}+2 q^{19}+3 q^{17}+3 q^{15}+2 q^{13}-q^9-2 q^7-q^5+q+ q^{-1} + q^{-3} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{60}+q^{58}+2 q^{56}+2 q^{54}+q^{52}-q^{50}-3 q^{48}-6 q^{46}-7 q^{44}-6 q^{42}-4 q^{40}-q^{38}+2 q^{36}+6 q^{34}+6 q^{32}+5 q^{30}+4 q^{28}+3 q^{26}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-2 q^{10}-2 q^8-q^6-q^4+2+2 q^{-2} + q^{-4} + q^{-6} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+2 q^{24}+3 q^{22}+4 q^{20}+3 q^{18}+2 q^{16}-q^{12}-2 q^{10}-2 q^8-q^6+q^2+1+ q^{-2} + q^{-4} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{46}-q^{42}-q^{38}+q^{30}+2 q^{26}+2 q^{22}+q^{18}-q^{10}-q^6+q^4+ q^{-2} + q^{-4} }[/math]
1,0	[math]\displaystyle{ q^{76}+q^{68}-q^{64}-q^{62}-q^{58}-2 q^{56}-q^{54}-q^{48}+q^{42}+q^{38}+q^{34}+q^{32}+q^{30}+q^{26}+2 q^{24}+q^{22}+q^{16}-q^{12}-q^{10}-q^4+1+ q^{-2} + q^{-6} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{62}+q^{58}+q^{54}-q^{52}-q^{50}-2 q^{48}-3 q^{46}-3 q^{44}-4 q^{42}-2 q^{40}-2 q^{38}+q^{36}+q^{34}+4 q^{32}+4 q^{30}+6 q^{28}+4 q^{26}+4 q^{24}+2 q^{22}+q^{20}-2 q^{16}-2 q^{14}-3 q^{12}-q^{10}-2 q^8+q^2+2+ q^{-2} + q^{-4} + q^{-6} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{108}+q^{104}-q^{102}-q^{96}+q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{82}-4 q^{80}-4 q^{70}+q^{68}+3 q^{66}-q^{62}-q^{60}+3 q^{58}+5 q^{56}+2 q^{54}-q^{52}+q^{50}+3 q^{48}+4 q^{46}-2 q^{42}+q^{40}+3 q^{38}+q^{36}-q^{34}-2 q^{30}+q^{28}-3 q^{24}-q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-q^8+q^6+2+ q^{-4} + q^{-6} + q^{-10} }[/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["10 140"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^6-2 a^6+z^4 a^4+4 z^2 a^4+4 a^4-z^2 a^2-2 a^2+1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^6 z^8+a^4 z^8+a^7 z^7+2 a^5 z^7+a^3 z^7-6 a^6 z^6-6 a^4 z^6-6 a^7 z^5-11 a^5 z^5-5 a^3 z^5+11 a^6 z^4+12 a^4 z^4+a^2 z^4+10 a^7 z^3+16 a^5 z^3+6 a^3 z^3-8 a^6 z^2-12 a^4 z^2-4 a^2 z^2-4 a^7 z-6 a^5 z-2 a^3 z+2 a^6+4 a^4+2 a^2+1 }[/math]