10 132

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2-t+1- t^{-1} + t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 5, 0 }
Jones polynomial	[math]\displaystyle{ q^{-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+3 a^4 }[/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^5-12 a^5 z^5-6 a^3 z^5+10 a^6 z^4+10 a^4 z^4+10 a^7 z^3+19 a^5 z^3+9 a^3 z^3-6 a^6 z^2-7 a^4 z^2-a^2 z^2-5 a^7 z-8 a^5 z-4 a^3 z-a z+2 a^6+3 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2 q^{80}-q^{78}-q^{76}-2 q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2 q^{52}+3 q^{50}+q^{48}+q^{46}+2 q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^4-q^2+1+ q^{-6} - q^{-8} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_132.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{15}+q^7+q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{44}-q^{40}-q^{30}-q^{24}+q^{16}+q^{14}+q^{10}+q^6+2- q^{-4} }[/math]
3	[math]\displaystyle{ -q^{87}+q^{83}+q^{81}-q^{77}+q^{67}-q^{63}+q^{59}+q^{57}-q^{55}-q^{53}-q^{45}-q^{43}+q^{39}-q^{35}+q^{31}-q^{27}-q^{25}+q^{17}+2 q^{15}+2 q^{13}+q^7+2 q^5+q^3-q- q^{-1} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{46}+q^{42}+q^{40}-2 q^{36}-2 q^{34}-4 q^{32}-5 q^{30}-2 q^{28}-q^{26}+2 q^{24}+3 q^{22}+6 q^{20}+4 q^{18}+3 q^{16}+2 q^{14}+q^{12}-q^{10}-q^8-q^4+1 }[/math]
1,0,0	[math]\displaystyle{ -q^{29}-q^{27}-2 q^{25}-q^{23}+q^{19}+2 q^{17}+2 q^{15}+2 q^{13}+q^{11}+q^9 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{60}+q^{58}+2 q^{56}+2 q^{54}+2 q^{52}+q^{50}-q^{48}-4 q^{46}-6 q^{44}-8 q^{42}-9 q^{40}-8 q^{38}-5 q^{36}+4 q^{32}+8 q^{30}+11 q^{28}+11 q^{26}+8 q^{24}+6 q^{22}+q^{20}-q^{18}-3 q^{16}-2 q^{14}-2 q^{12}-q^{10}+q^6+1 }[/math]
1,0,0,0	[math]\displaystyle{ -q^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{46}-q^{42}-q^{40}+q^{30}+q^{26}+q^{22}+q^{16}+q^{12}+q^{10}+q^8+q^4-1 }[/math]
1,0	[math]\displaystyle{ q^{76}+q^{68}-q^{58}-2 q^{56}-q^{54}-q^{52}-2 q^{50}-2 q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{18}-q^{12}-q^4+1 }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{62}+q^{58}+q^{56}+q^{54}-q^{50}-2 q^{48}-4 q^{46}-4 q^{44}-5 q^{42}-4 q^{40}-3 q^{38}+q^{34}+4 q^{32}+5 q^{30}+6 q^{28}+5 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18}-q^{16}-q^{14}-q^{12}-q^8+1 }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2 q^{80}-q^{78}-q^{76}-2 q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2 q^{52}+3 q^{50}+q^{48}+q^{46}+2 q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^4-q^2+1+ q^{-6} - q^{-8} }[/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["10 132"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 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]=

{ 5, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-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+3 a^4 }[/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^5-12 a^5 z^5-6 a^3 z^5+10 a^6 z^4+10 a^4 z^4+10 a^7 z^3+19 a^5 z^3+9 a^3 z^3-6 a^6 z^2-7 a^4 z^2-a^2 z^2-5 a^7 z-8 a^5 z-4 a^3 z-a z+2 a^6+3 a^4 }[/math]