K11a198

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K11a197.gif

K11a197

K11a199.gif

K11a199

K11a198.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a198 at Knotilus!

[edit K11a198 Quick Notes]


[edit K11a198 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,5,17,6 X14,7,15,8 X18,10,19,9 X2,12,3,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X10,20,11,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 12 16 14 18 2 22 6 20 10 8
A Braid Representative
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A Morse Link Presentation K11a198 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -2

[edit Notes for K11a198's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 115, 2 }
Jones polynomial [math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+17 q^3-18 q^2+18 q-15+11 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+z^4 a^{-2} -2 z^4 a^{-4} +3 z^4-2 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} +5 z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+8 z^9 a^{-1} +5 z^9 a^{-3} +3 a^2 z^8+16 z^8 a^{-2} +10 z^8 a^{-4} +9 z^8+a^3 z^7-6 a z^7-12 z^7 a^{-1} +6 z^7 a^{-3} +11 z^7 a^{-5} -12 a^2 z^6-47 z^6 a^{-2} -12 z^6 a^{-4} +8 z^6 a^{-6} -39 z^6-4 a^3 z^5-5 a z^5-15 z^5 a^{-1} -32 z^5 a^{-3} -14 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+38 z^4 a^{-2} +z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +45 z^4+5 a^3 z^3+14 a z^3+25 z^3 a^{-1} +24 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} -8 a^2 z^2-12 z^2 a^{-2} +z^2 a^{-4} +2 z^2 a^{-6} -19 z^2-2 a^3 z-6 a z-8 z a^{-1} -5 z a^{-3} -z a^{-5} +a^2+ a^{-2} +3 }[/math]
The A2 invariant Data:K11a198/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a198/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a198 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["K11a198"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+z^4+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]= { 115, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+17 q^3-18 q^2+18 q-15+11 q^{-1} -6 q^{-2} +3 q^{-3} - 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^6-a^2 z^4+z^4 a^{-2} -2 z^4 a^{-4} +3 z^4-2 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} +5 z^2-a^2- a^{-2} +3 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+8 z^9 a^{-1} +5 z^9 a^{-3} +3 a^2 z^8+16 z^8 a^{-2} +10 z^8 a^{-4} +9 z^8+a^3 z^7-6 a z^7-12 z^7 a^{-1} +6 z^7 a^{-3} +11 z^7 a^{-5} -12 a^2 z^6-47 z^6 a^{-2} -12 z^6 a^{-4} +8 z^6 a^{-6} -39 z^6-4 a^3 z^5-5 a z^5-15 z^5 a^{-1} -32 z^5 a^{-3} -14 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+38 z^4 a^{-2} +z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +45 z^4+5 a^3 z^3+14 a z^3+25 z^3 a^{-1} +24 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} -8 a^2 z^2-12 z^2 a^{-2} +z^2 a^{-4} +2 z^2 a^{-6} -19 z^2-2 a^3 z-6 a z-8 z a^{-1} -5 z a^{-3} -z a^{-5} +a^2+ a^{-2} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_121, K11a41, K11a183, K11a331,}

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["K11a198"];
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{ 2 t^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+17 q^3-18 q^2+18 q-15+11 q^{-1} -6 q^{-2} +3 q^{-3} - 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]= {10_121, K11a41, K11a183, K11a331,}
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, 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{ 4 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ -\frac{14}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{56}{3} }[/math] [math]\displaystyle{ -\frac{56}{3} }[/math] [math]\displaystyle{ \frac{1471}{30} }[/math] [math]\displaystyle{ -\frac{302}{15} }[/math] [math]\displaystyle{ \frac{2462}{45} }[/math] [math]\displaystyle{ -\frac{223}{18} }[/math] [math]\displaystyle{ \frac{991}{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]2 is the signature of K11a198. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
15           11
13          3 -3
11         51 4
9        83  -5
7       95   4
5      98    -1
3     99     0
1    710      3
-1   48       -4
-3  27        5
-5 14         -3
-7 2          2
-91           -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=3 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a197.gif

K11a197

K11a199.gif

K11a199

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