K11a266

K11a265

K11a267

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a266 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X12,6,13,5 X14,7,15,8 X16,10,17,9 X18,11,19,12 X22,13,1,14 X20,16,21,15 X4,18,5,17 X2,19,3,20 X8,21,9,22
Gauss code	1, -10, 2, -9, 3, -1, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code	6 10 12 14 16 18 22 20 4 2 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a266/ThurstonBennequinNumber Hyperbolic Volume 20.2286 A-Polynomial See Data:K11a266/A-polynomial

[edit Notes for K11a266's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a266's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-7 t^3+23 t^2-45 t+57-45 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+z^6+z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 209, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+5 q^4-12 q^3+21 q^2-29 q+34-34 q^{-1} +30 q^{-2} -22 q^{-3} +14 q^{-4} -6 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +6 z^4-z^2 a^{-2} +z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 6 a^2 z^{10}+6 z^{10}+15 a^3 z^9+31 a z^9+16 z^9 a^{-1} +14 a^4 z^8+20 a^2 z^8+18 z^8 a^{-2} +24 z^8+6 a^5 z^7-23 a^3 z^7-57 a z^7-16 z^7 a^{-1} +12 z^7 a^{-3} +a^6 z^6-26 a^4 z^6-65 a^2 z^6-26 z^6 a^{-2} +5 z^6 a^{-4} -69 z^6-7 a^5 z^5+2 a^3 z^5+19 a z^5-4 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} +10 a^4 z^4+37 a^2 z^4+14 z^4 a^{-2} -3 z^4 a^{-4} +44 z^4+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +4 z^3 a^{-3} +2 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}-3 q^{16}+3 q^{12}-5 q^{10}+7 q^8-q^6+4 q^2-7+6 q^{-2} -6 q^{-4} +2 q^{-6} +4 q^{-8} -4 q^{-10} +3 q^{-12} - q^{-14} }[/math]
The G2 invariant	Data:K11a266/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a266 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["K11a266"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^4-7 t^3+23 t^2-45 t+57-45 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+z^6+z^4+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]=

{ 209, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^5+5 q^4-12 q^3+21 q^2-29 q+34-34 q^{-1} +30 q^{-2} -22 q^{-3} +14 q^{-4} -6 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +6 z^4-z^2 a^{-2} +z^2-a^4+3 a^2+ a^{-2} -2 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ 6 a^2 z^{10}+6 z^{10}+15 a^3 z^9+31 a z^9+16 z^9 a^{-1} +14 a^4 z^8+20 a^2 z^8+18 z^8 a^{-2} +24 z^8+6 a^5 z^7-23 a^3 z^7-57 a z^7-16 z^7 a^{-1} +12 z^7 a^{-3} +a^6 z^6-26 a^4 z^6-65 a^2 z^6-26 z^6 a^{-2} +5 z^6 a^{-4} -69 z^6-7 a^5 z^5+2 a^3 z^5+19 a z^5-4 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} +10 a^4 z^4+37 a^2 z^4+14 z^4 a^{-2} -3 z^4 a^{-4} +44 z^4+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +4 z^3 a^{-3} +2 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/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["K11a266"];

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^4-7 t^3+23 t^2-45 t+57-45 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+5 q^4-12 q^3+21 q^2-29 q+34-34 q^{-1} +30 q^{-2} -22 q^{-3} +14 q^{-4} -6 q^{-5} + q^{-6} }[/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:

(0, -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{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ \frac{104}{3} }[/math] [math]\displaystyle{ -\frac{112}{3} }[/math] [math]\displaystyle{ 0 }[/math]

V_2,1 through V_6,9 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 K11a266. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     4   4 7                   8 1   -7 5                 13 4     9 3               16 8       -8 1             18 13         5 -1           17 17           0 -3         13 17             -4 -5       9 17               8 -7     5 13                 -8 -9   1 9                   8 -11   5                     -5 -13 1                       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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{17} }[/math] [math]\displaystyle{ {\mathbb Z}^{17} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{17} }[/math] [math]\displaystyle{ {\mathbb Z}^{18} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a265 K11a267