K11n32

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K11n31.gif

K11n31

K11n33.gif

K11n33

K11n32.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n32 at Knotilus!

[edit K11n32 Quick Notes]


[edit K11n32 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X2837 X16,9,17,10 X11,6,12,7 X20,14,21,13 X10,15,11,16 X22,17,1,18 X14,20,15,19 X18,21,19,22
Gauss code 1, -4, 2, -1, -3, 6, 4, -2, 5, -8, -6, 3, 7, -10, 8, -5, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 -12 2 16 -6 20 10 22 14 18
A Braid Representative {{{braid_table}}}
A Morse Link Presentation K11n32 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:K11n32/ThurstonBennequinNumber
Hyperbolic Volume 13.939
A-Polynomial See Data:K11n32/A-polynomial

[edit Notes for K11n32'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 K11n32's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 69, 0 }
Jones polynomial [math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-11 q+12-11 q^{-1} +8 q^{-2} -5 q^{-3} +2 q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6+a^2 z^4+2 z^4 a^{-2} -3 z^4+4 z^2 a^{-2} -z^2 a^{-4} -4 z^2+a^4-a^2+3 a^{-2} - a^{-4} -1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7+3 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} -a^2 z^6+z^6 a^{-2} +3 z^6 a^{-4} -3 z^6+3 a^3 z^5-9 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +3 a^4 z^4+4 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -3 z^4-6 a^3 z^3-8 a z^3-z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} -4 a^4 z^2-6 a^2 z^2+8 z^2 a^{-2} +4 z^2 a^{-4} +2 z^2+3 a^3 z+5 a z+3 z a^{-1} +2 z a^{-3} +z a^{-5} +a^4+a^2-3 a^{-2} - a^{-4} -1 }[/math]
The A2 invariant Data:K11n32/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n32/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n32 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["K11n32"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-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]= { 69, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-11 q+12-11 q^{-1} +8 q^{-2} -5 q^{-3} +2 q^{-4} }[/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 z^4+2 z^4 a^{-2} -3 z^4+4 z^2 a^{-2} -z^2 a^{-4} -4 z^2+a^4-a^2+3 a^{-2} - a^{-4} -1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7+3 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} -a^2 z^6+z^6 a^{-2} +3 z^6 a^{-4} -3 z^6+3 a^3 z^5-9 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +3 a^4 z^4+4 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -3 z^4-6 a^3 z^3-8 a z^3-z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} -4 a^4 z^2-6 a^2 z^2+8 z^2 a^{-2} +4 z^2 a^{-4} +2 z^2+3 a^3 z+5 a z+3 z a^{-1} +2 z a^{-3} +z a^{-5} +a^4+a^2-3 a^{-2} - a^{-4} -1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_34, K11n119,}

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["K11n32"];
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+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-11 q+12-11 q^{-1} +8 q^{-2} -5 q^{-3} +2 q^{-4} }[/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]= {9_34, K11n119,}
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: (-1, 2)
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{ -4 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{320}{3} }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{56}{3} }[/math] [math]\displaystyle{ \frac{5489}{30} }[/math] [math]\displaystyle{ \frac{862}{15} }[/math] [math]\displaystyle{ -\frac{302}{45} }[/math] [math]\displaystyle{ \frac{367}{18} }[/math] [math]\displaystyle{ -\frac{751}{30} }[/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 K11n32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
11         1-1
9        2 2
7       41 -3
5      62  4
3     54   -1
1    76    1
-1   56     1
-3  36      -3
-5 25       3
-7 3        -3
-92         2
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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/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.

K11n31.gif

K11n31

K11n33.gif

K11n33

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