K11a316

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K11a315.gif

K11a315

K11a317.gif

K11a317

K11a316.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a316 at Knotilus!

[edit K11a316 Quick Notes]


[edit K11a316 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,5,17,6 X20,8,21,7 X22,10,1,9 X4,12,5,11 X18,13,19,14 X10,15,11,16 X2,17,3,18 X14,19,15,20 X8,22,9,21
Gauss code 1, -9, 2, -6, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -7, 10, -4, 11, -5
Dowker-Thistlethwaite code 6 12 16 20 22 4 18 10 2 14 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11a316 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:K11a316/ThurstonBennequinNumber
Hyperbolic Volume 15.8582
A-Polynomial See Data:K11a316/A-polynomial

[edit Notes for K11a316's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ [0,4] }[/math]
Rasmussen s-Invariant 0

[edit Notes for K11a316's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-5 t^3+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+3 z^6+4 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 121, 0 }
Jones polynomial [math]\displaystyle{ q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-5 a^{-2} + a^{-4} +8 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+6 a z^9+11 z^9 a^{-1} +5 z^9 a^{-3} +8 a^2 z^8+7 z^8 a^{-2} +6 z^8 a^{-4} +9 z^8+6 a^3 z^7-10 a z^7-26 z^7 a^{-1} -6 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-21 a^2 z^6-26 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -36 z^6+a^5 z^5-13 a^3 z^5+4 a z^5+23 z^5 a^{-1} -5 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+27 a^2 z^4+27 z^4 a^{-2} +5 z^4 a^{-4} -2 z^4 a^{-6} +52 z^4-2 a^5 z^3+10 a^3 z^3+10 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} +6 z^3 a^{-5} -14 a^2 z^2-18 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -30 z^2-3 a^3 z-5 a z-3 z a^{-1} -z a^{-3} +3 a^2+5 a^{-2} + a^{-4} +8 }[/math]
The A2 invariant [math]\displaystyle{ -q^{14}+q^{12}-3 q^{10}+q^8+q^6-2 q^4+6 q^2-1+4 q^{-2} -2 q^{-6} +2 q^{-8} -4 q^{-10} + q^{-12} - q^{-16} + q^{-18} }[/math]
The G2 invariant Data:K11a316/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a316 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["K11a316"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-5 t^3+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+3 z^6+4 z^4+2 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]= { 121, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 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^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-5 a^{-2} + a^{-4} +8 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+6 a z^9+11 z^9 a^{-1} +5 z^9 a^{-3} +8 a^2 z^8+7 z^8 a^{-2} +6 z^8 a^{-4} +9 z^8+6 a^3 z^7-10 a z^7-26 z^7 a^{-1} -6 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-21 a^2 z^6-26 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -36 z^6+a^5 z^5-13 a^3 z^5+4 a z^5+23 z^5 a^{-1} -5 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+27 a^2 z^4+27 z^4 a^{-2} +5 z^4 a^{-4} -2 z^4 a^{-6} +52 z^4-2 a^5 z^3+10 a^3 z^3+10 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} +6 z^3 a^{-5} -14 a^2 z^2-18 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -30 z^2-3 a^3 z-5 a z-3 z a^{-1} -z a^{-3} +3 a^2+5 a^{-2} + a^{-4} +8 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a35,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a35, K11a36,}

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["K11a316"];
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+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 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]= {K11a35,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a35, K11a36,}

Vassiliev invariants

V2 and V3: (2, -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{ 8 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{28}{3} }[/math] [math]\displaystyle{ -\frac{52}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{464}{3} }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ -168 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ -\frac{416}{3} }[/math] [math]\displaystyle{ \frac{751}{15} }[/math] [math]\displaystyle{ \frac{4156}{15} }[/math] [math]\displaystyle{ -\frac{15116}{45} }[/math] [math]\displaystyle{ -\frac{415}{9} }[/math] [math]\displaystyle{ -\frac{1649}{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]0 is the signature of K11a316. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          3 -3
9         51 4
7        83  -5
5       95   4
3      108    -2
1     109     1
-1    711      4
-3   59       -4
-5  27        5
-7 15         -4
-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=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/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}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/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.

K11a315.gif

K11a315

K11a317.gif

K11a317

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