K11n146

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K11n145.gif

K11n145

K11n147.gif

K11n147

K11n146.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n146 at Knotilus!

[edit K11n146 Quick Notes]


[edit K11n146 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X11,19,12,18 X2,14,3,13 X8,15,9,16 X6,18,7,17 X19,13,20,12 X16,22,17,21
Gauss code 1, -7, 2, -1, 3, -9, 4, -8, 5, -3, -6, 10, 7, -2, 8, -11, 9, 6, -10, -4, 11, -5
Dowker-Thistlethwaite code 4 14 10 20 22 -18 2 8 6 -12 16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n146 ML.gif

Three dimensional invariants

Symmetry type Reversible
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:K11n146/ThurstonBennequinNumber
Hyperbolic Volume 14.497
A-Polynomial See Data:K11n146/A-polynomial

[edit Notes for K11n146'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 K11n146's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-5 t^2+15 t-21+15 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 63, 2 }
Jones polynomial [math]\displaystyle{ q^9-4 q^8+6 q^7-9 q^6+11 q^5-10 q^4+10 q^3-7 q^2+4 q-1 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +8 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +9 z^8 a^{-6} +5 z^8 a^{-8} +2 z^7 a^{-3} +2 z^7 a^{-7} +4 z^7 a^{-9} -13 z^6 a^{-4} -29 z^6 a^{-6} -15 z^6 a^{-8} +z^6 a^{-10} -3 z^5 a^{-3} -11 z^5 a^{-5} -20 z^5 a^{-7} -12 z^5 a^{-9} +4 z^4 a^{-2} +24 z^4 a^{-4} +33 z^4 a^{-6} +11 z^4 a^{-8} -2 z^4 a^{-10} +z^3 a^{-1} +8 z^3 a^{-3} +18 z^3 a^{-5} +18 z^3 a^{-7} +7 z^3 a^{-9} -3 z^2 a^{-2} -17 z^2 a^{-4} -18 z^2 a^{-6} -4 z^2 a^{-8} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -4 z a^{-7} +z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6} }[/math]
The A2 invariant [math]\displaystyle{ -1+2 q^{-2} -2 q^{-4} +3 q^{-8} +4 q^{-12} + q^{-16} -3 q^{-20} + q^{-22} -2 q^{-24} - q^{-26} + q^{-28} }[/math]
The G2 invariant Data:K11n146/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n146 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["K11n146"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-5 t^2+15 t-21+15 t^{-1} -5 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+z^4+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]= { 63, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^9-4 q^8+6 q^7-9 q^6+11 q^5-10 q^4+10 q^3-7 q^2+4 q-1 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +8 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +9 z^8 a^{-6} +5 z^8 a^{-8} +2 z^7 a^{-3} +2 z^7 a^{-7} +4 z^7 a^{-9} -13 z^6 a^{-4} -29 z^6 a^{-6} -15 z^6 a^{-8} +z^6 a^{-10} -3 z^5 a^{-3} -11 z^5 a^{-5} -20 z^5 a^{-7} -12 z^5 a^{-9} +4 z^4 a^{-2} +24 z^4 a^{-4} +33 z^4 a^{-6} +11 z^4 a^{-8} -2 z^4 a^{-10} +z^3 a^{-1} +8 z^3 a^{-3} +18 z^3 a^{-5} +18 z^3 a^{-7} +7 z^3 a^{-9} -3 z^2 a^{-2} -17 z^2 a^{-4} -18 z^2 a^{-6} -4 z^2 a^{-8} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -4 z a^{-7} +z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n167,}

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["K11n146"];
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^3-5 t^2+15 t-21+15 t^{-1} -5 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ q^9-4 q^8+6 q^7-9 q^6+11 q^5-10 q^4+10 q^3-7 q^2+4 q-1 }[/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]= {K11n167,}
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: (4, 7)
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{ 16 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{872}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ 896 }[/math] [math]\displaystyle{ \frac{4784}{3} }[/math] [math]\displaystyle{ \frac{896}{3} }[/math] [math]\displaystyle{ 248 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ \frac{13952}{3} }[/math] [math]\displaystyle{ \frac{2560}{3} }[/math] [math]\displaystyle{ \frac{132422}{15} }[/math] [math]\displaystyle{ -\frac{688}{15} }[/math] [math]\displaystyle{ \frac{179408}{45} }[/math] [math]\displaystyle{ \frac{346}{9} }[/math] [math]\displaystyle{ \frac{8822}{15} }[/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 K11n146. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -1012345678χ
19         11
17        3 -3
15       31 2
13      63  -3
11     53   2
9    56    1
7   55     0
5  25      3
3 25       -3
1 3        3
-11         -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=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=8 }[/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.

K11n145.gif

K11n145

K11n147.gif

K11n147

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