K11n148

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K11n147.gif

K11n147

K11n149.gif

K11n149

K11n148.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n148 at Knotilus!

[edit K11n148 Quick Notes]


[edit K11n148 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X5,19,6,18 X7,20,8,21 X9,1,10,22 X11,6,12,7 X2,14,3,13 X15,9,16,8 X17,10,18,11 X19,13,20,12 X21,16,22,17
Gauss code 1, -7, 2, -1, -3, 6, -4, 8, -5, 9, -6, 10, 7, -2, -8, 11, -9, 3, -10, 4, -11, 5
Dowker-Thistlethwaite code 4 14 -18 -20 -22 -6 2 -8 -10 -12 -16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n148 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{1,2,3\} }[/math]
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n148/ThurstonBennequinNumber
Hyperbolic Volume 15.4617
A-Polynomial See Data:K11n148/A-polynomial

[edit Notes for K11n148'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 K11n148's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+5 t^3-10 t^2+14 t-15+14 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-3 z^6+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \left\{5,t^2+2 t+1\right\} }[/math]
Determinant and Signature { 75, 2 }
Jones polynomial [math]\displaystyle{ -2 q^6+5 q^5-9 q^4+12 q^3-12 q^2+13 q-10+7 q^{-1} -4 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +z^6-7 z^4 a^{-2} +4 z^4 a^{-4} +3 z^4-z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} +z^2+3 a^{-2} - a^{-6} -1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 z^9 a^{-1} +3 z^9 a^{-3} +12 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8+4 a z^7-z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-36 z^6 a^{-2} -16 z^6 a^{-4} +z^6 a^{-6} -18 z^6-11 a z^5-15 z^5 a^{-1} -10 z^5 a^{-3} -6 z^5 a^{-5} -2 a^2 z^4+33 z^4 a^{-2} +21 z^4 a^{-4} +4 z^4 a^{-6} +14 z^4+6 a z^3+12 z^3 a^{-1} +12 z^3 a^{-3} +9 z^3 a^{-5} +3 z^3 a^{-7} -8 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -4 z^2-z a^{-1} -2 z a^{-3} -4 z a^{-5} -3 z a^{-7} -3 a^{-2} + a^{-6} -1 }[/math]
The A2 invariant [math]\displaystyle{ q^8-2 q^6+q^4-2 q^2+2 q^{-2} - q^{-4} +6 q^{-6} - q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} + q^{-16} -2 q^{-18} + q^{-20} - q^{-22} }[/math]
The G2 invariant Data:K11n148/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n148 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["K11n148"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+5 t^3-10 t^2+14 t-15+14 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-3 z^6+3 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{ \left\{5,t^2+2 t+1\right\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 75, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -2 q^6+5 q^5-9 q^4+12 q^3-12 q^2+13 q-10+7 q^{-1} -4 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} -5 z^6 a^{-2} +z^6 a^{-4} +z^6-7 z^4 a^{-2} +4 z^4 a^{-4} +3 z^4-z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} +z^2+3 a^{-2} - a^{-6} -1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 z^9 a^{-1} +3 z^9 a^{-3} +12 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8+4 a z^7-z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-36 z^6 a^{-2} -16 z^6 a^{-4} +z^6 a^{-6} -18 z^6-11 a z^5-15 z^5 a^{-1} -10 z^5 a^{-3} -6 z^5 a^{-5} -2 a^2 z^4+33 z^4 a^{-2} +21 z^4 a^{-4} +4 z^4 a^{-6} +14 z^4+6 a z^3+12 z^3 a^{-1} +12 z^3 a^{-3} +9 z^3 a^{-5} +3 z^3 a^{-7} -8 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -4 z^2-z a^{-1} -2 z a^{-3} -4 z a^{-5} -3 z a^{-7} -3 a^{-2} + a^{-6} -1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a223,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n168,}

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["K11n148"];
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+5 t^3-10 t^2+14 t-15+14 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -2 q^6+5 q^5-9 q^4+12 q^3-12 q^2+13 q-10+7 q^{-1} -4 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]= {K11a223,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n168,}

Vassiliev invariants

V2 and V3: (3, 4)
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{ 12 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 158 }[/math] [math]\displaystyle{ 34 }[/math] [math]\displaystyle{ 384 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ \frac{320}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ 1896 }[/math] [math]\displaystyle{ 408 }[/math] [math]\displaystyle{ \frac{32911}{10} }[/math] [math]\displaystyle{ -\frac{462}{5} }[/math] [math]\displaystyle{ \frac{23102}{15} }[/math] [math]\displaystyle{ \frac{401}{6} }[/math] [math]\displaystyle{ \frac{2351}{10} }[/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 K11n148. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
13         2-2
11        3 3
9       62 -4
7      63  3
5     66   0
3    76    1
1   47     3
-1  36      -3
-3 14       3
-5 3        -3
-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11n147.gif

K11n147

K11n149.gif

K11n149

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