K11n27

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K11n26.gif

K11n26

K11n28.gif

K11n28

K11n27.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n27 at Knotilus!

[edit K11n27 Quick Notes]


[edit K11n27 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X9,15,10,14 X11,18,12,19 X13,7,14,6 X15,21,16,20 X17,1,18,22 X19,10,20,11 X21,17,22,16
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 3, -7, 5, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 -12 2 -14 -18 -6 -20 -22 -10 -16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n27 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant -6

[edit Notes for K11n27's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+3 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-5 z^6-5 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 19, 6 }
Jones polynomial [math]\displaystyle{ q^9-2 q^8+2 q^7-3 q^6+3 q^5-3 q^4+3 q^3-q^2+q }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -19 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +6 a^{-4} -9 a^{-6} +5 a^{-8} - a^{-10} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -2 z^7 a^{-7} +3 z^7 a^{-9} -7 z^6 a^{-4} -24 z^6 a^{-6} -16 z^6 a^{-8} +z^6 a^{-10} +5 z^5 a^{-5} -10 z^5 a^{-7} -15 z^5 a^{-9} +17 z^4 a^{-4} +45 z^4 a^{-6} +24 z^4 a^{-8} -4 z^4 a^{-10} +4 z^3 a^{-5} +22 z^3 a^{-7} +18 z^3 a^{-9} -17 z^2 a^{-4} -33 z^2 a^{-6} -15 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-5} -10 z a^{-7} -6 z a^{-9} -z a^{-11} +6 a^{-4} +9 a^{-6} +5 a^{-8} + a^{-10} }[/math]
The A2 invariant [math]\displaystyle{ q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-16} - q^{-18} -3 q^{-20} + q^{-26} + q^{-28} + q^{-32} - q^{-34} }[/math]
The G2 invariant Data:K11n27/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n27 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["K11n27"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+3 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-5 z^6-5 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]= { 19, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^9-2 q^8+2 q^7-3 q^6+3 q^5-3 q^4+3 q^3-q^2+q }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -19 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +6 a^{-4} -9 a^{-6} +5 a^{-8} - a^{-10} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -2 z^7 a^{-7} +3 z^7 a^{-9} -7 z^6 a^{-4} -24 z^6 a^{-6} -16 z^6 a^{-8} +z^6 a^{-10} +5 z^5 a^{-5} -10 z^5 a^{-7} -15 z^5 a^{-9} +17 z^4 a^{-4} +45 z^4 a^{-6} +24 z^4 a^{-8} -4 z^4 a^{-10} +4 z^3 a^{-5} +22 z^3 a^{-7} +18 z^3 a^{-9} -17 z^2 a^{-4} -33 z^2 a^{-6} -15 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-5} -10 z a^{-7} -6 z a^{-9} -z a^{-11} +6 a^{-4} +9 a^{-6} +5 a^{-8} + a^{-10} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {10_133,}

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["K11n27"];
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+3 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^9-2 q^8+2 q^7-3 q^6+3 q^5-3 q^4+3 q^3-q^2+q }[/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]= {10_133,}

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{110}{3} }[/math] [math]\displaystyle{ \frac{130}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 320 }[/math] [math]\displaystyle{ 160 }[/math] [math]\displaystyle{ 160 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ \frac{520}{3} }[/math] [math]\displaystyle{ \frac{44191}{30} }[/math] [math]\displaystyle{ \frac{338}{15} }[/math] [math]\displaystyle{ \frac{56462}{45} }[/math] [math]\displaystyle{ \frac{2081}{18} }[/math] [math]\displaystyle{ \frac{2911}{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]6 is the signature of K11n27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-10123456χ
19        11
17       1 -1
15      22 0
13     21  -1
11    121  0
9   22    0
7  11     0
5 13      2
3         0
11        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=5 }[/math] [math]\displaystyle{ i=7 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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]
[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.

K11n26.gif

K11n26

K11n28.gif

K11n28

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