K11n6

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K11n5.gif

K11n5

K11n7.gif

K11n7

K11n6.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n6 at Knotilus!

[edit K11n6 Quick Notes]


[edit K11n6 Further Notes and Views]
Knot K11n6.
A graph, knot K11n6.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,12,18,13 X19,22,20,1 X21,16,22,17
Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 9, 7, -4, -8, 11, -9, 6, -10, 8, -11, 10
Dowker-Thistlethwaite code 4 8 10 14 2 -18 6 -20 -12 -22 -16
A Braid Representative
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A Morse Link Presentation K11n6 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number [math]\displaystyle{ \{1,2\} }[/math]
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n6/ThurstonBennequinNumber
Hyperbolic Volume 11.407
A-Polynomial See Data:K11n6/A-polynomial

[edit Notes for K11n6's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n6's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 17, 0 }
Jones polynomial [math]\displaystyle{ q^3-q^2+1-2 q^{-1} +3 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^2 a^6-2 a^6+2 z^4 a^4+7 z^2 a^4+5 a^4-z^6 a^2-5 z^4 a^2-7 z^2 a^2-3 a^2+z^2 a^{-2} + a^{-2} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^5 z^9+a^3 z^9+2 a^6 z^8+4 a^4 z^8+2 a^2 z^8+a^7 z^7-2 a^5 z^7-3 a^3 z^7+a z^7+z^7 a^{-1} -10 a^6 z^6-21 a^4 z^6-12 a^2 z^6+z^6 a^{-2} -5 a^7 z^5-8 a^5 z^5-5 a^3 z^5-8 a z^5-6 z^5 a^{-1} +14 a^6 z^4+31 a^4 z^4+19 a^2 z^4-5 z^4 a^{-2} -3 z^4+7 a^7 z^3+15 a^5 z^3+13 a^3 z^3+13 a z^3+8 z^3 a^{-1} -8 a^6 z^2-19 a^4 z^2-12 a^2 z^2+5 z^2 a^{-2} +4 z^2-3 a^7 z-6 a^5 z-6 a^3 z-6 a z-3 z a^{-1} +2 a^6+5 a^4+3 a^2- a^{-2} }[/math]
The A2 invariant Data:K11n6/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n6/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n6 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["K11n6"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-3 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]= { 17, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-q^2+1-2 q^{-1} +3 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^2 a^6-2 a^6+2 z^4 a^4+7 z^2 a^4+5 a^4-z^6 a^2-5 z^4 a^2-7 z^2 a^2-3 a^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{ a^5 z^9+a^3 z^9+2 a^6 z^8+4 a^4 z^8+2 a^2 z^8+a^7 z^7-2 a^5 z^7-3 a^3 z^7+a z^7+z^7 a^{-1} -10 a^6 z^6-21 a^4 z^6-12 a^2 z^6+z^6 a^{-2} -5 a^7 z^5-8 a^5 z^5-5 a^3 z^5-8 a z^5-6 z^5 a^{-1} +14 a^6 z^4+31 a^4 z^4+19 a^2 z^4-5 z^4 a^{-2} -3 z^4+7 a^7 z^3+15 a^5 z^3+13 a^3 z^3+13 a z^3+8 z^3 a^{-1} -8 a^6 z^2-19 a^4 z^2-12 a^2 z^2+5 z^2 a^{-2} +4 z^2-3 a^7 z-6 a^5 z-6 a^3 z-6 a z-3 z a^{-1} +2 a^6+5 a^4+3 a^2- a^{-2} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_2,}

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["K11n6"];
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+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^3-q^2+1-2 q^{-1} +3 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} }[/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]= {8_2,}
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, -3)
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{ -24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -112 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 632 }[/math] [math]\displaystyle{ -\frac{40}{3} }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ 56 }[/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 K11n6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-101234χ
7           11
5            0
3        111 -1
1       21   1
-1      221   -1
-3     221    1
-5    22      0
-7   221      1
-9  12        1
-11 12         -1
-13 1          1
-151           -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/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.

K11n5.gif

K11n5

K11n7.gif

K11n7

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