K11n11

From Knot Atlas
Jump to navigationJump to search

K11n10.gif

K11n10

K11n12.gif

K11n12

K11n11.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n11 at Knotilus!

[edit K11n11 Quick Notes]


[edit K11n11 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X11,19,12,18 X13,6,14,7 X15,21,16,20 X17,1,18,22 X19,13,20,12 X21,17,22,16
Gauss code 1, -5, 2, -1, 3, 7, -4, -2, 5, -3, -6, 10, -7, 4, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 10 -14 2 -18 -6 -20 -22 -12 -16
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation K11n11 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:K11n11/ThurstonBennequinNumber
Hyperbolic Volume 13.0518
A-Polynomial See Data:K11n11/A-polynomial

[edit Notes for K11n11'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 K11n11's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 55, 2 }
Jones polynomial [math]\displaystyle{ -q^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-4} -2 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -4 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +z^2-2 a^{-2} +4 a^{-4} -2 a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +4 z^8 a^{-4} +3 z^8 a^{-6} -2 z^7 a^{-3} +2 z^7 a^{-5} +4 z^7 a^{-7} -2 z^6 a^{-2} -9 z^6 a^{-4} -4 z^6 a^{-6} +3 z^6 a^{-8} +3 z^5 a^{-1} +8 z^5 a^{-3} -3 z^5 a^{-5} -7 z^5 a^{-7} +z^5 a^{-9} +8 z^4 a^{-2} +15 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4 a^{-8} +z^4-4 z^3 a^{-1} -7 z^3 a^{-3} +z^3 a^{-7} -2 z^3 a^{-9} -8 z^2 a^{-2} -12 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} -2 z^2+z a^{-1} +2 z a^{-3} +z a^{-9} +2 a^{-2} +4 a^{-4} +2 a^{-6} +1 }[/math]
The A2 invariant [math]\displaystyle{ q^4-1+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} +3 q^{-12} - q^{-14} +2 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} - q^{-24} }[/math]
The G2 invariant Data:K11n11/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n11 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["K11n11"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+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]= { 55, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-4} -2 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -4 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +z^2-2 a^{-2} +4 a^{-4} -2 a^{-6} +1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +4 z^8 a^{-4} +3 z^8 a^{-6} -2 z^7 a^{-3} +2 z^7 a^{-5} +4 z^7 a^{-7} -2 z^6 a^{-2} -9 z^6 a^{-4} -4 z^6 a^{-6} +3 z^6 a^{-8} +3 z^5 a^{-1} +8 z^5 a^{-3} -3 z^5 a^{-5} -7 z^5 a^{-7} +z^5 a^{-9} +8 z^4 a^{-2} +15 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4 a^{-8} +z^4-4 z^3 a^{-1} -7 z^3 a^{-3} +z^3 a^{-7} -2 z^3 a^{-9} -8 z^2 a^{-2} -12 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} -2 z^2+z a^{-1} +2 z a^{-3} +z a^{-9} +2 a^{-2} +4 a^{-4} +2 a^{-6} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_31, K11n22, K11n112, K11n127,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {9_39, K11n112,}

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["K11n11"];
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^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -q^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1} }[/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]= {9_31, K11n22, K11n112, K11n127,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {9_39, K11n112,}

Vassiliev invariants

V2 and V3: (2, 4)
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{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{316}{3} }[/math] [math]\displaystyle{ \frac{20}{3} }[/math] [math]\displaystyle{ 256 }[/math] [math]\displaystyle{ \frac{1376}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{2528}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ \frac{30991}{15} }[/math] [math]\displaystyle{ \frac{676}{15} }[/math] [math]\displaystyle{ \frac{30604}{45} }[/math] [math]\displaystyle{ \frac{161}{9} }[/math] [math]\displaystyle{ \frac{991}{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 K11n11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-101234567χ
17         1-1
15        2 2
13       41 -3
11      42  2
9     54   -1
7    54    1
5   35     2
3  35      -2
1 14       3
-1 2        -2
-31         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=3 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=7 }[/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.

K11n10.gif

K11n10

K11n12.gif

K11n12

Retrieved from "https://katlas.org/index.php?title=K11n11&oldid=110691"