K11a39
From Knot Atlas
Jump to navigationJump to search
|
|
|
|
Visit K11a39's page at Knotilus!
Visit K11a39's page at the original Knot Atlas! |
| K11a39 Quick Notes |
K11a39 Further Notes and Views
Knot presentations
| Planar diagram presentation | X4251 X8493 X14,6,15,5 X2837 X18,9,19,10 X20,11,21,12 X6,14,7,13 X22,16,1,15 X12,17,13,18 X10,19,11,20 X16,22,17,21 |
| Gauss code | 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -9, 7, -3, 8, -11, 9, -5, 10, -6, 11, -8 |
| Dowker-Thistlethwaite code | 4 8 14 2 18 20 6 22 12 10 16 |
| Conway Notation | [312,21,2] |
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3 t^3+12 t^2-22 t+27-22 t^{-1} +12 t^{-2} -3 t^{-3} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3 z^6-6 z^4-z^2+1} |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 101, 0 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+3 q^6-6 q^5+10 q^4-13 q^3+16 q^2-16 q+14-11 q^{-1} +7 q^{-2} -3 q^{-3} + q^{-4} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^6 a^{-2} -z^6+a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} -2 z^4+2 a^2 z^2-11 z^2 a^{-2} +9 z^2 a^{-4} -z^2 a^{-6} +a^2-5 a^{-2} +6 a^{-4} -2 a^{-6} +1} |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +7 z^9 a^{-3} +3 z^9 a^{-5} +11 z^8 a^{-2} +7 z^8 a^{-4} +3 z^8 a^{-6} +7 z^8+8 a z^7+2 z^7 a^{-1} -14 z^7 a^{-3} -7 z^7 a^{-5} +z^7 a^{-7} +6 a^2 z^6-36 z^6 a^{-2} -34 z^6 a^{-4} -12 z^6 a^{-6} -8 z^6+3 a^3 z^5-12 a z^5-18 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-7 a^2 z^4+37 z^4 a^{-2} +43 z^4 a^{-4} +15 z^4 a^{-6} +z^4-2 a^3 z^3+10 a z^3+15 z^3 a^{-1} +10 z^3 a^{-3} +12 z^3 a^{-5} +5 z^3 a^{-7} -a^4 z^2+5 a^2 z^2-22 z^2 a^{-2} -24 z^2 a^{-4} -7 z^2 a^{-6} +z^2-3 a z-5 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} -a^2+5 a^{-2} +6 a^{-4} +2 a^{-6} +1} |
| The A2 invariant | Data:K11a39/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a39/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
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["K11a39"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3 t^3+12 t^2-22 t+27-22 t^{-1} +12 t^{-2} -3 t^{-3} } |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3 z^6-6 z^4-z^2+1} |
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]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 101, 0 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+3 q^6-6 q^5+10 q^4-13 q^3+16 q^2-16 q+14-11 q^{-1} +7 q^{-2} -3 q^{-3} + q^{-4} } |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^6 a^{-2} -z^6+a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} -2 z^4+2 a^2 z^2-11 z^2 a^{-2} +9 z^2 a^{-4} -z^2 a^{-6} +a^2-5 a^{-2} +6 a^{-4} -2 a^{-6} +1} |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +7 z^9 a^{-3} +3 z^9 a^{-5} +11 z^8 a^{-2} +7 z^8 a^{-4} +3 z^8 a^{-6} +7 z^8+8 a z^7+2 z^7 a^{-1} -14 z^7 a^{-3} -7 z^7 a^{-5} +z^7 a^{-7} +6 a^2 z^6-36 z^6 a^{-2} -34 z^6 a^{-4} -12 z^6 a^{-6} -8 z^6+3 a^3 z^5-12 a z^5-18 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-7 a^2 z^4+37 z^4 a^{-2} +43 z^4 a^{-4} +15 z^4 a^{-6} +z^4-2 a^3 z^3+10 a z^3+15 z^3 a^{-1} +10 z^3 a^{-3} +12 z^3 a^{-5} +5 z^3 a^{-7} -a^4 z^2+5 a^2 z^2-22 z^2 a^{-2} -24 z^2 a^{-4} -7 z^2 a^{-6} +z^2-3 a z-5 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} -a^2+5 a^{-2} +6 a^{-4} +2 a^{-6} +1} |
Vassiliev invariants
| V2 and V3: | (-1, 2) |
| V2,1 through V6,9: |
|
V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
| The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of K11a39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 17, 2005, 14:44:34)... | |
In[2]:= | Crossings[Knot[11, Alternating, 39]] |
Out[2]= | 11 |
In[3]:= | PD[Knot[11, Alternating, 39]] |
Out[3]= | PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7],X[18, 9, 19, 10], X[20, 11, 21, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], X[12, 17, 13, 18], X[10, 19, 11, 20],X[16, 22, 17, 21]] |
In[4]:= | GaussCode[Knot[11, Alternating, 39]] |
Out[4]= | GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -9, 7, -3, 8, -11, 9, -5, 10, -6, 11, -8] |
In[5]:= | BR[Knot[11, Alternating, 39]] |
Out[5]= | BR[Knot[11, Alternating, 39]] |
In[6]:= | alex = Alexander[Knot[11, Alternating, 39]][t] |
Out[6]= | 3 12 22 2 3 |
In[7]:= | Conway[Knot[11, Alternating, 39]][z] |
Out[7]= | 2 4 6 1 - z - 6 z - 3 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[11, Alternating, 39]} |
In[9]:= | {KnotDet[Knot[11, Alternating, 39]], KnotSignature[Knot[11, Alternating, 39]]} |
Out[9]= | {101, 0} |
In[10]:= | J=Jones[Knot[11, Alternating, 39]][q] |
Out[10]= | -4 3 7 11 2 3 4 5 6 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[11, Alternating, 39]} |
In[12]:= | A2Invariant[Knot[11, Alternating, 39]][q] |
Out[12]= | -12 -10 -8 2 2 3 2 6 8 10 |
In[13]:= | Kauffman[Knot[11, Alternating, 39]][a, z] |
Out[13]= | 22 6 5 2 2 z 4 z 4 z 5 z 2 7 z |
In[14]:= | {Vassiliev[2][Knot[11, Alternating, 39]], Vassiliev[3][Knot[11, Alternating, 39]]} |
Out[14]= | {0, 2} |
In[15]:= | Kh[Knot[11, Alternating, 39]][q, t] |
Out[15]= | 8 1 2 1 5 2 6 5 |
