K11a305

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K11a304.gif

K11a304

K11a306.gif

K11a306

K11a305.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a305 at Knotilus!

[edit K11a305 Quick Notes]


[edit K11a305 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X10,5,11,6 X22,8,1,7 X16,9,17,10 X18,12,19,11 X20,13,21,14 X8,15,9,16 X4,18,5,17 X2,19,3,20 X14,21,15,22
Gauss code 1, -10, 2, -9, 3, -1, 4, -8, 5, -3, 6, -2, 7, -11, 8, -5, 9, -6, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 12 10 22 16 18 20 8 4 2 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a305 ML.gif

Three dimensional invariants

Symmetry type Reversible
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:K11a305/ThurstonBennequinNumber
Hyperbolic Volume 17.2512
A-Polynomial See Data:K11a305/A-polynomial

[edit Notes for K11a305's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ [2,4] }[/math]
Rasmussen s-Invariant 2

[edit Notes for K11a305's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 135, -2 }
Jones polynomial [math]\displaystyle{ -q^4+4 q^3-9 q^2+14 q-18+22 q^{-1} -21 q^{-2} +19 q^{-3} -14 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^2 z^8+a^4 z^6-5 a^2 z^6+2 z^6+3 a^4 z^4-9 a^2 z^4-z^4 a^{-2} +7 z^4+2 a^4 z^2-5 a^2 z^2-2 z^2 a^{-2} +7 z^2-a^4+a^2- a^{-2} +2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+10 a^3 z^9+16 a z^9+6 z^9 a^{-1} +14 a^4 z^8+15 a^2 z^8+4 z^8 a^{-2} +5 z^8+12 a^5 z^7-14 a^3 z^7-44 a z^7-17 z^7 a^{-1} +z^7 a^{-3} +8 a^6 z^6-27 a^4 z^6-64 a^2 z^6-13 z^6 a^{-2} -42 z^6+4 a^7 z^5-15 a^5 z^5-5 a^3 z^5+26 a z^5+9 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+20 a^4 z^4+64 a^2 z^4+13 z^4 a^{-2} +50 z^4-2 a^7 z^3+4 a^5 z^3+9 a^3 z^3+3 a z^3+3 z^3 a^{-1} +3 z^3 a^{-3} -6 a^4 z^2-19 a^2 z^2-6 z^2 a^{-2} -19 z^2+a^5 z-2 a z-2 z a^{-1} -z a^{-3} -a^4-a^2+ a^{-2} +2 }[/math]
The A2 invariant [math]\displaystyle{ q^{20}-2 q^{18}+2 q^{16}-3 q^{14}-2 q^{12}+3 q^{10}-3 q^8+6 q^6-q^4+2 q^2+2-3 q^{-2} +3 q^{-4} -2 q^{-6} + q^{-10} - q^{-12} }[/math]
The G2 invariant Data:K11a305/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a305 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["K11a305"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 z^6+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]= { 135, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^4+4 q^3-9 q^2+14 q-18+22 q^{-1} -21 q^{-2} +19 q^{-3} -14 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^2 z^8+a^4 z^6-5 a^2 z^6+2 z^6+3 a^4 z^4-9 a^2 z^4-z^4 a^{-2} +7 z^4+2 a^4 z^2-5 a^2 z^2-2 z^2 a^{-2} +7 z^2-a^4+a^2- a^{-2} +2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+10 a^3 z^9+16 a z^9+6 z^9 a^{-1} +14 a^4 z^8+15 a^2 z^8+4 z^8 a^{-2} +5 z^8+12 a^5 z^7-14 a^3 z^7-44 a z^7-17 z^7 a^{-1} +z^7 a^{-3} +8 a^6 z^6-27 a^4 z^6-64 a^2 z^6-13 z^6 a^{-2} -42 z^6+4 a^7 z^5-15 a^5 z^5-5 a^3 z^5+26 a z^5+9 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+20 a^4 z^4+64 a^2 z^4+13 z^4 a^{-2} +50 z^4-2 a^7 z^3+4 a^5 z^3+9 a^3 z^3+3 a z^3+3 z^3 a^{-1} +3 z^3 a^{-3} -6 a^4 z^2-19 a^2 z^2-6 z^2 a^{-2} -19 z^2+a^5 z-2 a z-2 z a^{-1} -z a^{-3} -a^4-a^2+ a^{-2} +2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a157, K11a264,}

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["K11a305"];
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+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+4 q^3-9 q^2+14 q-18+22 q^{-1} -21 q^{-2} +19 q^{-3} -14 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }[/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]= {K11a157, K11a264,}
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: (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{172}{3} }[/math] [math]\displaystyle{ \frac{44}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{368}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{1376}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ \frac{8791}{15} }[/math] [math]\displaystyle{ \frac{476}{15} }[/math] [math]\displaystyle{ \frac{9604}{45} }[/math] [math]\displaystyle{ \frac{425}{9} }[/math] [math]\displaystyle{ \frac{871}{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 K11a305. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
9           1-1
7          3 3
5         61 -5
3        83  5
1       106   -4
-1      128    4
-3     1011     1
-5    911      -2
-7   510       5
-9  39        -6
-11 15         4
-13 3          -3
-151           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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a304.gif

K11a304

K11a306.gif

K11a306

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