K11a227

K11a226

K11a228

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a227 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X18,6,19,5 X22,8,1,7 X16,10,17,9 X2,12,3,11 X8,14,9,13 X20,16,21,15 X10,18,11,17 X6,20,7,19 X14,22,15,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -7, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -8, 11, -4
Dowker-Thistlethwaite code	4 12 18 22 16 2 8 20 10 6 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a227/ThurstonBennequinNumber Hyperbolic Volume 17.3495 A-Polynomial See Data:K11a227/A-polynomial

[edit Notes for K11a227's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant -6

[edit Notes for K11a227's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 5 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +5 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 5 z^6+13 z^4+8 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 143, 6 }
Jones polynomial	[math]\displaystyle{ -q^{14}+4 q^{13}-9 q^{12}+15 q^{11}-21 q^{10}+23 q^9-23 q^8+20 q^7-14 q^6+9 q^5-3 q^4+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-12} +3 z^2 a^{-6} +12 z^2 a^{-8} -6 z^2 a^{-10} -z^2 a^{-12} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +12 z^9 a^{-11} +7 z^9 a^{-13} +6 z^8 a^{-8} +12 z^8 a^{-10} +16 z^8 a^{-12} +10 z^8 a^{-14} +3 z^7 a^{-7} -z^7 a^{-9} -13 z^7 a^{-11} -z^7 a^{-13} +8 z^7 a^{-15} +z^6 a^{-6} -14 z^6 a^{-8} -34 z^6 a^{-10} -38 z^6 a^{-12} -15 z^6 a^{-14} +4 z^6 a^{-16} -6 z^5 a^{-7} -18 z^5 a^{-9} -14 z^5 a^{-11} -15 z^5 a^{-13} -12 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +15 z^4 a^{-8} +29 z^4 a^{-10} +25 z^4 a^{-12} +9 z^4 a^{-14} -5 z^4 a^{-16} +3 z^3 a^{-7} +20 z^3 a^{-9} +24 z^3 a^{-11} +15 z^3 a^{-13} +7 z^3 a^{-15} -z^3 a^{-17} +3 z^2 a^{-6} -12 z^2 a^{-8} -16 z^2 a^{-10} -4 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -8 z a^{-9} -10 z a^{-11} -4 z a^{-13} -2 z a^{-15} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12} }[/math]
The A2 invariant	Data:K11a227/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a227/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a227 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["K11a227"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 5 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +5 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 5 z^6+13 z^4+8 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]=

{ 143, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{14}+4 q^{13}-9 q^{12}+15 q^{11}-21 q^{10}+23 q^9-23 q^8+20 q^7-14 q^6+9 q^5-3 q^4+q^3 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-12} +3 z^2 a^{-6} +12 z^2 a^{-8} -6 z^2 a^{-10} -z^2 a^{-12} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +12 z^9 a^{-11} +7 z^9 a^{-13} +6 z^8 a^{-8} +12 z^8 a^{-10} +16 z^8 a^{-12} +10 z^8 a^{-14} +3 z^7 a^{-7} -z^7 a^{-9} -13 z^7 a^{-11} -z^7 a^{-13} +8 z^7 a^{-15} +z^6 a^{-6} -14 z^6 a^{-8} -34 z^6 a^{-10} -38 z^6 a^{-12} -15 z^6 a^{-14} +4 z^6 a^{-16} -6 z^5 a^{-7} -18 z^5 a^{-9} -14 z^5 a^{-11} -15 z^5 a^{-13} -12 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +15 z^4 a^{-8} +29 z^4 a^{-10} +25 z^4 a^{-12} +9 z^4 a^{-14} -5 z^4 a^{-16} +3 z^3 a^{-7} +20 z^3 a^{-9} +24 z^3 a^{-11} +15 z^3 a^{-13} +7 z^3 a^{-15} -z^3 a^{-17} +3 z^2 a^{-6} -12 z^2 a^{-8} -16 z^2 a^{-10} -4 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -8 z a^{-9} -10 z a^{-11} -4 z a^{-13} -2 z a^{-15} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12} }[/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["K11a227"];

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{ 5 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +5 t^{-3} }[/math], [math]\displaystyle{ -q^{14}+4 q^{13}-9 q^{12}+15 q^{11}-21 q^{10}+23 q^9-23 q^8+20 q^7-14 q^6+9 q^5-3 q^4+q^3 }[/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:

(8, 21)

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{ 32 }[/math] [math]\displaystyle{ 168 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3424}{3} }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ 5376 }[/math] [math]\displaystyle{ 8688 }[/math] [math]\displaystyle{ 1440 }[/math] [math]\displaystyle{ 936 }[/math] [math]\displaystyle{ \frac{16384}{3} }[/math] [math]\displaystyle{ 14112 }[/math] [math]\displaystyle{ \frac{109568}{3} }[/math] [math]\displaystyle{ \frac{14080}{3} }[/math] [math]\displaystyle{ \frac{1024804}{15} }[/math] [math]\displaystyle{ \frac{61144}{15} }[/math] [math]\displaystyle{ \frac{992536}{45} }[/math] [math]\displaystyle{ \frac{3884}{9} }[/math] [math]\displaystyle{ \frac{39364}{15} }[/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]6 is the signature of K11a227. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 29                       1 -1 27                     3   3 25                   6 1   -5 23                 9 3     6 21               12 6       -6 19             11 9         2 17           12 12           0 15         8 11             -3 13       6 12               6 11     3 8                 -5 9     6                   6 7 1 3                     -2 5 1                       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=7 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=11 }[/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.

K11a226 K11a228