K11a273

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K11a272.gif

K11a272

K11a274.gif

K11a274

K11a273.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a273 at Knotilus!

[edit K11a273 Quick Notes]


[edit K11a273 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X14,5,15,6 X16,8,17,7 X20,9,21,10 X4,11,5,12 X18,13,19,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code 1, -8, 2, -6, 3, -1, 4, -11, 5, -2, 6, -10, 7, -3, 8, -4, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 10 14 16 20 4 18 2 22 12 8
A Braid Representative
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A Morse Link Presentation K11a273 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number [math]\displaystyle{ \{1,2\} }[/math]
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a273/ThurstonBennequinNumber
Hyperbolic Volume 18.4008
A-Polynomial See Data:K11a273/A-polynomial

[edit Notes for K11a273's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a273's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^3-16 t^2+37 t-47+37 t^{-1} -16 t^{-2} +3 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^6+2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 159, -2 }
Jones polynomial [math]\displaystyle{ -q^2+4 q-9+17 q^{-1} -22 q^{-2} +26 q^{-3} -26 q^{-4} +22 q^{-5} -17 q^{-6} +10 q^{-7} -4 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^8+a^8-3 z^4 a^6-5 z^2 a^6-2 a^6+2 z^6 a^4+5 z^4 a^4+4 z^2 a^4+z^6 a^2+z^4 a^2+z^2 a^2+2 a^2-z^4-z^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-8 z^5 a^9+5 z^3 a^9-z a^9+8 z^8 a^8-17 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+8 z^9 a^7-11 z^7 a^7-z^5 a^7+z^3 a^7+z a^7+3 z^{10} a^6+14 z^8 a^6-45 z^6 a^6+39 z^4 a^6-16 z^2 a^6+2 a^6+17 z^9 a^5-30 z^7 a^5+15 z^5 a^5-8 z^3 a^5+5 z a^5+3 z^{10} a^4+17 z^8 a^4-47 z^6 a^4+40 z^4 a^4-13 z^2 a^4+9 z^9 a^3-7 z^7 a^3-3 z^5 a^3+z^3 a^3+3 z a^3+11 z^8 a^2-16 z^6 a^2+10 z^4 a^2-z^2 a^2-2 a^2+8 z^7 a-10 z^5 a+4 z^3 a+4 z^6-5 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-1} }[/math]
The A2 invariant Data:K11a273/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a273/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a273 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["K11a273"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^3-16 t^2+37 t-47+37 t^{-1} -16 t^{-2} +3 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^6+2 z^4+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]= { 159, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^2+4 q-9+17 q^{-1} -22 q^{-2} +26 q^{-3} -26 q^{-4} +22 q^{-5} -17 q^{-6} +10 q^{-7} -4 q^{-8} + q^{-9} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^8+a^8-3 z^4 a^6-5 z^2 a^6-2 a^6+2 z^6 a^4+5 z^4 a^4+4 z^2 a^4+z^6 a^2+z^4 a^2+z^2 a^2+2 a^2-z^4-z^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-8 z^5 a^9+5 z^3 a^9-z a^9+8 z^8 a^8-17 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+8 z^9 a^7-11 z^7 a^7-z^5 a^7+z^3 a^7+z a^7+3 z^{10} a^6+14 z^8 a^6-45 z^6 a^6+39 z^4 a^6-16 z^2 a^6+2 a^6+17 z^9 a^5-30 z^7 a^5+15 z^5 a^5-8 z^3 a^5+5 z a^5+3 z^{10} a^4+17 z^8 a^4-47 z^6 a^4+40 z^4 a^4-13 z^2 a^4+9 z^9 a^3-7 z^7 a^3-3 z^5 a^3+z^3 a^3+3 z a^3+11 z^8 a^2-16 z^6 a^2+10 z^4 a^2-z^2 a^2-2 a^2+8 z^7 a-10 z^5 a+4 z^3 a+4 z^6-5 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-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["K11a273"];
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{ 3 t^3-16 t^2+37 t-47+37 t^{-1} -16 t^{-2} +3 t^{-3} }[/math], [math]\displaystyle{ -q^2+4 q-9+17 q^{-1} -22 q^{-2} +26 q^{-3} -26 q^{-4} +22 q^{-5} -17 q^{-6} +10 q^{-7} -4 q^{-8} + q^{-9} }[/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, 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{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -80 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{640}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -136 }[/math] [math]\displaystyle{ 144 }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -\frac{296}{3} }[/math] [math]\displaystyle{ 8 }[/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 K11a273. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-10123χ
5           1-1
3          3 3
1         61 -5
-1        113  8
-3       127   -5
-5      1410    4
-7     1212     0
-9    1014      -4
-11   712       5
-13  310        -7
-15 17         6
-17 3          -3
-191           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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/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.

K11a272.gif

K11a272

K11a274.gif

K11a274

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