K11a281

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K11a280.gif

K11a280

K11a282.gif

K11a282

K11a281.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a281 at Knotilus!

[edit K11a281 Quick Notes]


[edit K11a281 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,6,17,5 X12,8,13,7 X20,10,21,9 X2,11,3,12 X18,13,19,14 X4,16,5,15 X22,17,1,18 X8,20,9,19 X14,21,15,22
Gauss code 1, -6, 2, -8, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 10 16 12 20 2 18 4 22 8 14
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a281 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{1,2\} }[/math]
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a281/ThurstonBennequinNumber
Hyperbolic Volume 18.3111
A-Polynomial See Data:K11a281/A-polynomial

[edit Notes for K11a281's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a281's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-18 t^2+33 t-39+33 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6-2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 155, 2 }
Jones polynomial [math]\displaystyle{ q^7-4 q^6+10 q^5-17 q^4+22 q^3-25 q^2+25 q-21+16 q^{-1} -9 q^{-2} +4 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-11 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-12 z^2 a^{-2} +4 z^2 a^{-4} +9 z^2-a^2-5 a^{-2} +2 a^{-4} +5 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10}+6 a z^9+18 z^9 a^{-1} +12 z^9 a^{-3} +4 a^2 z^8+24 z^8 a^{-2} +19 z^8 a^{-4} +9 z^8+a^3 z^7-15 a z^7-41 z^7 a^{-1} -8 z^7 a^{-3} +17 z^7 a^{-5} -13 a^2 z^6-81 z^6 a^{-2} -33 z^6 a^{-4} +10 z^6 a^{-6} -51 z^6-3 a^3 z^5+6 a z^5+12 z^5 a^{-1} -25 z^5 a^{-3} -24 z^5 a^{-5} +4 z^5 a^{-7} +15 a^2 z^4+71 z^4 a^{-2} +19 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +59 z^4+3 a^3 z^3+7 a z^3+12 z^3 a^{-1} +22 z^3 a^{-3} +14 z^3 a^{-5} -7 a^2 z^2-28 z^2 a^{-2} -7 z^2 a^{-4} +3 z^2 a^{-6} -25 z^2-a^3 z-3 a z-5 z a^{-1} -6 z a^{-3} -3 z a^{-5} +a^2+5 a^{-2} +2 a^{-4} +5 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}+q^{10}-2 q^6+5 q^4-2 q^2+3+2 q^{-2} -4 q^{-4} +4 q^{-6} -6 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +4 q^{-16} -2 q^{-18} + q^{-20} }[/math]
The G2 invariant Data:K11a281/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a281 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["K11a281"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-18 t^2+33 t-39+33 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 z^6-2 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]= { 155, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^7-4 q^6+10 q^5-17 q^4+22 q^3-25 q^2+25 q-21+16 q^{-1} -9 q^{-2} +4 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^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-11 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-12 z^2 a^{-2} +4 z^2 a^{-4} +9 z^2-a^2-5 a^{-2} +2 a^{-4} +5 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10}+6 a z^9+18 z^9 a^{-1} +12 z^9 a^{-3} +4 a^2 z^8+24 z^8 a^{-2} +19 z^8 a^{-4} +9 z^8+a^3 z^7-15 a z^7-41 z^7 a^{-1} -8 z^7 a^{-3} +17 z^7 a^{-5} -13 a^2 z^6-81 z^6 a^{-2} -33 z^6 a^{-4} +10 z^6 a^{-6} -51 z^6-3 a^3 z^5+6 a z^5+12 z^5 a^{-1} -25 z^5 a^{-3} -24 z^5 a^{-5} +4 z^5 a^{-7} +15 a^2 z^4+71 z^4 a^{-2} +19 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +59 z^4+3 a^3 z^3+7 a z^3+12 z^3 a^{-1} +22 z^3 a^{-3} +14 z^3 a^{-5} -7 a^2 z^2-28 z^2 a^{-2} -7 z^2 a^{-4} +3 z^2 a^{-6} -25 z^2-a^3 z-3 a z-5 z a^{-1} -6 z a^{-3} -3 z a^{-5} +a^2+5 a^{-2} +2 a^{-4} +5 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a19, K11a25,}

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["K11a281"];
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^3-18 t^2+33 t-39+33 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-4 q^6+10 q^5-17 q^4+22 q^3-25 q^2+25 q-21+16 q^{-1} -9 q^{-2} +4 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]= {K11a19, K11a25,}
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{62}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{512}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{248}{3} }[/math] [math]\displaystyle{ \frac{10289}{30} }[/math] [math]\displaystyle{ \frac{314}{5} }[/math] [math]\displaystyle{ -\frac{542}{45} }[/math] [math]\displaystyle{ \frac{2287}{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]2 is the signature of K11a281. 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         71 6
9        103  -7
7       127   5
5      1310    -3
3     1212     0
1    1014      4
-1   611       -5
-3  310        7
-5 16         -5
-7 3          3
-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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.

K11a280.gif

K11a280

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K11a282

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