K11n24

From Knot Atlas
Jump to navigationJump to search

K11n23.gif

K11n23

K11n25.gif

K11n25

K11n24.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n24 at Knotilus!

[edit K11n24 Quick Notes]


[edit K11n24 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X11,18,12,19 X6,13,7,14 X15,21,16,20 X17,1,18,22 X19,10,20,11 X21,17,22,16
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, 10, -6, -3, 7, -5, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 12 2 14 -18 6 -20 -22 -10 -16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n24 ML.gif

Three dimensional invariants

Symmetry type Chiral
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:K11n24/ThurstonBennequinNumber
Hyperbolic Volume 9.90549
A-Polynomial See Data:K11n24/A-polynomial

[edit Notes for K11n24's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant 0

[edit Notes for K11n24's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+3 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 23, 2 }
Jones polynomial [math]\displaystyle{ -q^4+2 q^3-3 q^2+4 q-3+4 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6-a^2 z^4-z^4 a^{-2} +5 z^4-3 a^2 z^2-3 z^2 a^{-2} +8 z^2-2 a^2-2 a^{-2} +5 }[/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+a^3 z^7-2 a z^7-2 z^7 a^{-1} +z^7 a^{-3} -10 a^2 z^6-10 z^6 a^{-2} -20 z^6-5 a^3 z^5-8 a z^5-8 z^5 a^{-1} -5 z^5 a^{-3} +14 a^2 z^4+14 z^4 a^{-2} +28 z^4+7 a^3 z^3+16 a z^3+16 z^3 a^{-1} +7 z^3 a^{-3} -8 a^2 z^2-8 z^2 a^{-2} -16 z^2-3 a^3 z-7 a z-7 z a^{-1} -3 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}-q^8+q^4+q^2+3+ q^{-2} + q^{-4} - q^{-8} - q^{-12} }[/math]
The G2 invariant Data:K11n24/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n24 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["K11n24"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+3 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]= { 23, 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+4 q-3+4 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6-a^2 z^4-z^4 a^{-2} +5 z^4-3 a^2 z^2-3 z^2 a^{-2} +8 z^2-2 a^2-2 a^{-2} +5 }[/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+a^3 z^7-2 a z^7-2 z^7 a^{-1} +z^7 a^{-3} -10 a^2 z^6-10 z^6 a^{-2} -20 z^6-5 a^3 z^5-8 a z^5-8 z^5 a^{-1} -5 z^5 a^{-3} +14 a^2 z^4+14 z^4 a^{-2} +28 z^4+7 a^3 z^3+16 a z^3+16 z^3 a^{-1} +7 z^3 a^{-3} -8 a^2 z^2-8 z^2 a^{-2} -16 z^2-3 a^3 z-7 a z-7 z a^{-1} -3 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_7,}

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["K11n24"];
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^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -q^4+2 q^3-3 q^2+4 q-3+4 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }[/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_7,}
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, 0)
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{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ -\frac{28}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{608}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ \frac{3031}{15} }[/math] [math]\displaystyle{ \frac{1396}{15} }[/math] [math]\displaystyle{ -\frac{4436}{45} }[/math] [math]\displaystyle{ \frac{137}{9} }[/math] [math]\displaystyle{ -\frac{809}{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 K11n24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123χ
9        1-1
7       1 1
5      21 -1
3     21  1
1    23   1
-1   221   1
-3  12     1
-5 12      -1
-7 1       1
-91        -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{ i=3 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/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.

K11n23.gif

K11n23

K11n25.gif

K11n25

Retrieved from "https://katlas.org/index.php?title=K11n24&oldid=110544"