K11n86

From Knot Atlas
Jump to navigationJump to search

K11n85.gif

K11n85

K11n87.gif

K11n87

K11n86.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n86 at Knotilus!

[edit K11n86 Quick Notes]


[edit K11n86 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X7,15,8,14 X9,16,10,17 X2,11,3,12 X13,19,14,18 X15,20,16,21 X17,1,18,22 X19,6,20,7 X21,9,22,8
Gauss code 1, -6, 2, -1, 3, 10, -4, 11, -5, -2, 6, -3, -7, 4, -8, 5, -9, 7, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 10 12 -14 -16 2 -18 -20 -22 -6 -8
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n86 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant 0

[edit Notes for K11n86's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+4 t^2-7 t+9-7 t^{-1} +4 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 33, 0 }
Jones polynomial [math]\displaystyle{ q^6-3 q^5+4 q^4-5 q^3+6 q^2-5 q+5-3 q^{-1} + q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +z^4-4 z^2 a^{-2} +2 z^2 a^{-4} +2 z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +3 z^8 a^{-4} +z^8-3 z^7 a^{-1} +3 z^7 a^{-5} -17 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -5 z^6-11 z^5 a^{-3} -11 z^5 a^{-5} +22 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} +10 z^4+2 a z^3+7 z^3 a^{-1} +13 z^3 a^{-3} +8 z^3 a^{-5} -9 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -5 z^2-a z-3 z a^{-1} -3 z a^{-3} -z a^{-5} +1 }[/math]
The A2 invariant [math]\displaystyle{ q^6-q^4+q^2+2 q^{-4} +2 q^{-8} - q^{-10} - q^{-12} - q^{-16} + q^{-18} }[/math]
The G2 invariant Data:K11n86/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n86 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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["K11n86"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+4 t^2-7 t+9-7 t^{-1} +4 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-2 z^4+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]= { 33, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-3 q^5+4 q^4-5 q^3+6 q^2-5 q+5-3 q^{-1} + q^{-2} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +z^4-4 z^2 a^{-2} +2 z^2 a^{-4} +2 z^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +3 z^8 a^{-4} +z^8-3 z^7 a^{-1} +3 z^7 a^{-5} -17 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -5 z^6-11 z^5 a^{-3} -11 z^5 a^{-5} +22 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} +10 z^4+2 a z^3+7 z^3 a^{-1} +13 z^3 a^{-3} +8 z^3 a^{-5} -9 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -5 z^2-a z-3 z a^{-1} -3 z a^{-3} -z a^{-5} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["K11n86"];
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+4 t^2-7 t+9-7 t^{-1} +4 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^6-3 q^5+4 q^4-5 q^3+6 q^2-5 q+5-3 q^{-1} + q^{-2} }[/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]= {}
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: (0, 0)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ \frac{176}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ \frac{104}{3} }[/math] [math]\displaystyle{ -8 }[/math]

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]0 is the signature of K11n86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-10123456χ
13        11
11       2 -2
9      21 1
7     32  -1
5    32   1
3   23    1
1  33     0
-1 13      2
-3 2       -2
-51        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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.

K11n85.gif

K11n85

K11n87.gif

K11n87

Retrieved from "https://katlas.org/index.php?title=K11n86&oldid=110864"