K11a293

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K11a292.gif

K11a292

K11a294.gif

K11a294

K11a293.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a293 at Knotilus!

[edit K11a293 Quick Notes]


[edit K11a293 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X16,5,17,6 X22,8,1,7 X4,10,5,9 X18,11,19,12 X20,13,21,14 X8,15,9,16 X2,17,3,18 X12,19,13,20 X14,21,15,22
Gauss code 1, -9, 2, -5, 3, -1, 4, -8, 5, -2, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 10 16 22 4 18 20 8 2 12 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11a293 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{2,3\} }[/math]
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a293/ThurstonBennequinNumber
Hyperbolic Volume 13.4596
A-Polynomial See Data:K11a293/A-polynomial

[edit Notes for K11a293's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a293's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-5 t^3+12 t^2-15 t+15-15 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+3 z^6+2 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature { 81, -4 }
Jones polynomial [math]\displaystyle{ q^2-3 q+5-8 q^{-1} +11 q^{-2} -11 q^{-3} +13 q^{-4} -11 q^{-5} +8 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^4 z^8-a^6 z^6+6 a^4 z^6-2 a^2 z^6-4 a^6 z^4+14 a^4 z^4-9 a^2 z^4+z^4-5 a^6 z^2+17 a^4 z^2-11 a^2 z^2+3 z^2-4 a^6+8 a^4-4 a^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-5 z^3 a^9+2 z a^9+8 z^6 a^8-10 z^4 a^8+2 z^2 a^8+9 z^7 a^7-16 z^5 a^7+4 z^3 a^7-2 z a^7+9 z^8 a^6-24 z^6 a^6+18 z^4 a^6-11 z^2 a^6+4 a^6+6 z^9 a^5-15 z^7 a^5+z^5 a^5+10 z^3 a^5-6 z a^5+2 z^{10} a^4+3 z^8 a^4-36 z^6 a^4+51 z^4 a^4-25 z^2 a^4+8 a^4+9 z^9 a^3-40 z^7 a^3+50 z^5 a^3-15 z^3 a^3-2 z a^3+2 z^{10} a^2-5 z^8 a^2-9 z^6 a^2+28 z^4 a^2-17 z^2 a^2+4 a^2+3 z^9 a-16 z^7 a+27 z^5 a-15 z^3 a+z^8-5 z^6+8 z^4-5 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}+q^{24}-2 q^{22}-q^{20}-q^{18}-q^{16}+4 q^{14}+4 q^{10}+q^8+q^4-2 q^2- q^{-2} + q^{-6} }[/math]
The G2 invariant Data:K11a293/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a293 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["K11a293"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-5 t^3+12 t^2-15 t+15-15 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+3 z^6+2 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{ \{3,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 81, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^2-3 q+5-8 q^{-1} +11 q^{-2} -11 q^{-3} +13 q^{-4} -11 q^{-5} +8 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^4 z^8-a^6 z^6+6 a^4 z^6-2 a^2 z^6-4 a^6 z^4+14 a^4 z^4-9 a^2 z^4+z^4-5 a^6 z^2+17 a^4 z^2-11 a^2 z^2+3 z^2-4 a^6+8 a^4-4 a^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-5 z^3 a^9+2 z a^9+8 z^6 a^8-10 z^4 a^8+2 z^2 a^8+9 z^7 a^7-16 z^5 a^7+4 z^3 a^7-2 z a^7+9 z^8 a^6-24 z^6 a^6+18 z^4 a^6-11 z^2 a^6+4 a^6+6 z^9 a^5-15 z^7 a^5+z^5 a^5+10 z^3 a^5-6 z a^5+2 z^{10} a^4+3 z^8 a^4-36 z^6 a^4+51 z^4 a^4-25 z^2 a^4+8 a^4+9 z^9 a^3-40 z^7 a^3+50 z^5 a^3-15 z^3 a^3-2 z a^3+2 z^{10} a^2-5 z^8 a^2-9 z^6 a^2+28 z^4 a^2-17 z^2 a^2+4 a^2+3 z^9 a-16 z^7 a+27 z^5 a-15 z^3 a+z^8-5 z^6+8 z^4-5 z^2+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a293"];
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+12 t^2-15 t+15-15 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^2-3 q+5-8 q^{-1} +11 q^{-2} -11 q^{-3} +13 q^{-4} -11 q^{-5} +8 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9} }[/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]= {}
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, -8)
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{ -64 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{920}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ -1024 }[/math] [math]\displaystyle{ -\frac{5440}{3} }[/math] [math]\displaystyle{ -\frac{544}{3} }[/math] [math]\displaystyle{ -416 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ \frac{14720}{3} }[/math] [math]\displaystyle{ \frac{2176}{3} }[/math] [math]\displaystyle{ \frac{152942}{15} }[/math] [math]\displaystyle{ -\frac{3976}{5} }[/math] [math]\displaystyle{ \frac{226328}{45} }[/math] [math]\displaystyle{ \frac{1810}{9} }[/math] [math]\displaystyle{ \frac{10382}{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]-4 is the signature of K11a293. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-101234χ
5           11
3          2 -2
1         31 2
-1        52  -3
-3       63   3
-5      66    0
-7     75     2
-9    46      2
-11   47       -3
-13  24        2
-15 14         -3
-17 2          2
-191           -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}_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.

K11a292.gif

K11a292

K11a294.gif

K11a294

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