K11a267

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K11a266.gif

K11a266

K11a268.gif

K11a268

K11a267.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a267 at Knotilus!

[edit K11a267 Quick Notes]


[edit K11a267 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X14,7,15,8 X20,10,21,9 X18,11,19,12 X22,13,1,14 X8,15,9,16 X4,18,5,17 X2,19,3,20 X16,21,17,22
Gauss code 1, -10, 2, -9, 3, -1, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -6, 10, -5, 11, -7
Dowker-Thistlethwaite code 6 10 12 14 20 18 22 8 4 2 16
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a267 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant 2

[edit Notes for K11a267's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+7 t^3-22 t^2+41 t-49+41 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-z^6+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 191, -2 }
Jones polynomial [math]\displaystyle{ q^3-5 q^2+12 q-19+27 q^{-1} -31 q^{-2} +31 q^{-3} -27 q^{-4} +20 q^{-5} -12 q^{-6} +5 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^2 z^8+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+5 a^4 z^4-6 a^2 z^4+2 z^4-a^6 z^2+3 a^4 z^2-3 a^2 z^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 5 a^4 z^{10}+5 a^2 z^{10}+14 a^5 z^9+26 a^3 z^9+12 a z^9+17 a^6 z^8+21 a^4 z^8+15 a^2 z^8+11 z^8+12 a^7 z^7-13 a^5 z^7-51 a^3 z^7-21 a z^7+5 z^7 a^{-1} +5 a^8 z^6-25 a^6 z^6-60 a^4 z^6-54 a^2 z^6+z^6 a^{-2} -23 z^6+a^9 z^5-14 a^7 z^5-5 a^5 z^5+26 a^3 z^5+8 a z^5-8 z^5 a^{-1} -3 a^8 z^4+12 a^6 z^4+42 a^4 z^4+42 a^2 z^4-z^4 a^{-2} +14 z^4+4 a^7 z^3+4 a^5 z^3-2 a^3 z^3+2 z^3 a^{-1} -3 a^6 z^2-9 a^4 z^2-9 a^2 z^2-3 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}+2 q^{22}-3 q^{18}+5 q^{16}-5 q^{14}+2 q^{12}+q^{10}-4 q^8+6 q^6-6 q^4+6 q^2-2 q^{-2} +4 q^{-4} -3 q^{-6} + q^{-8} }[/math]
The G2 invariant Data:K11a267/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a267 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["K11a267"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+7 t^3-22 t^2+41 t-49+41 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-z^6+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]= { 191, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-5 q^2+12 q-19+27 q^{-1} -31 q^{-2} +31 q^{-3} -27 q^{-4} +20 q^{-5} -12 q^{-6} +5 q^{-7} - q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^2 z^8+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+5 a^4 z^4-6 a^2 z^4+2 z^4-a^6 z^2+3 a^4 z^2-3 a^2 z^2+z^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 5 a^4 z^{10}+5 a^2 z^{10}+14 a^5 z^9+26 a^3 z^9+12 a z^9+17 a^6 z^8+21 a^4 z^8+15 a^2 z^8+11 z^8+12 a^7 z^7-13 a^5 z^7-51 a^3 z^7-21 a z^7+5 z^7 a^{-1} +5 a^8 z^6-25 a^6 z^6-60 a^4 z^6-54 a^2 z^6+z^6 a^{-2} -23 z^6+a^9 z^5-14 a^7 z^5-5 a^5 z^5+26 a^3 z^5+8 a z^5-8 z^5 a^{-1} -3 a^8 z^4+12 a^6 z^4+42 a^4 z^4+42 a^2 z^4-z^4 a^{-2} +14 z^4+4 a^7 z^3+4 a^5 z^3-2 a^3 z^3+2 z^3 a^{-1} -3 a^6 z^2-9 a^4 z^2-9 a^2 z^2-3 z^2+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["K11a267"];
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^4+7 t^3-22 t^2+41 t-49+41 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-5 q^2+12 q-19+27 q^{-1} -31 q^{-2} +31 q^{-3} -27 q^{-4} +20 q^{-5} -12 q^{-6} +5 q^{-7} - q^{-8} }[/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{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/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{ -96 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/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]-2 is the signature of K11a267. 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          4 -4
3         81 7
1        114  -7
-1       168   8
-3      1612    -4
-5     1515     0
-7    1216      4
-9   815       -7
-11  412        8
-13 18         -7
-15 4          4
-171           -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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a266.gif

K11a266

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K11a268

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