4 1
From Knot Atlas
Jump to navigationJump to search
Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -t- t^{-1} +3 }[/math] |
| Conway polynomial | [math]\displaystyle{ 1-z^2 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 5, 0 } |
| Jones polynomial | [math]\displaystyle{ q^2+ q^{-2} -q- q^{-1} +1 }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ a^2+ a^{-2} -z^2-1 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ a^2 z^2+z^2 a^{-2} -a^2- a^{-2} +a z^3+z^3 a^{-1} -a z-z a^{-1} +2 z^2-1 }[/math] |
| The A2 invariant | [math]\displaystyle{ q^8+q^6-1+ q^{-6} + q^{-8} }[/math] |
| The G2 invariant | [math]\displaystyle{ q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^4-1- q^{-4} - q^{-10} + q^{-16} + q^{-18} + q^{-24} + q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-38} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 4_1.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | [math]\displaystyle{ q^5+ q^{-5} }[/math] |
| 2 | [math]\displaystyle{ q^{14}-q^{10}+q^2+1+ q^{-2} - q^{-10} + q^{-14} }[/math] |
| 3 | [math]\displaystyle{ q^{27}-q^{23}-q^{21}+q^{17}+q^{11}+q^9+ q^{-9} + q^{-11} + q^{-17} - q^{-21} - q^{-23} + q^{-27} }[/math] |
| 4 | [math]\displaystyle{ q^{44}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}+q^{24}+q^{22}-q^{18}-q^{16}+q^4+q^2+1+ q^{-2} + q^{-4} - q^{-16} - q^{-18} + q^{-22} + q^{-24} - q^{-26} + q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} - q^{-40} + q^{-44} }[/math] |
| 5 | [math]\displaystyle{ q^{65}-q^{61}-q^{59}-q^{57}+q^{53}+2 q^{51}+q^{49}-q^{45}-q^{43}+q^{39}+q^{37}-q^{35}-2 q^{33}-q^{31}+q^{27}+q^{25}+q^{17}+q^{15}+q^{13}+ q^{-13} + q^{-15} + q^{-17} + q^{-25} + q^{-27} - q^{-31} -2 q^{-33} - q^{-35} + q^{-37} + q^{-39} - q^{-43} - q^{-45} + q^{-49} +2 q^{-51} + q^{-53} - q^{-57} - q^{-59} - q^{-61} + q^{-65} }[/math] |
| 6 | [math]\displaystyle{ q^{90}-q^{86}-q^{84}-q^{82}+2 q^{76}+2 q^{74}+q^{72}-q^{68}-2 q^{66}-2 q^{64}+q^{62}+q^{60}+q^{58}-q^{54}-2 q^{52}-2 q^{50}+q^{48}+2 q^{46}+2 q^{44}+q^{42}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}-q^{24}-q^{22}+q^6+q^4+q^2+1+ q^{-2} + q^{-4} + q^{-6} - q^{-22} - q^{-24} - q^{-26} + q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} + q^{-42} +2 q^{-44} +2 q^{-46} + q^{-48} -2 q^{-50} -2 q^{-52} - q^{-54} + q^{-58} + q^{-60} + q^{-62} -2 q^{-64} -2 q^{-66} - q^{-68} + q^{-72} +2 q^{-74} +2 q^{-76} - q^{-82} - q^{-84} - q^{-86} + q^{-90} }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^8+q^6-1+ q^{-6} + q^{-8} }[/math] |
| 1,1 | [math]\displaystyle{ q^{20}+2 q^{16}-2 q^{10}-2 q^8+4 q^2+2+4 q^{-2} -2 q^{-8} -2 q^{-10} +2 q^{-16} + q^{-20} }[/math] |
| 2,0 | [math]\displaystyle{ q^{20}+q^{18}+q^{16}-q^{14}-q^{12}-q^{10}-q^8+q^4+2 q^2+2+2 q^{-2} + q^{-4} - q^{-8} - q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-18} + q^{-20} }[/math] |
