K11n103
From Knot Atlas
Jump to navigationJump to search
|
|
|
 (Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X14,5,15,6 X12,8,13,7 X9,20,10,21 X2,11,3,12 X16,13,17,14 X6,15,7,16 X17,22,18,1 X19,8,20,9 X21,18,22,19 |
| Gauss code | 1, -6, 2, -1, 3, -8, 4, 10, -5, -2, 6, -4, 7, -3, 8, -7, -9, 11, -10, 5, -11, 9 |
| Dowker-Thistlethwaite code | 4 10 14 12 -20 2 16 6 -22 -8 -18 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -t^3+7 t^2-15 t+19-15 t^{-1} +7 t^{-2} - t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ -z^6+z^4+4 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 65, -4 } |
| Jones polynomial | [math]\displaystyle{ 1-3 q^{-1} +6 q^{-2} -8 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -2 q^{-9} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -2 z^2 a^8-2 a^8+3 z^4 a^6+7 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-3 z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ 3 z^3 a^{11}-2 z a^{11}+z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}+3 z^7 a^9-2 z^5 a^9+3 z^3 a^9-z a^9+3 z^8 a^8-2 z^6 a^8+3 z^2 a^8-2 a^8+z^9 a^7+6 z^7 a^7-14 z^5 a^7+7 z^3 a^7-z a^7+6 z^8 a^6-9 z^6 a^6-5 z^4 a^6+9 z^2 a^6-3 a^6+z^9 a^5+6 z^7 a^5-21 z^5 a^5+14 z^3 a^5-3 z a^5+3 z^8 a^4-5 z^6 a^4-4 z^4 a^4+6 z^2 a^4-a^4+3 z^7 a^3-9 z^5 a^3+7 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2 }[/math] |
| The A2 invariant | Data:K11n103/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n103/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["K11n103"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ -t^3+7 t^2-15 t+19-15 t^{-1} +7 t^{-2} - t^{-3} }[/math] |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
[math]\displaystyle{ -z^6+z^4+4 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]=
|
{ 65, -4 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
[math]\displaystyle{ 1-3 q^{-1} +6 q^{-2} -8 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -2 q^{-9} }[/math] |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ -2 z^2 a^8-2 a^8+3 z^4 a^6+7 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-3 z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2 }[/math] |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ 3 z^3 a^{11}-2 z a^{11}+z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}+3 z^7 a^9-2 z^5 a^9+3 z^3 a^9-z a^9+3 z^8 a^8-2 z^6 a^8+3 z^2 a^8-2 a^8+z^9 a^7+6 z^7 a^7-14 z^5 a^7+7 z^3 a^7-z a^7+6 z^8 a^6-9 z^6 a^6-5 z^4 a^6+9 z^2 a^6-3 a^6+z^9 a^5+6 z^7 a^5-21 z^5 a^5+14 z^3 a^5-3 z a^5+3 z^8 a^4-5 z^6 a^4-4 z^4 a^4+6 z^2 a^4-a^4+3 z^7 a^3-9 z^5 a^3+7 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2 }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n10, K11n144,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n175,}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(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 May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["K11n103"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ [math]\displaystyle{ -t^3+7 t^2-15 t+19-15 t^{-1} +7 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ 1-3 q^{-1} +6 q^{-2} -8 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -2 q^{-9} }[/math] } |
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
|
{K11n10, K11n144,} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{K11n175,} |
Vassiliev invariants
| V2 and V3: | (4, -9) |
| 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-4 is the signature of K11n103. 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.
Modifying This Page
| Read me first: Modifying Knot Pages.
See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top. |
|
