K11n120

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K11n119.gif

K11n119

K11n121.gif

K11n121

K11n120.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n120 at Knotilus!

[edit K11n120 Quick Notes]


[edit K11n120 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X5,15,6,14 X7,20,8,21 X2,10,3,9 X11,17,12,16 X13,18,14,19 X15,9,16,8 X17,1,18,22 X19,6,20,7 X21,12,22,13
Gauss code 1, -5, 2, -1, -3, 10, -4, 8, 5, -2, -6, 11, -7, 3, -8, 6, -9, 7, -10, 4, -11, 9
Dowker-Thistlethwaite code 4 10 -14 -20 2 -16 -18 -8 -22 -6 -12
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation K11n120 ML.gif

Three dimensional invariants

Symmetry type Chiral
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:K11n120/ThurstonBennequinNumber
Hyperbolic Volume 12.7751
A-Polynomial See Data:K11n120/A-polynomial

[edit Notes for K11n120'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 K11n120's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+4 t^3-7 t^2+8 t-7+8 t^{-1} -7 t^{-2} +4 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-4 z^6-3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 47, 2 }
Jones polynomial [math]\displaystyle{ -q^6+3 q^5-5 q^4+7 q^3-8 q^2+8 q-6+5 q^{-1} -3 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +z^6-12 z^4 a^{-2} +5 z^4 a^{-4} +4 z^4-10 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} +4 z^2-3 a^{-2} +3 a^{-4} - a^{-6} +2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +7 z^8 a^{-2} +3 z^8 a^{-4} +4 z^8+3 a z^7-4 z^7 a^{-1} -6 z^7 a^{-3} +z^7 a^{-5} +a^2 z^6-29 z^6 a^{-2} -13 z^6 a^{-4} -15 z^6-10 a z^5-4 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} -3 a^2 z^4+40 z^4 a^{-2} +25 z^4 a^{-4} +3 z^4 a^{-6} +15 z^4+6 a z^3+7 z^3 a^{-1} +3 z^3 a^{-3} +3 z^3 a^{-5} +z^3 a^{-7} +a^2 z^2-20 z^2 a^{-2} -16 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2-a z-2 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2 }[/math]
The A2 invariant [math]\displaystyle{ q^8-q^6+q^4+1+ q^{-2} -2 q^{-4} +2 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-16} - q^{-18} + q^{-20} - q^{-22} }[/math]
The G2 invariant Data:K11n120/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n120 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["K11n120"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+4 t^3-7 t^2+8 t-7+8 t^{-1} -7 t^{-2} +4 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-4 z^6-3 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]= { 47, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^6+3 q^5-5 q^4+7 q^3-8 q^2+8 q-6+5 q^{-1} -3 q^{-2} + q^{-3} }[/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} -6 z^6 a^{-2} +z^6 a^{-4} +z^6-12 z^4 a^{-2} +5 z^4 a^{-4} +4 z^4-10 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} +4 z^2-3 a^{-2} +3 a^{-4} - a^{-6} +2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +7 z^8 a^{-2} +3 z^8 a^{-4} +4 z^8+3 a z^7-4 z^7 a^{-1} -6 z^7 a^{-3} +z^7 a^{-5} +a^2 z^6-29 z^6 a^{-2} -13 z^6 a^{-4} -15 z^6-10 a z^5-4 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} -3 a^2 z^4+40 z^4 a^{-2} +25 z^4 a^{-4} +3 z^4 a^{-6} +15 z^4+6 a z^3+7 z^3 a^{-1} +3 z^3 a^{-3} +3 z^3 a^{-5} +z^3 a^{-7} +a^2 z^2-20 z^2 a^{-2} -16 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2-a z-2 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2 }[/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["K11n120"];
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+4 t^3-7 t^2+8 t-7+8 t^{-1} -7 t^{-2} +4 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^6+3 q^5-5 q^4+7 q^3-8 q^2+8 q-6+5 q^{-1} -3 q^{-2} + q^{-3} }[/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, 1)
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{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{272}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 168 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ -88 }[/math] [math]\displaystyle{ 104 }[/math] [math]\displaystyle{ -40 }[/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 K11n120. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
13         1-1
11        2 2
9       31 -2
7      42  2
5     43   -1
3    44    0
1   35     2
-1  23      -1
-3 13       2
-5 2        -2
-71         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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/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]

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.

K11n119.gif

K11n119

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K11n121

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