K11n184
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 | X6271 X3,15,4,14 X10,6,11,5 X18,7,19,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X15,5,16,4 X12,18,13,17 X8,19,9,20 X16,22,17,21 |
| Gauss code | 1, -5, -2, 8, 3, -1, 4, -10, 5, -3, 6, -9, 7, 2, -8, -11, 9, -4, 10, -7, 11, -6 |
| Dowker-Thistlethwaite code | 6 -14 10 18 2 22 20 -4 12 8 16 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
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 2 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 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 2 z^6+3 z^4+2 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 | { 87, 2 } |
| 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^9-4 q^8+8 q^7-12 q^6+14 q^5-15 q^4+14 q^3-10 q^2+7 q-2} |
| 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^{-4} -2 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} -3 z^2 a^{-2} +11 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} +4 a^{-4} -4 a^{-6} + a^{-8} } |
| 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 3 z^9 a^{-5} +3 z^9 a^{-7} +8 z^8 a^{-4} +14 z^8 a^{-6} +6 z^8 a^{-8} +6 z^7 a^{-3} +6 z^7 a^{-5} +4 z^7 a^{-7} +4 z^7 a^{-9} +z^6 a^{-2} -20 z^6 a^{-4} -36 z^6 a^{-6} -14 z^6 a^{-8} +z^6 a^{-10} -9 z^5 a^{-3} -27 z^5 a^{-5} -28 z^5 a^{-7} -10 z^5 a^{-9} +8 z^4 a^{-2} +29 z^4 a^{-4} +29 z^4 a^{-6} +6 z^4 a^{-8} -2 z^4 a^{-10} +3 z^3 a^{-1} +11 z^3 a^{-3} +25 z^3 a^{-5} +24 z^3 a^{-7} +7 z^3 a^{-9} -7 z^2 a^{-2} -19 z^2 a^{-4} -14 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -3 z a^{-3} -7 z a^{-5} -7 z a^{-7} -2 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8} } |
| The A2 invariant | 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+3 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +3 q^{-12} -3 q^{-14} + q^{-16} -3 q^{-20} +2 q^{-22} -2 q^{-24} + q^{-28} } |
| The G2 invariant | Data:K11n184/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["K11n184"];
|
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 2 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 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 2 z^6+3 z^4+2 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]=
|
{ 87, 2 } |
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^9-4 q^8+8 q^7-12 q^6+14 q^5-15 q^4+14 q^3-10 q^2+7 q-2} |
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^{-4} -2 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} -3 z^2 a^{-2} +11 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} +4 a^{-4} -4 a^{-6} + a^{-8} } |
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 3 z^9 a^{-5} +3 z^9 a^{-7} +8 z^8 a^{-4} +14 z^8 a^{-6} +6 z^8 a^{-8} +6 z^7 a^{-3} +6 z^7 a^{-5} +4 z^7 a^{-7} +4 z^7 a^{-9} +z^6 a^{-2} -20 z^6 a^{-4} -36 z^6 a^{-6} -14 z^6 a^{-8} +z^6 a^{-10} -9 z^5 a^{-3} -27 z^5 a^{-5} -28 z^5 a^{-7} -10 z^5 a^{-9} +8 z^4 a^{-2} +29 z^4 a^{-4} +29 z^4 a^{-6} +6 z^4 a^{-8} -2 z^4 a^{-10} +3 z^3 a^{-1} +11 z^3 a^{-3} +25 z^3 a^{-5} +24 z^3 a^{-7} +7 z^3 a^{-9} -7 z^2 a^{-2} -19 z^2 a^{-4} -14 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -3 z a^{-3} -7 z a^{-5} -7 z a^{-7} -2 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_84, K11a46,}
Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {}
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["K11n184"];
|
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]=
|
{ 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 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 t^{-3} } , 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^9-4 q^8+8 q^7-12 q^6+14 q^5-15 q^4+14 q^3-10 q^2+7 q-2} } |
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]=
|
{10_84, K11a46,} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{} |
Vassiliev invariants
| V2 and V3: | (2, 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=} 2 is the signature of K11n184. 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. |
|