| 3,0 | [math]\displaystyle{ q^{36}+q^{34}+q^{32}-2 q^{28}-2 q^{26}-2 q^{24}+q^{18}+2 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^8-q^4-2 q^2-2-2 q^{-2} - q^{-4} + q^{-8} +2 q^{-10} +3 q^{-12} +3 q^{-14} +2 q^{-16} + q^{-18} -2 q^{-24} -2 q^{-26} -2 q^{-28} + q^{-32} + q^{-34} + q^{-36} }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ q^{16}+q^{12}+q^{10}+ q^{-10} + q^{-12} + q^{-16} }[/math] |
| 1,0,0 | [math]\displaystyle{ q^{11}+q^9+q^7-q- q^{-1} + q^{-7} + q^{-9} + q^{-11} }[/math] |
| 1,0,1 | [math]\displaystyle{ q^{26}+2 q^{22}+2 q^{20}+q^{18}+2 q^{16}-2 q^{14}-2 q^{12}-4 q^{10}-4 q^8+2 q^4+6 q^2+7+6 q^{-2} +2 q^{-4} -4 q^{-8} -4 q^{-10} -2 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} +2 q^{-20} +2 q^{-22} + q^{-26} }[/math] |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{10}-q^8+q^2+2+ q^{-2} - q^{-8} - q^{-10} + q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^{14}+q^{12}+q^{10}+q^8-q^2-1- q^{-2} + q^{-8} + q^{-10} + q^{-12} + q^{-14} }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ q^{16}+q^{12}+q^{10}-2+ q^{-10} + q^{-12} + q^{-16} }[/math] |
| 1,0 | [math]\displaystyle{ q^{26}+q^{18}+1+ q^{-18} + q^{-26} }[/math] |
B3 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0 | [math]\displaystyle{ q^{38}+q^{30}+q^{26}+q^{22}-q^2+1- q^{-2} + q^{-22} + q^{-26} + q^{-30} + q^{-38} }[/math] |
B4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{50}+q^{42}+q^{38}+q^{34}+q^{30}+q^{26}-q^6-q^2+1- q^{-2} - q^{-6} + q^{-26} + q^{-30} + q^{-34} + q^{-38} + q^{-42} + q^{-50} }[/math] |
C3 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0 | [math]\displaystyle{ q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^2-2- q^{-2} + q^{-12} + q^{-14} + q^{-16} + q^{-18} + q^{-22} }[/math] |
C4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{28}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^4-q^2-2- q^{-2} - q^{-4} + q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} + q^{-24} + q^{-28} }[/math] |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{38}+q^{34}+3 q^{32}+2 q^{30}+q^{28}+4 q^{26}+q^{24}-2 q^{18}-5 q^{16}-3 q^{14}-3 q^{12}-4 q^{10}+3 q^6+5 q^4+6 q^2+8+6 q^{-2} +5 q^{-4} +3 q^{-6} -4 q^{-10} -3 q^{-12} -3 q^{-14} -5 q^{-16} -2 q^{-18} + q^{-24} +4 q^{-26} + q^{-28} +2 q^{-30} +3 q^{-32} + q^{-34} + q^{-38} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^2- q^{-2} + q^{-12} + q^{-14} + q^{-16} + q^{-18} + q^{-22} }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^4-1- q^{-4} - q^{-10} + q^{-16} + q^{-18} + q^{-24} + q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-38} }[/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.
|
In[3]:=
|
K = Knot["4 1"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ -t- t^{-1} +3 }[/math] |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
[math]\displaystyle{ 1-z^2 }[/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^{-2} -q- q^{-1} +1 }[/math] |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ a^2+ a^{-2} -z^2-1 }[/math] |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ a^2 z^2+z^2 a^{-2} -a^2- a^{-2} +a z^3+z^3 a^{-1} -a z-z a^{-1} +2 z^2-1 }[/math] |
