K11a282

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K11a281.gif

K11a281

K11a283.gif

K11a283

K11a282.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a282 at Knotilus!

[edit K11a282 Quick Notes]


[edit K11a282 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X16,5,17,6 X14,8,15,7 X4,10,5,9 X18,11,19,12 X20,13,21,14 X22,16,1,15 X2,17,3,18 X12,19,13,20 X8,21,9,22
Gauss code 1, -9, 2, -5, 3, -1, 4, -11, 5, -2, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 10 16 14 4 18 20 22 2 12 8
A Braid Representative
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A Morse Link Presentation K11a282 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 K11a282's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-16 t^2+26 t-29+26 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 127, -2 }
Jones polynomial [math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-17+20 q^{-1} -20 q^{-2} +18 q^{-3} -13 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^2 z^8+a^4 z^6-5 a^2 z^6+2 z^6+3 a^4 z^4-9 a^2 z^4-z^4 a^{-2} +7 z^4+2 a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} +6 z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+9 a^3 z^9+15 a z^9+6 z^9 a^{-1} +12 a^4 z^8+10 a^2 z^8+4 z^8 a^{-2} +2 z^8+11 a^5 z^7-14 a^3 z^7-46 a z^7-20 z^7 a^{-1} +z^7 a^{-3} +8 a^6 z^6-20 a^4 z^6-47 a^2 z^6-14 z^6 a^{-2} -33 z^6+4 a^7 z^5-13 a^5 z^5+a^3 z^5+39 a z^5+18 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-7 a^6 z^4+10 a^4 z^4+45 a^2 z^4+14 z^4 a^{-2} +41 z^4-2 a^7 z^3+3 a^5 z^3+3 a^3 z^3-9 a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-2 a^4 z^2-12 a^2 z^2-5 z^2 a^{-2} -14 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ q^{20}-2 q^{18}+2 q^{16}-2 q^{14}-q^{12}+3 q^{10}-3 q^8+5 q^6-2 q^4+q^2+1-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} }[/math]
The G2 invariant Data:K11a282/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a282 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["K11a282"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-16 t^2+26 t-29+26 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 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]= { 127, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-17+20 q^{-1} -20 q^{-2} +18 q^{-3} -13 q^{-4} +8 q^{-5} -4 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{ -a^2 z^8+a^4 z^6-5 a^2 z^6+2 z^6+3 a^4 z^4-9 a^2 z^4-z^4 a^{-2} +7 z^4+2 a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} +6 z^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+9 a^3 z^9+15 a z^9+6 z^9 a^{-1} +12 a^4 z^8+10 a^2 z^8+4 z^8 a^{-2} +2 z^8+11 a^5 z^7-14 a^3 z^7-46 a z^7-20 z^7 a^{-1} +z^7 a^{-3} +8 a^6 z^6-20 a^4 z^6-47 a^2 z^6-14 z^6 a^{-2} -33 z^6+4 a^7 z^5-13 a^5 z^5+a^3 z^5+39 a z^5+18 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-7 a^6 z^4+10 a^4 z^4+45 a^2 z^4+14 z^4 a^{-2} +41 z^4-2 a^7 z^3+3 a^5 z^3+3 a^3 z^3-9 a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-2 a^4 z^2-12 a^2 z^2-5 z^2 a^{-2} -14 z^2+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a81,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a81,}

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["K11a282"];
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+6 t^3-16 t^2+26 t-29+26 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-17+20 q^{-1} -20 q^{-2} +18 q^{-3} -13 q^{-4} +8 q^{-5} -4 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]= {K11a81,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a81,}

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{ -64 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 160 }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -32 }[/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 K11a282. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
9           1-1
7          3 3
5         51 -4
3        83  5
1       95   -4
-1      118    3
-3     1010     0
-5    810      -2
-7   510       5
-9  38        -5
-11 15         4
-13 3          -3
-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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a281.gif

K11a281

K11a283.gif

K11a283

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