K11n58

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K11n57.gif

K11n57

K11n59.gif

K11n59

K11n58.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n58 at Knotilus!

[edit K11n58 Quick Notes]


[edit K11n58 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 14 2 -16 -18 -20 6 -22 -12 -10
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n58 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:K11n58/ThurstonBennequinNumber
Hyperbolic Volume 9.97833
A-Polynomial See Data:K11n58/A-polynomial

[edit Notes for K11n58'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 K11n58's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 35, -2 }
Jones polynomial [math]\displaystyle{ -q^4+2 q^3-3 q^2+5 q-5+6 q^{-1} -5 q^{-2} +4 q^{-3} -3 q^{-4} + q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6-2 a^2 z^4-z^4 a^{-2} +5 z^4+a^4 z^2-6 a^2 z^2-3 z^2 a^{-2} +9 z^2+a^4-4 a^2-2 a^{-2} +6 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7-a z^7-2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 a^2 z^6-10 z^6 a^{-2} -17 z^6-5 a^3 z^5-5 a z^5-5 z^5 a^{-1} -5 z^5 a^{-3} +8 a^2 z^4+15 z^4 a^{-2} +23 z^4+3 a^5 z^3+6 a^3 z^3+6 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} +a^6 z^2-10 a^2 z^2-8 z^2 a^{-2} -17 z^2-a^5 z-3 a^3 z-4 a z-4 z a^{-1} -2 z a^{-3} +a^4+4 a^2+2 a^{-2} +6 }[/math]
The A2 invariant [math]\displaystyle{ q^{16}-q^{12}-2 q^8+q^2+3+ q^{-2} +2 q^{-4} - q^{-8} - q^{-12} }[/math]
The G2 invariant Data:K11n58/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n58 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["K11n58"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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]= { 35, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^4+2 q^3-3 q^2+5 q-5+6 q^{-1} -5 q^{-2} +4 q^{-3} -3 q^{-4} + q^{-5} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6-2 a^2 z^4-z^4 a^{-2} +5 z^4+a^4 z^2-6 a^2 z^2-3 z^2 a^{-2} +9 z^2+a^4-4 a^2-2 a^{-2} +6 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7-a z^7-2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 a^2 z^6-10 z^6 a^{-2} -17 z^6-5 a^3 z^5-5 a z^5-5 z^5 a^{-1} -5 z^5 a^{-3} +8 a^2 z^4+15 z^4 a^{-2} +23 z^4+3 a^5 z^3+6 a^3 z^3+6 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} +a^6 z^2-10 a^2 z^2-8 z^2 a^{-2} -17 z^2-a^5 z-3 a^3 z-4 a z-4 z a^{-1} -2 z a^{-3} +a^4+4 a^2+2 a^{-2} +6 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_16, 10_156, K11n15, K11n56,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n56,}

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["K11n58"];
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^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -q^4+2 q^3-3 q^2+5 q-5+6 q^{-1} -5 q^{-2} +4 q^{-3} -3 q^{-4} + q^{-5} }[/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]= {8_16, 10_156, K11n15, K11n56,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n56,}

Vassiliev invariants

V2 and V3: (1, 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{ 4 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{176}{3} }[/math] [math]\displaystyle{ -\frac{64}{3} }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ -\frac{209}{30} }[/math] [math]\displaystyle{ \frac{1258}{15} }[/math] [math]\displaystyle{ -\frac{3898}{45} }[/math] [math]\displaystyle{ -\frac{463}{18} }[/math] [math]\displaystyle{ -\frac{1169}{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 K11n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
9         1-1
7        1 1
5       21 -1
3      31  2
1     22   0
-1    43    1
-3   23     1
-5  23      -1
-7 12       1
-9 2        -2
-111         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=-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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

K11n57.gif

K11n57

K11n59.gif

K11n59

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