K11a72
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 X10,4,11,3 X12,5,13,6 X14,8,15,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X20,15,21,16 X8,18,9,17 X6,19,7,20 X16,21,17,22 |
| Gauss code | 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -6 |
| Dowker-Thistlethwaite code | 4 10 12 14 2 22 18 20 8 6 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 t^4-6 t^3+18 t^2-32 t+39-32 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} } |
| 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 z^8+2 z^6+2 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 | { 153, 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^6-5 q^5+10 q^4-16 q^3+22 q^2-24 q+25-21 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} } |
| 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 z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2} |
| 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 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+14 z^9 a^{-1} +7 z^9 a^{-3} +10 a^2 z^8+19 z^8 a^{-2} +9 z^8 a^{-4} +20 z^8+8 a^3 z^7+a z^7-16 z^7 a^{-1} -4 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-15 a^2 z^6-55 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -53 z^6+a^5 z^5-12 a^3 z^5-20 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+11 a^2 z^4+44 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +47 z^4-a^5 z^3+7 a^3 z^3+20 a z^3+22 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-5 a^2 z^2-12 z^2 a^{-2} -2 z^2 a^{-4} -16 z^2-2 a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2} |
| 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 -q^{14}+2 q^{12}-3 q^{10}+2 q^8+q^6-4 q^4+5 q^2-4+4 q^{-2} + q^{-4} +5 q^{-8} -4 q^{-10} + q^{-12} - q^{-14} -2 q^{-16} + q^{-18} } |
| The G2 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 q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+16 q^{72}-16 q^{70}+7 q^{68}+16 q^{66}-45 q^{64}+83 q^{62}-114 q^{60}+115 q^{58}-80 q^{56}-9 q^{54}+139 q^{52}-278 q^{50}+389 q^{48}-410 q^{46}+295 q^{44}-41 q^{42}-306 q^{40}+643 q^{38}-839 q^{36}+789 q^{34}-473 q^{32}-53 q^{30}+605 q^{28}-976 q^{26}+1019 q^{24}-679 q^{22}+96 q^{20}+491 q^{18}-837 q^{16}+777 q^{14}-340 q^{12}-279 q^{10}+797 q^8-952 q^6+644 q^4+28 q^2-796+1345 q^{-2} -1425 q^{-4} +976 q^{-6} -156 q^{-8} -748 q^{-10} +1412 q^{-12} -1590 q^{-14} +1240 q^{-16} -491 q^{-18} -358 q^{-20} +997 q^{-22} -1192 q^{-24} +906 q^{-26} -281 q^{-28} -387 q^{-30} +813 q^{-32} -819 q^{-34} +412 q^{-36} +228 q^{-38} -798 q^{-40} +1054 q^{-42} -878 q^{-44} +334 q^{-46} +339 q^{-48} -892 q^{-50} +1112 q^{-52} -949 q^{-54} +500 q^{-56} +44 q^{-58} -492 q^{-60} +703 q^{-62} -662 q^{-64} +443 q^{-66} -155 q^{-68} -92 q^{-70} +226 q^{-72} -253 q^{-74} +196 q^{-76} -106 q^{-78} +32 q^{-80} +21 q^{-82} -39 q^{-84} +35 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94} } |
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["K11a72"];
|
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 t^4-6 t^3+18 t^2-32 t+39-32 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} } |
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 z^8+2 z^6+2 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]=
|
{ 153, 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^6-5 q^5+10 q^4-16 q^3+22 q^2-24 q+25-21 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} } |
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 z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2} |
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 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+14 z^9 a^{-1} +7 z^9 a^{-3} +10 a^2 z^8+19 z^8 a^{-2} +9 z^8 a^{-4} +20 z^8+8 a^3 z^7+a z^7-16 z^7 a^{-1} -4 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-15 a^2 z^6-55 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -53 z^6+a^5 z^5-12 a^3 z^5-20 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+11 a^2 z^4+44 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +47 z^4-a^5 z^3+7 a^3 z^3+20 a z^3+22 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-5 a^2 z^2-12 z^2 a^{-2} -2 z^2 a^{-4} -16 z^2-2 a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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["K11a72"];
|
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 t^4-6 t^3+18 t^2-32 t+39-32 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-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 q^6-5 q^5+10 q^4-16 q^3+22 q^2-24 q+25-21 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} } } |
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]=
|
{} |
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, 1) |
| 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 K11a72. 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. |
|
